Integrability of pushforward measures by analytic maps

Itay Glazer Department of Mathematics, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter (550), Woodstock Road, Oxford, OX2 6GG, UK and Department of Mathematics, Technion – Israel Institute of Technology, Haifa 3200003, Israel [email protected] https://sites.google.com/view/itay-glazer , Yotam I. Hendel KU Leuven, Department of Mathematics, B-3001 Leuven, Belgium and Ben Gurion University of the Negev, Department of Mathematics, Be’er Sheva 8410501, Israel [email protected] https://sites.google.com/view/yotam-hendel and Sasha Sodin School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK and Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel [email protected] https://webspace.maths.qmul.ac.uk/a.sodin/

Abstract.

Given a map $\phi:X\rightarrow Y$ between $F$ -analytic manifolds over a local field $F$ of characteristic $0$ , we introduce an invariant $\epsilon_{\star}(\phi)$ which quantifies the integrability of pushforwards of smooth compactly supported measures by $\phi$ . We further define a local version $\epsilon_{\star}(\phi,x)$ near $x\in X$ . These invariants have a strong connection to the singularities of $\phi$ .

When $Y$ is one-dimensional, we give an explicit formula for $\epsilon_{\star}(\phi,x)$ , and show it is asymptotically equivalent to other known singularity invariants such as the $F$ -log-canonical threshold $\operatorname{lct}_{F}(\phi-\phi(x);x)$ at $x$ .

In the general case, we show that $\epsilon_{\star}(\phi,x)$ is bounded from below by the $F$ -log-canonical threshold $\lambda=\operatorname{lct}_{F}(\mathcal{J}_{\phi};x)$ of the Jacobian ideal $\mathcal{J}_{\phi}$ near $x$ . If $\dim Y=\dim X$ , equality is attained. If $\dim Y<\dim X$ , the inequality can be strict; however, for $F=\mathbb{C}$ , we establish the upper bound $\epsilon_{\star}(\phi,x)\leq\lambda/(1-\lambda)$ , whenever $\lambda<1$ .

Finally, we specialize to polynomial maps $\varphi:X\rightarrow Y$ between smooth algebraic $\mathbb{Q}$ -varieties $X$ and $Y$ . We geometrically characterize the condition that $\epsilon_{\star}(\varphi_{F})=\infty$ over a large family of local fields, by showing it is equivalent to $\varphi$ being flat with fibers of semi-log-canonical singularities.

Key words and phrases:

L^{p}

-spaces, log-canonical threshold, analytic maps, pushforward measures, constructible functions, Young’s convolution inequality, regularization by convolution

1991 Mathematics Subject Classification:

14B05, 03C98, 14E15, 32B20, 60B15

1. Introduction

The goal of this paper is to explore a singularity invariant $\epsilon_{\star}(\phi)$ of a map $\phi$ between two manifolds over a local field. This invariant quantifies the integrability of pushforward measures by $\phi$ ; we define it in (1.2) below after introducing some notation that will also be used in the sequel.

Throughout this paper, we fix a local field $F$ of characteristic $0$ , i.e., $\mathbb{R}$ , $\mathbb{C}$ or a finite extension of $\mathbb{Q}_{p}$ . If $X$ is an $F$ -analytic manifold of dimension $n$ , let $(U_{\alpha}\subset X,\psi_{\alpha}:U_{\alpha}\to F^{n})_{\alpha\in\mathcal{A}}$ be an atlas. We denote by $C^{\infty}(X)$ the space of smooth functions on $X$ , i.e. functions $f:X\to\mathbb{C}$ such that $f\circ\psi_{\alpha}^{-1}|_{\psi_{\alpha}(U_{\alpha})}$ is smooth for each $\alpha\in\mathcal{A}$ , and by $C_{c}^{\infty}(X)$ the subspace of compactly supported smooth functions (if $F$ is non-Archimedean, smooth means locally constant). We similarly write $\mathcal{M}^{\infty}(X)$ for the space of smooth measures on $X$ , i.e. measures such that each $(\psi_{\alpha})_{*}(\mu|_{U_{\alpha}})$ has a smooth density with respect to the Haar measure. We use $\mathcal{M}_{c}^{\infty}(X)$ to denote the space of smooth compactly supported measures on $X$ . For $1\leq q\leq\infty$ , consider the class $\mathcal{M}_{c,q}(X)$ of finite Radon measures $\mu$ on $X$ that are compactly supported and such that for any $\alpha\in\mathcal{A}$ the measure $(\psi_{\alpha})_{*}(\mu|_{U_{\alpha}})$ is absolutely continuous with density in $L^{q}(F^{n})$ . All these classes do not depend on the choice of the atlas. For $\mu\in\mathcal{M}_{c,1}(X)$ we define

\epsilon_{\star}(\mu):=\sup\left\{\epsilon\geq 0\,|\,\mu\in\mathcal{M}_{c,1+\epsilon}(X)\right\}.

Note that by Jensen’s inequality $\mu\in\mathcal{M}_{c,1+\epsilon}(X)$ for all $0\leq\epsilon<\epsilon_{\star}(\mu)$ .

Now let $\phi:X\to Y$ be an analytic map between $F$ -analytic manifolds $X,Y$ . If $\phi$ is locally dominant, i.e. $\phi(U)$ contains a non-empty open set for each open subset $U\subseteq X$ , then $\phi_{*}\mu\in\mathcal{M}_{c,1}(Y)$ whenever $\mu\in\mathcal{M}_{c,1}(X)$ . We now set for each $x\in X$ ,

(1.1)

\epsilon_{\star}(\phi;x):=\sup_{U\ni x}\inf_{\mu\in\mathcal{M}_{c}^{\infty}(U)}\epsilon_{\star}(\phi_{*}\mu)=\sup_{U\ni x}\inf_{\mu\in\mathcal{M}_{c,\infty}(U)}\epsilon_{\star}(\phi_{*}\mu),

where the supremum is over all open neighborhoods $U$ of $x$ . Finally, we can define

(1.2)

\epsilon_{\star}(\phi):=\inf_{\mu\in\mathcal{M}_{c}^{\infty}(X)}\epsilon_{\star}(\phi_{*}\mu)=\inf_{x\in X}\epsilon_{\star}(\phi;x).

Note that if there exists $U\ni x$ such that $\phi_{*}\mu$ lies in $\mathcal{M}_{c,q}(Y)$ for all $1<q<\infty$ and all $\mu\in\mathcal{M}_{c}^{\infty}(U)$ , then $\epsilon_{\star}(\phi;x)=\infty$ . The best case scenario in this setting is obtained when $\phi_{*}\mu\in\mathcal{M}_{c,\infty}(Y)$ for every $\mu\in\mathcal{M}_{c}^{\infty}(X)$ . In this case we say that $\phi$ is an $L^{\infty}$ -map.

The main motivation for $\epsilon_{\star}(\phi;x)$ comes from singularity theory. In general, bad singularities of $\phi$ should manifest themselves in poor analytic behavior of the pushforward $\phi_{*}\mu$ of $\mu\in\mathcal{M}_{c}^{\infty}(X)$ . This phenomenon has been extensively studied in the case $Y=F^{m}$ , through the analysis of the Fourier transform $\mathcal{F}(\phi_{*}\mu)$ , which takes the form of an oscillatory integral in the Archimedean case, and of an exponential sum in the $p$ -adic case. When the dimension of the target space is equal to $m=1$ , the rate of decay of $\mathcal{F}(\varphi_{*}\mu)$ is closely related to singularity invariants such as the log-canonical threshold (see e.g. [Igu78], [Den91, Corollary 1.4.5] and [CMN19, Theorem 1.5] for the non-Archimedean case, and [AGZV88, Chapter 7] for the Archimedean case). In higher dimension, the connections are less explicit, however, milder singularities of $\phi$ still result in faster decay rates of $\mathcal{F}(\phi_{*}\mu)$ . For more information on $\mathcal{F}(\phi_{*}\mu)$ , we refer to Igusa’s work [Igu78], the surveys of Denef, Meuser and León-Cardenal [Den91, LC22, Meu16], as well as the book [AGZV88, Parts II, III] by Arnold, Guseĭn-Zade, and Varchenko, and also the discussion in $\mathsection$ 3.2 below.

The study of the integrability properties of $\phi_{*}\mu$ and their relation to the singularities of $\phi$ has received less treatment. The invariant $\epsilon_{\star}(\phi;x)$ is a natural step in this direction, and is more robust than Fourier-type invariants as it is also meaningful when $Y$ is any smooth manifold, which is especially important for applications (see $\mathsection$ 1.1).

The invariant $\epsilon_{\star}(\phi;x)$ tends to be small whenever the singularities of $\phi$ are bad near $x$ . When $\phi$ is a submersion, the pushforward $\phi_{*}\mu$ of any $\mu\in\mathcal{M}_{c}^{\infty}(X)$ is smooth and in particular lies in $\mathcal{M}_{c,\infty}(Y)$ . Moreover, Aizenbud and Avni [AA16] have shown that for algebraic maps $\varphi:X\rightarrow Y$ between smooth algebraic $\mathbb{Q}$ -varieties, the condition that the corresponding map $\varphi_{\mathbb{Q}_{p}}:X(\mathbb{Q}_{p})\rightarrow Y(\mathbb{Q}_{p})$ of $\mathbb{Q}_{p}$ -analytic varieties is an $L^{\infty}$ -map is equivalent to a certain mild singularity property, namely that $\varphi$ is flat with fibers of rational singularities (abbreviated (FRS), see Definition 1.8).

When analyzing $\epsilon_{\star}(\phi;x)$ , one can further restrict the infima in (1.1)–(1.2) to the class of compactly supported measures $\mu$ which are constructible, in the sense of [CGH14, Section 3] and [CM11, Definition 1.2] (see also [LR97]). This class is preserved under pushforward by analytic maps, and therefore $\phi_{*}\mu$ is constructible as well. Moreover, constructible measures admit a well behaved structure theory and have tame analytic behavior (see e.g. [CL08, CL10, CM11, CM13, CGH14, CGH18]), and, in particular, it follows from [GH21, CM13] that $\epsilon_{\star}(\phi_{*}\mu)>0$ . Positivity of $\epsilon_{\star}(\phi_{*}\mu)$ in the real case can further be deduced from [RS88, Section 2].

Our goal in this paper is to explore in more detail the properties of $\epsilon_{\star}(\phi;x)$ , and in particular to obtain upper and lower bounds on $\epsilon_{\star}(\phi;x)$ in terms of other singularity invariants which may be easier to compute, such as the log-canonical threshold of certain ideals. In Theorem 1.1 (proved in $\mathsection$ 4), we give a lower bound on $\epsilon_{\star}(\phi;x)$ . In particular, this provides a proof for the positivity of $\epsilon_{\star}$ , without using the theory of motivic integration. In Theorem 1.5 (proved in $\mathsection$ 5) an upper bound on $\epsilon_{\star}(\phi;x)$ is given in the complex case. In Theorem 1.2 (proved in $\mathsection$ 3) an explicit formula is given for $\epsilon_{\star}(\phi;x)$ when $Y$ is one-dimensional, over any local field. Finally, in Theorem 1.12 (proved in $\mathsection$ 6), we specialize to polynomial maps between smooth algebraic varieties, and geometrically characterize the condition $\epsilon_{\star}(\phi;x)=\infty$ .

1.1. Application: regularization by convolution

Apart from the geometric motivation discussed above, an additional source of motivation comes from the study of random walks on groups. Assume that $G$ is an $F$ -analytic group, and take a finite measure $\nu$ on $G$ . Can one find a number $k\in\mathbb{N}$ such that the $k$ -th convolution power $\nu^{*k}$ lies in $\mathcal{M}_{c,\infty}(G)$ , and if so, what is the smallest such number $k_{\star}(\nu)$ ?

An important class of examples comes from the realm of word maps. Given a word $w$ in $r$ letters, by which we mean an element either of the free group $F_{r}$ , or of the free Lie algebra $\mathcal{L}_{r}$ , one can consider the corresponding word maps $w_{G}:G^{r}\rightarrow G$ or $w_{\mathfrak{g}}:\mathfrak{g}^{r}\rightarrow\mathfrak{g}$ , where $\mathfrak{g}$ is the Lie algebra of $G$ . When $G$ is a compact real or $p$ -adic Lie group, $w$ induces a natural measure $\nu:=(w_{G})_{*}\pi_{G}$ where $\pi_{G}$ is the normalized Haar measure on $G$ . Here, one may further ask what is the $L^{\infty}$ -mixing time of the word measure $\nu$ , namely, how large should $k$ be to ensure that $\left\|\nu^{*k}-\pi_{G}\right\|_{\infty}\ll 1$ . Questions of this kind have been studied e.g. in [AA16, LST19, GH24, AG, AGL].

Similar problems have also appeared in a variety of other applications; we mention the work of Ricci and Stein on singular integrals on non-abelian groups (see the ICM survey of Stein [Ste87]). In the study of one-dimensional random operators, the regularity of the distribution of transfer matrices (lying in SL( $2,\mathbb{R}$ ), or, more generally, in Sp( $2W,\mathbb{R}$ )) plays a role in the work of Shubin–Vakilian–Wolff [SVW98] as well as in the recent work [GS22].

Motivated by the above examples, we focus on the following setting: $G$ and $X$ are analytic, and $\nu=\phi_{*}\mu$ is the pushforward of a measure $\mu\in\mathcal{M}_{c,\infty}(X)$ under a locally dominant analytic map $\phi:X\to G$ . For $x\in X$ , let

k_{\star}(\phi;x)=\min_{U\ni x}\max_{\mu\in\mathcal{M}_{c}^{\infty}(U)}k_{\star}(\phi_{*}\mu),

be the smallest $k$ that works for any $\mu$ supported in a sufficiently small neighborhood $U$ of $x$ .

According to Young’s convolution inequality for locally compact groups (see [KR78, Corollary 2.3] and Remark 1.10),

\nu\in\mathcal{M}_{c,1+\epsilon}(G)\Longrightarrow\nu^{*k}\in\mathcal{M}_{c,1+r}(G),\quad\text{where}\quad r=\begin{cases}\frac{k\epsilon}{1-(k-1)\epsilon}&\text{if}\,k<\frac{1+\epsilon}{\epsilon},\\ \infty&\text{if}\,k\geq\frac{1+\epsilon}{\epsilon}.\end{cases}

This implies

(1.3)

k_{\star}(\phi;x)\leq\left\lfloor\frac{1+\epsilon_{\star}(\phi;x)}{\epsilon_{\star}(\phi;x)}\right\rfloor+1<\infty.

As mentioned above, a more classical approach to bounding $k_{\star}(\phi;x)$ relies on the study of the decay of the Fourier transform of $\phi_{*}\mu$ . While the Fourier-analytic approach often provides sharper bounds (see [GH24, Proposition 5.7]), it is mainly applicable for abelian groups such as $G=F^{n}$ , or mildly non-abelian groups such as the Heisenberg model. One can use non-commutative Fourier transform to analyze compact Lie groups such as $\mathrm{SO}_{n}(\mathbb{R})$ , but such representation theoretic techniques are much less effective for compact $p$ -adic groups and non-compact Lie groups. To treat the latter cases, one can use algebro-geometric techniques as in [AA16, GH24]; however, this method requires some assumptions on $\phi$ . Thus one can argue that the approach to regularization via (1.3) is currently the most efficient one for treating $p$ -adic analytic groups and non-compact Lie groups such as $SL_{n}(\mathbb{R})$ .

1.2. Main results

We now discuss the main results in this paper.

1.2.1. A lower bound on $\epsilon_{\star}$

While the mere positivity of $\epsilon_{\star}$ (and the mere finiteness of $k_{\star}$ ) are sufficient for some applications, other ones require explicit bounds. Our first result provides a bound in terms of an important exponent known as the log-canonical threshold, see e.g. [Mus12, Kol] for $F=\mathbb{C}$ . For an analytic map $\psi:X\to F$ , define the $F$ -log-canonical threshold

(1.4)

\operatorname{lct}_{F}(\psi;x):=\sup\left\{s>0:\exists U\ni x\text{ s.t.}\,\forall\mu\in\mathcal{M}_{c,\infty}(U),\,\int_{X}\left|\psi(x)\right|_{F}^{-s}d\mu(x)<\infty\right\},

where $U$ runs over all open neighborhoods of $x$ , and $\left|\,\cdot\,\right|_{F}$ is the absolute value on $F$ , normalized such that $\mu_{F}(aA)=\left|a\right|_{F}\cdot\mu_{F}(A)$ , for all $a\in F^{\times}$ , $A\subseteq F$ , and where $\mu_{F}$ is a Haar measure on $F$ . In particular, $\left|\cdot\right|_{\mathbb{C}}=\left|\cdot\right|^{2}$ is the square of the usual absolute value $\left|\cdot\right|$ on $\mathbb{C}$ . More generally, if $J$ is a non-zero ideal of analytic functions generated by $\psi_{1},\cdots,\psi_{\ell}$ , define

(1.5)

\operatorname{lct}_{F}(J;x):=\sup\left\{s>0:\exists U\ni x\text{ \,s.t.\, }\forall\mu\in\mathcal{M}_{c,\infty}(U),\,\int_{X}\min_{1\leq i\leq l}\Big{[}\left|\psi_{i}(x)\right|_{F}^{-s}\Big{]}d\mu(x)<\infty\right\}.

This definition does not depend on the choice of the generators, and thus it extends in a straightforward way to sheaves of ideals. Furthermore, the log-canonical threshold is always strictly positive (see $\mathsection$ 2).

Given a locally dominant analytic map $\phi:X\to Y$ between two $F$ -analytic manifolds, we define the Jacobian ideal sheaf $\mathcal{J}_{\phi}$ as follows. If $X\subseteq F^{n}$ and $Y\subseteq F^{m}$ are open subsets, we define $\mathcal{J}_{\phi}$ to be the ideal in the algebra of analytic functions on $X$ , generated by the $m\times m$ -minors of the differential $d_{x}(\phi)$ of $\phi$ . Note that if $\psi_{1}:X\rightarrow X^{\prime}\subseteq F^{n}$ and $\psi_{2}:Y\rightarrow Y^{\prime}\subseteq F^{m}$ are analytic diffeomorphisms, then $\psi_{1}^{*}\left(\mathcal{J}_{\psi_{2}\circ\phi\circ\psi_{1}^{-1}}\right)=\mathcal{J}_{\phi}$ . Hence, the definition of $\mathcal{J}_{\phi}$ can be generalized (or glued) to an ideal sheaf on $X$ , if $X$ and $Y$ are $F$ -analytic manifolds.

We now describe the first main result, which provides a lower bound on $\epsilon_{\star}(\phi;x)$ in terms of the Jacobian ideal $\mathcal{J}_{\phi}$ of $\phi$ . The proof is given in $\mathsection$ 4.

Theorem 1.1.

Let $X,Y$ be analytic $F$ -manifolds, $\dim X=n\geq\dim Y=m$ , and let $\phi:X\to Y$ be a locally dominant analytic map. Then for every $x\in X$ ,

(1.6)

\epsilon_{\star}(\phi;x)\geq\operatorname{lct}_{F}(\mathcal{J}_{\phi};x);

if $m=n$ , equality is achieved.

As we will see later in Remark 1.6(2), for $\dim X>\dim Y$ the inequality (1.6) may be strict. In the next paragraph we discuss an additional case in which $\epsilon_{\star}$ can be computed explicitly: $\dim Y=1$ .

1.2.2. A formula in the one-dimensional case and a reverse Young inequality

When the target space $Y$ is one-dimensional, Hironaka’s theorem on the embedded resolution of singularities [Hir64] provides a powerful tool to study the structural properties of algebraic and analytic maps. This theorem, as well the asymptotic expansion of pushforward measures about a critical value of the map, allows us to obtain the following much more detailed results, the proofs of which are given in $\mathsection$ 3.

The first one is an exact formula relating $\epsilon_{\star}$ to the log-canonical threshold.

Theorem 1.2.

Let $X$ be an analytic $F$ -manifold, and let $\phi:X\to F$ be a locally dominant analytic map. Then for each $x\in X$ , we have:

(1.7)

\epsilon_{\star}(\phi;x)=\begin{cases}\infty&\text{if }\operatorname{lct}_{F}(\phi-\phi(x);x)\geq 1,\\ \frac{\operatorname{lct}_{F}(\phi-\phi(x);x)}{1-\operatorname{lct}_{F}(\phi-\phi(x);x)}&\text{if }\operatorname{lct}_{F}(\phi-\phi(x);x)<1.\end{cases}

By Theorem 1.2, by (1.3) and by a Thom–Sebastiani type result for $\operatorname{lct}_{F}$ (Proposition 3.11(1)), one can further show:

(1.8)

\left\lceil\frac{1}{\operatorname{lct}_{F}(\phi_{x};x)}\right\rceil\leq k_{\star}(\phi;x)\leq\left\lfloor\frac{1}{\operatorname{lct}_{F}(\phi_{x};x)}\right\rfloor+1.

We therefore see that that $\epsilon_{\star}(\phi;x),\operatorname{lct}_{F}(\phi_{x};x)$ and $1/k_{\star}(\phi;x)$ are asymptotically equivalent as $\operatorname{lct}_{F}(\phi_{x};x)\rightarrow 0$ . In $\mathsection$ 3.2, we shall see that these invariants are also tightly related to an invariant $\delta_{\star}$ quantifying Fourier decay. We will further see in $\mathsection$ 3.4 that the close relation between all these quantities is a special feature of the one-dimensional case, and does not generalize to higher dimensions.

We next provide a reverse Young result for pushforward measures by analytic maps. Recall that Young’s convolution inequality (see e.g. [Wei40, pp. 54-55]) implies that

(1.9)

\frac{\epsilon_{1}}{1+\epsilon_{1}}+\frac{\epsilon_{2}}{1+\epsilon_{2}}\geq\frac{\epsilon}{1+\epsilon}\Longrightarrow\mathcal{M}_{c,1+\epsilon_{1}}(F)*\mathcal{M}_{c,1+\epsilon_{2}}(F)\subseteq\mathcal{M}_{c,1+\epsilon}(F).

Using the connection between $\epsilon_{\star}(\phi;x)$ and $k_{\star}(\phi;x)$ as well as the structure of pushforward measures, we show the following converse to (1.9):

Theorem 1.3 (Reverse Young inequality).

Let $\nu_{1},\nu_{2}\in\mathcal{M}_{c,1}(F)$ be pushforward measures of the form $\nu_{j}=(\phi_{j})_{*}\mu_{j}$ , where $\mu_{j}\in\mathcal{M}_{c}^{\infty}(F^{n_{j}})$ and $\phi_{j}:F^{n_{j}}\to F$ are analytic, locally dominant. If $\nu_{1}*\nu_{2}\in\mathcal{M}_{c,1+\epsilon}(F)$ for some $\epsilon>0$ , then

(1.10)

\frac{\epsilon_{\star}(\nu_{1})}{1+\epsilon_{\star}(\nu_{1})}+\frac{\epsilon_{\star}(\nu_{2})}{1+\epsilon_{\star}(\nu_{2})}>\frac{\epsilon}{1+\epsilon}.

In particular, if $\nu_{1}$ is equal to $\nu_{2}$ , it lies in $\mathcal{M}_{c,1+\frac{\epsilon}{2+\epsilon}}(F)$ .

Remark 1.4.

Under the assumptions of Theorem 1.3, if $\nu_{1}*\nu_{2}\in\mathcal{M}_{c,\infty}(F)$ then (1.10) (applied with $\epsilon\to+\infty$ ) implies

(1.11)

\frac{\epsilon_{\star}(\nu_{1})}{1+\epsilon_{\star}(\nu_{1})}+\frac{\epsilon_{\star}(\nu_{2})}{1+\epsilon_{\star}(\nu_{2})}\geq 1.

In general, one cannot hope for a strict inequality in (1.11); indeed taking $F=\mathbb{\mathbb{R}}$ or $F=\mathbb{Q}_{p}$ for $p\equiv 3\mod 4$ , $\phi_{1}=\phi_{2}=x^{2}$ and $\mu_{1}=\mu_{2}$ a uniform measure on some ball around $0$ , one has $\epsilon_{\star}(\nu_{1})=\epsilon_{\star}(\nu_{2})=1$ , so that (1.11) $=1$ . On the other hand $\nu_{1}*\nu_{2}\in\mathcal{M}_{c,\infty}(F)$ .

When $F=\mathbb{C}$ , or more generally when the codimension of $(\phi_{1}*\phi_{2})^{-1}(0)$ in $F^{n_{1}+n_{2}}$ is $1$ , we expect (1.11) to hold with a strict inequality.

1.2.3. Upper bounds on $\epsilon_{\star}$

For $n>m$ , the lower bound (1.6) may in general not be an equality. However, the next result shows that when $F=\mathbb{C}$ , (1.6) is asymptotically sharp as $\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};x)\rightarrow 0$ .

Theorem 1.5.

Let $X,Y$ be analytic $\mathbb{C}$ -manifolds, and let $\phi:X\to Y$ be a locally dominant analytic map. Then, whenever $\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};x)<1$ ,

(1.12)

\epsilon_{\star}(\phi;x)\leq\frac{\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};x)}{1-\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};x)}.

Remark 1.6.

(1)

When $Y=F$ , we have $\operatorname{lct}_{F}(\phi-\phi(x);x)\leq\operatorname{lct}_{F}(\mathcal{J}_{\phi};x)$ (see Proposition 3.13), so that (1.12) holds also for $F\neq\mathbb{C}$ , whenever $\operatorname{lct}_{F}(\mathcal{J}_{\phi};x)<1$ . In particular, Theorem 1.2 implies Theorem 1.5 in the case that $Y=\mathbb{C}$ .
(2)

The upper bound (1.12) is asymptotically tight, in the sense that the value of $\epsilon_{\star}$ can be arbitrarily close to the upper bound (1.12), as seen from the following family of examples. Let $\phi:=x_{1}^{m}x_{2}^{m}...x_{n}^{m}$ . Then $\nabla\phi=\langle x_{1}^{m-1}x_{2}^{m}...x_{n}^{m},...,x_{1}^{m}x_{2}^{m}...x_{n}^{m-1}\rangle$ , and thus by [How01, Main Theorem and Example 5], it follows that

$\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};0)=\frac{1}{m-\frac{1}{n}},$

so that the upper bound in (1.12) becomes $\epsilon_{\star}(\phi;0)\leq\frac{1}{m-1-\frac{1}{n}}$ , whereas the lower bound (1.6) is $\epsilon_{\star}(\phi;0)\geq\frac{1}{m-\frac{1}{n}}$ . We see that the true value $\epsilon_{\star}(\phi;0)=\frac{1}{m-1}$ is closer to the upper bound than to the lower bound.

One may wonder whether Theorem 1.5 can be extended to $F\neq\mathbb{C}$ . In the current proof, the volumes of balls in complex manifolds are bounded from below using Lelong’s monotonicity theorem, and the latter fails for $F=\mathbb{\mathbb{R}}$ and for any $\mathbb{Q}_{p}$ . If $\varphi:X\to Y$ is a polynomial map between smooth varieties, defined over $\mathbb{Q}$ , we expect the upper bound in Theorem 1.5 to hold for $\varphi_{\mathbb{Q}_{p}}:X(\mathbb{Q}_{p})\to Y(\mathbb{Q}_{p})$ for infinitely many primes $p$ . This would follow from a positive answer to Question 1.14.

1.2.4. Applications to convolutions of algebraic morphisms

Throughout this and the next subsections we assume $K$ is a number field. In [GH19, GH21] and [GH24], the first two authors have studied the following convolution operation in algebraic geometry:

Definition 1.7.

Let $\varphi:X\rightarrow G$ and $\psi:Y\rightarrow G$ be morphisms from algebraic $K$ -varieties $X,Y$ to an algebraic $K$ -group $G$ . We define their convolution by

\varphi*\psi:X\times Y\rightarrow G,\text{ }(x,y)\mapsto\varphi(x)\cdot_{G}\psi(y).

We denote by $\varphi^{*k}:X^{k}\rightarrow G$ the $k$ -th self convolution of $\varphi$ .

We restrict ourselves to the setting where $X,Y$ are smooth algebraic $K$ -varieties and $G$ is a connected algebraic $K$ -group. The main motto is that the algebraic convolution operation has a smoothing effect on morphisms, similarly to the usual convolution operation in analysis (see [GH19, Proposition 1.3] and [GH21, Proposition 3.1]). For example, starting from any dominant map $\varphi:X\rightarrow G$ , the $k$ -th self convolution $\varphi^{*k}:X^{k}\rightarrow G$ is flat for every $k\geq\dim G$ ([GH21, Theorem B]). To explain the connection to this work, we introduce the following property:

Definition 1.8 ([AA16, Definition II]).

(1)

A $K$ -variety $Z$ has rational singularities if it is normal and for every resolution of singularities $\pi:\widetilde{Z}\rightarrow Z$ , the pushforward $\pi_{*}(\mathcal{O}_{\widetilde{Z}})$ of the structure sheaf has no higher cohomologies.
(2)

A morphism $\varphi:X\rightarrow Y$ between smooth $K$ -varieties is called (FRS) if it is flat and if every fiber of $\varphi$ has rational singularities.

In [AA16, Theorem 3.4] (see Theorem 6.1 below), Aizenbud and Avni proved the following. A morphism $\varphi:X\rightarrow Y$ between smooth $K$ -varieties, is (FRS) if and only if for every non-Archimedean local field $F\supseteq K$ , one has $\left(\varphi_{F}\right)_{*}\mu\in\mathcal{M}_{c,\infty}(Y(F))$ for every $\mu\in\mathcal{M}_{c}^{\infty}(X(F))$ . A similar characterization can be given for $F=\mathbb{C}$ , see Corollary 6.2.

This characterization of $L^{\infty}$ -maps allows one to study random walks on analytic groups as in $\mathsection$ 1.1 in an algebro-geometric way, via the above algebraic convolution operation. Starting from a pushforward $\nu=\left(\varphi_{F}\right)_{*}\mu$ of $\mu\in\mathcal{M}_{c,\infty}(X(F))$ by an algebraic map $\varphi:X\to G$ , instead of showing that $\nu^{*k}\in\mathcal{M}_{c,\infty}(G(F))$ , it is enough to show that $\varphi^{*k}:X^{k}\rightarrow G$ is an (FRS) morphism. This method was used in [AA16, GH24] to study word maps. Moreover, in [GH19, GH21] it was shown that any locally dominant morphism $\varphi:X\rightarrow G$ becomes (FRS) after sufficiently many self-convolutions.

Using (1.3), Theorem 1.1 and Corollary 6.2, explicit bounds can be given on the required number of self convolutions, in terms of the Jacobian ideal of $\varphi$ .

Corollary 1.9.

Let $X$ be a smooth $K$ -variety, $G$ be a connected $K$ -algebraic group and let $\varphi:X\to G$ be a locally dominant morphism. Then $\varphi^{*k}$ is (FRS) for any $k\geq\left\lfloor\frac{1+\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\varphi})}{\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\varphi})}\right\rfloor+1$ .

Remark 1.10.

In the setting of locally compact groups, Young’s convolution inequality is commonly stated under the assumption that the group $G$ is unimodular. In [KR78, Lemma 2.1 and Corollary 2.3] a version for non-unimodular groups is given; if $G$ is a locally compact group, with modular character $\triangle:G\rightarrow\mathbb{\mathbb{R}}_{>0}$ , and if $1\leq p,q,r\leq\infty$ satisfy $\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$ , then we have $\left\|\mu*(\triangle^{1-\frac{1}{p}}\nu^{\prime})\right\|_{r}\leq\left\|\mu\right\|_{p}\left\|\nu^{\prime}\right\|_{q}$ whenever $\mu\in\mathcal{M}_{c,p}(G)$ and $\nu^{\prime}\in\mathcal{M}_{c,q}(G)$ . However, since the modular character $\triangle:G\rightarrow\mathbb{\mathbb{R}}_{>0}$ is a continuous homomorphism, it bounded on the compact support of $\nu^{\prime}$ . Hence, for every $\mu\in\mathcal{M}_{c,p}(G)$ and $\nu\in\mathcal{M}_{c,q}(G)$ , we deduce that

\left\|\mu*\nu\right\|_{r}=\left\|\mu*\left(\triangle^{1-\frac{1}{p}}(\triangle^{\frac{1}{p}-1}\cdot\nu)\right)\right\|_{r}\leq\left\|\mu\right\|_{p}\cdot\left\|\triangle^{\frac{1}{p}-1}\cdot\nu\right\|_{q}\leq C\cdot\left\|\mu\right\|_{p}\cdot\left\|\nu\right\|_{q},

for some constant $C$ depending on $G$ and $\nu$ . In particular, $\mu*\nu\in\mathcal{M}_{c,r}(G)$ .

1.2.5. An algebraic characterization of $\epsilon_{\star}=\infty$

Let $\varphi:X\rightarrow Y$ be a morphism between smooth $K$ -varieties. We would like to characterize the condition that $\epsilon_{\star}(\varphi_{F})=\infty$ for all $F$ in certain families of local fields, in terms of the singularities of $\varphi$ . The singularity properties we consider play a central role in birational geometry (see [Kol97]).

Let $X$ be a normal $K$ -variety, and let $\omega\in\Omega^{\mathrm{top}}(X_{\mathrm{sm}})$ be a rational top form on the smooth locus $X_{\mathrm{sm}}$ of $X$ . The zeros and poles of $\omega$ give rise to a divisor $\operatorname{div}(\omega)$ on $X$ . Let $\pi:\widetilde{X}\rightarrow X$ be a resolution of singularities, namely, a proper morphism from a smooth variety $\widetilde{X}$ , which is an isomorphism over $X_{\mathrm{sm}}$ . Then $\pi^{*}\omega$ defines a unique rational top form on $\widetilde{X}$ . Moreover, when $X$ is nice enough (e.g. if $X$ is a local complete intersection), $\operatorname{div}(\omega)$ is $\mathbb{Q}$ -Cartier, and we can define its pullback $\pi^{*}\operatorname{div}(\omega)$ . The divisor $K_{\widetilde{X}/X}:=\operatorname{div}(\pi^{*}\omega)-\pi^{*}\operatorname{div}(\omega)$ on $\widetilde{X}$ is called the relative canonical divisor, and one can verify that it does not depend on the choice of $\omega$ . $K_{\widetilde{X}/X}$ can be written as $K_{\widetilde{X}/X}=\sum_{i=1}^{M}a_{i}E_{i}$ , for some prime divisors $E_{i}$ , $a_{i}\in\mathbb{Q}$ . We say that $X$ has canonical singularities (resp. log-canonical singularities), if $a_{i}\geq 0$ (resp. $a_{i}\geq-1$ ) for all $1\leq i\leq M$ . When $X$ is a local complete intersection (e.g. a fiber of a flat morphism between smooth schemes), canonical singularities are equivalent to rational singularities. Let us give an example:

Example 1.11.

Let $X\subseteq\mathbb{A}_{\mathbb{C}}^{n}$ be the variety defined by $\sum_{i=1}^{n}x_{i}^{d_{i}}=0$ , with $n\geq 3$ . Then $X$ has canonical singularities if and only if $\sum_{i=1}^{n}\frac{1}{d_{i}}>1$ , and log-canonical singularities if and only if $\sum_{i=1}^{n}\frac{1}{d_{i}}\geq 1$ .

As seen from Example 1.11, log-canonical singularities are very close to being canonical, so one could suspect being flat with fibers of log-canonical singularities is equivalent to $\epsilon_{\star}(\varphi_{\mathbb{Q}_{p}})=\infty$ , that is to being almost in $L^{\infty}$ . Unfortunately, the normality hypothesis required for log-canonical singularities turns out to be too strong. For example, the map $\varphi(x,y)=xy$ satisfies $\epsilon_{\star}(\varphi_{\mathbb{Q}_{p}})=\infty$ for all $p$ , but the fiber over $0$ is not normal (see $\mathsection$ 6, after Corollary 6.2). This technical issue can be resolved by considering the slightly weaker notion of semi-log-canonical singularities, which is an analogue of log-canonical singularities for demi-normal schemes (see [KSB88, Section 4]). Indeed, the variety $\left\{xy=0\right\}$ is demi-normal and has semi-log-canonical singularities. We can now state the main result of this section:

Theorem 1.12.

Let $\varphi:X\rightarrow Y$ be a map between smooth $K$ -varieties. Then the following are equivalent:

(1)

$\varphi$ is flat with fibers of semi-log-canonical singularities.
(2)

For every local field $F$ containing $K$ , we have $\epsilon_{\star}(\varphi_{F})=\infty$ , that is, for every $\mu\in\mathcal{M}_{c}^{\infty}(X(F))$ , the measure $\varphi_{*}\mu$ lies in $\mathcal{M}_{c,q}(Y(F))$ for all $1<q<\infty$ .
(3)

For every large enough prime $p$ , such that $\mathbb{Q}_{p}\supseteq K$ , we have $\epsilon_{\star}(\varphi_{\mathbb{Q}_{p}})=\infty$ .
(4)

We have $\epsilon_{\star}(\varphi_{\mathbb{C}})=\infty$ .

We prove Theorem 1.12 by showing the implications $(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(1)$ and $(1)\Rightarrow(2)\Rightarrow(4)\Rightarrow(1)$ . The implications $(2)\Rightarrow(3)$ and $(2)\Rightarrow(4)$ are immediate. In the proof of $(3)\Rightarrow(1)$ and $(4)\Rightarrow(1)$ , we reduce to the case $Y=\mathbb{A}^{n}$ , and show that $\varphi:X\rightarrow Y$ satisfies that $\varphi*\psi$ is (FRS) for every dominant map $\psi:Y^{\prime}\rightarrow Y$ . By analyzing the jets of $\varphi$ , and using a jet-scheme interpretation of semi-log canonical singularities (Lemma 6.5), we deduce Item (1). The proof also uses the Archimedean counterpart of [AA16, Theorem 3.4], which is stated in Corollary 6.2.

In the proof of $(1)\Rightarrow(2)$ , we reduce to showing Item (2) for constructible measures, and utilize their structure theory, namely, we use [CM11, CM12] for the Archimedean case, and [CGH18] for the non-Archimedean case.

1.3. Future directions and further applications

1.3.1. $\epsilon_{\star}$ as an invariant of singularities

Similarly to the analytic definition of $\epsilon_{\star}(\phi)$ in (1.2), one can also define an algebro-geometric invariant. Let $\operatorname{Loc}_{0}$ be the collection of all non-Archimedean local fields $F$ of characteristic $0$ .

Definition 1.13.

Let $\varphi:X\rightarrow Y$ be a morphism between smooth $\mathbb{Q}$ -varieties. We define $\epsilon_{\star}(\varphi):=\min\limits_{F\in\operatorname{Loc}_{0}}\epsilon_{\star}(\varphi_{F})$ , and $\epsilon_{\star}(\varphi;x)=\sup\limits_{U\ni x}\,\epsilon_{\star}(\varphi|_{U})$ , where $U$ varies over all Zariski open neighborhoods of $x\in X(\overline{\mathbb{Q}})$ .

It is a consequence of Theorem 1.1 that $\epsilon_{\star}(\varphi)>0$ for any $\varphi$ , and it essentially follows from [GH21] that $\epsilon_{\star}(\varphi)\in\mathbb{Q}$ . We further expect $\epsilon_{\star}(\varphi;x)$ to have a purely algebro-geometric formula, and to have a good behavior in families, which means the following. Suppose that $\varphi:\widetilde{X}\rightarrow\widetilde{Y}$ is a morphism over $\mathbb{A}_{\mathbb{C}}^{1}$ , where $\pi_{1}:\widetilde{X}\rightarrow\mathbb{A}_{\mathbb{C}}^{1}$ and $\pi_{2}:\widetilde{Y}\rightarrow\mathbb{A}_{\mathbb{C}}^{1}$ are smooth morphisms. This gives a family $\left\{\varphi_{t}:\widetilde{X}_{t}\rightarrow\widetilde{Y}_{t}\right\}_{t\in\mathbb{A}_{\mathbb{C}}^{1}}$ of morphisms between smooth varieties. It follows from [Var83], that the function $x\mapsto\operatorname{lct}(\varphi_{\pi_{1}(x)};x)$ is lower semicontinuous. By Theorem 1.2, $x\mapsto\epsilon_{\star}(\varphi_{\pi_{1}(x)};x)$ is lower semicontinuous as well, if $\pi_{2}:\widetilde{Y}\rightarrow\mathbb{A}^{1}$ has fibers of dimension $1$ . We expect $x\mapsto\epsilon_{\star}(\varphi_{\pi_{1}(x)};x)$ to be lower semicontinuous in general. We further expect the following question to have a positive answer.

Question 1.14.

Let $\varphi:X\rightarrow Y$ be as in Definition 1.13. Is it true that $\epsilon_{\star}(\varphi_{\mathbb{C}})=\epsilon_{\star}(\varphi)$ ?

Given a $\mathbb{Q}$ -morphism $\varphi:X\rightarrow Y$ as above, one may further wonder whether the quantity $\epsilon_{\star}(\varphi_{\mathbb{C}};x)=\sup_{U\ni x}\inf_{\mu\in\mathcal{M}_{c,\infty}(U)}\epsilon_{\star}((\varphi_{\mathbb{C}})_{*}\mu)$ in (1.1) stays the same when the supremum is taken over all Zariski open neighborhoods $U$ of $x$ instead of analytic ones.

1.3.2. $\epsilon_{\star}$ as an invariant of words

As discussed in $\mathsection$ 1.1, one particularly interesting potential application is to the study of word map on semisimple algebraic groups. In [GH24, Theorem A] it was shown that Lie algebra word maps $w_{\mathfrak{g}}:\mathfrak{g}^{r}\rightarrow\mathfrak{g}$ , where $\mathfrak{g}$ is a simple Lie algebra, become (FRS) after $\sim\operatorname{deg}(w)^{4}$ self convolutions, where $\operatorname{deg}(w)$ is the degree of $w$ .

Question 1.15.

Can we find $\alpha,C>0$ such that for any $w\in F_{r}$ of length $\ell(w)$ , and every simple algebraic group $G$ , the word map $w_{G}^{*C\ell(w)^{\alpha}}$ is (FRS)?

A potential way to tackle Question 1.15 is by studying $\epsilon_{\star}(w_{G})$ in the sense of Definition 1.13; For each $w\in F_{r}$ (resp. $w\in\mathcal{L}_{r}$ ), we define $\epsilon_{\star}(w):=\inf_{G}\,\epsilon_{\star}(w_{G})$ (resp. $\epsilon_{\star}(w):=\inf_{\mathfrak{g}}\,\epsilon_{\star}(w_{\mathfrak{g}})$ ), where $G$ runs over all simple, simply connected algebraic groups, and $\mathfrak{g}=\operatorname{Lie}(G)$ . We can now ask the following:

Question 1.16.

Can we find for each $l\in\mathbb{N}$ , a constant $\epsilon(l)>0$ such that:

(1)

For every $w\in F_{r}$ of length $\ell(w)=l$ , we have $\epsilon_{\star}(w)\geq\epsilon(l)$ ?
(2)

For every $w\in\mathcal{L}_{r}$ of degree $\operatorname{deg}(w)=l$ , we have $\epsilon_{\star}(w)\geq\epsilon(l)$ ?

1.3.3. $\epsilon_{\star}$ as an invariant of representations

In a recent work [GGH] of the first two authors with Julia Gordon, we apply the results and point of view of this paper to the realm of representation theory, and define and study a new invariant of representations of reductive groups $G$ over local fields. Harish-Chandra’s regularity theorem says that every character $\Theta(\pi)$ of an irreducible representation $\pi$ of $G$ is given by a locally $L^{1}$ -function. Since characters are of motivic nature, a variant of [GH21, Theorem F] suggests that they should in fact be locally in $L^{1+\epsilon}$ , for some $\epsilon>0$ . This gives rise to a new invariant $\epsilon_{\star}(\pi)$ , which is not equivalent to previously known invariants of representations, such as the Gelfand–Kirillov dimension (see e.g. [Vog78]). We use a geometric construction and Theorem 1.1 to provide a formula for $\epsilon_{\star}(\pi)$ in terms of the nilpotent orbits appearing the local character expansion of $\pi$ (see [GGH, Theorems A and D]).

1.4. Conventions

(1)

By $\mathbb{N}$ we mean the set $\{0,1,2,...\}$ .
(2)

We use $K$ (resp. $F$ ) to denote number (resp. local) fields and $\mathcal{O}_{K}$ (resp. $\mathcal{O}_{F}$ ) for their rings of integers.
(3)

Given an algebraic map $\varphi:X\rightarrow Y$ between smooth $K$ -varieties, and a local field $F$ , we denote by $\varphi_{F}:X(F)\rightarrow Y(F)$ the corresponding $F$ -analytic map. We sometimes write $\varphi$ instead of $\varphi_{F}$ if the local field $F$ is clear from the setting.
(4)

If $f,g:X\to\mathbb{R}$ are two functions, we write $f\lesssim g$ if there exists a number $C>0$ , possibly depending on $X$ , $f$ and $g$ , such that $f\leq Cg$ . We write $f\sim g$ if $g\lesssim f\lesssim g$ .
(5)

We write $d_{x}\phi$ for the differential of an analytic map $\phi:X\rightarrow Y$ at $x\in X$ , and we denote by $\operatorname{Jac}_{x}\phi:=\det(d_{x}\phi)$ the Jacobian of $\phi$ if $\dim X=\dim Y$ . The generalization of the Jacobian to the case of unequal dimensions is defined in $\mathsection$ 1.2.
(6)

Throughout the paper, we write $\left|\,\cdot\,\right|_{F}$ for the absolute value on $F$ , normalized so that $\mu_{F}(aA)=\left|a\right|_{F}\cdot\mu_{F}(A)$ , for all $a\in F^{\times}$ , $A\subseteq F$ , and where $\mu_{F}$ is a Haar measure on $F$ . Note that $\left|\,\cdot\,\right|_{\mathbb{C}}$ is the square of the usual absolute value on $\mathbb{C}$ .

(7)

We write $H_{n}$ for the $n$ -dimensional Hausdorff measure. We recall that given a metric space $X$ and a subset $Z\subseteq X$ , we define $H_{n}(Z):=\underset{\delta\rightarrow 0}{\sup}H_{n}^{\delta}(Z)$ , where

H_{n}^{\delta}(Z)=\inf\left\{\sum_{i=1}^{\infty}\alpha(n)\left(\mathrm{diam}(U_{i})\right)^{n}:\bigcup_{i=1}^{\infty}U_{i}\supseteq Z,\mathrm{diam}(U_{i})<\delta\right\},

where $\mathrm{diam}(A)$ denotes the diameter of a set $A\subseteq X$ and $\alpha(n):=\frac{(\pi/4)^{n/2}}{\Gamma(\frac{n}{2}+1)}$ . The normalization constant $\alpha(n)$ is chosen such that $H_{n}$ coincides with the Lebesgue measure in the case of $X=\mathbb{\mathbb{R}}^{n}$ .

(8)

All the measures we consider are non-negative, unless stated otherwise.

Acknowledgement.

We thank Nir Avni, Joseph Bernstein, Lev Buhovski, Raf Cluckers and Stephan Snegirov for useful conversations and correspondences. We thank David Kazhdan for suggesting that the condition of semi-log-canonical singularities should play a role in Theorem 1.12. We are particularly grateful to Rami Aizenbud for numerous discussions on this project, and for initiating the collaboration between the authors.

SS was supported in part by the European Research Council starting grant 639305 (SPECTRUM), a Royal Society Wolfson Research Merit Award (WM170012), and a Philip Leverhulme Prize of the Leverhulme Trust (PLP-2020-064).

2. Preliminaries: embedded resolution of singularities

Let $F$ be a local field of characteristic zero. We use the following analytic version of Hironaka’s theorem [Hir64] on embedded resolution of singularities. The map $\pi:\widetilde{X}\rightarrow U$ below is called a log-principalization (or uniformization) of $J$ .

Theorem 2.1 (See [VZnG08, Theorem 2.3], [DvdD88, Theorem 2.2], [BM89] and [Wlo09]).

Let $U\subseteq F^{n}$ be an open subset, and let $f_{1},...,f_{r}:U\rightarrow F$ be $F$ -analytic maps, generating a non-zero ideal $J$ in the algebra of $F$ -analytic functions on $U$ . Then there exist an $F$ -analytic manifold $\widetilde{X}$ , a proper $F$ -analytic map $\pi:\widetilde{X}\rightarrow U$ and a collection of closed submanifolds $\left\{E_{i}\right\}_{i\in T}$ of $\widetilde{X}$ of codimension $1$ , equipped with pairs of non-negative integers $\left\{(a_{i},b_{i})\right\}_{i\in T}$ , such that the following hold:

(1)

$\pi$ is locally a composition of a finite number of blow-ups at closed submanifolds, and is an isomorphism over the complement of the common zero set $\mathrm{V}(J)$ of $J$ in $U$ .
(2)

For every $c\in\widetilde{X}$ , there are local coordinates $(\widetilde{x}_{1},...,\widetilde{x}_{n})$ in a neighborhood $V\ni c$ , such that each $E_{i}$ containing $c$ is given by the equation $\widetilde{x}_{i}=0$ . Moreover, if without loss of generality $E_{1},...,E_{m}$ contain $c$ , then there exists an $F$ -analytic unit $v:V\rightarrow F$ , such that the pullback of $J$ is the principal ideal

(2.1) $\pi^{*}J=\langle\widetilde{x}_{1}^{a_{1}}\cdots\widetilde{x}_{m}^{a_{m}}\rangle$

and such that the Jacobian of $\pi$ (i.e. $\det\,d_{\widetilde{x}}\pi$ ) is given by:

(2.2) $\operatorname{Jac}_{\widetilde{x}}(\pi)=v(\widetilde{x})\cdot\widetilde{x}_{1}^{b_{1}}\cdots\widetilde{x}_{m}^{b_{m}}.$

Remark 2.2.

(1)

Condition (2.1) means that for each $f_{i}:U\rightarrow F$ , one can write $f_{i}\circ\pi(\widetilde{x})=u_{i}(\widetilde{x})\widetilde{x}_{1}^{a_{1}}\cdots\widetilde{x}_{m}^{a_{m}}$ for some analytic functions $u_{i}$ , and $u_{j}(0)\neq 0$ for at least one $j\in\{1,...,r\}$ . Note that the $a_{1},...,a_{m}$ are the same for all the $f_{i}$ ’s.
(2)

If $r=1$ , so that $J=\langle f\rangle$ , we may further assume that $f\circ\pi(\widetilde{x}_{1},...,\widetilde{x}_{n})=C\cdot\widetilde{x}_{1}^{a_{1}}\cdots\widetilde{x}_{m}^{a_{m}}$ locally on each chart, for some constant $C\neq 0$ . Indeed, $u(0)\neq 0$ . If $u(0)$ is an $a_{1}$ -th power in $F$ then the same holds for $u(\widetilde{x})$ in a small neighborhood of $0$ . In this case, we may apply the change of coordinates

$(\widetilde{x}_{1},...,\widetilde{x}_{n})\mapsto(u(\widetilde{x})^{-\frac{1}{a_{1}}}\widetilde{x}_{1},...,\widetilde{x}_{n}).$

If $u(0)$ is not an $a_{1}$ -th power we may multiply it by $b\in F^{\times}$ such that $u(\widetilde{x})b$ is an $a_{1}$ -th power, and apply a similar change of coordinates.

The next lemma follows directly by changing coordinates using a log-principalization of $J$ and computing the integral $\int\min\limits_{1\leq i\leq l}\big{[}\left|f_{i}(x)\right|^{-s}\big{]}d\mu(s)$ with respect to the new coordinates.

Lemma 2.3 (See e.g. [Mus12, Theorem 1.2] and [VZnG08, Theorem 2.7]).

Let $J=\langle f_{1},...,f_{r}\rangle$ be an ideal of $F$ -analytic functions on $U\subseteq F^{n}$ . Let $\pi:\widetilde{X}\rightarrow U$ be a log-principalization of $J$ , with data $\left\{E_{i}\right\}_{i\in T}$ and $\left\{(a_{i},b_{i})\right\}_{i\in T}$ as in Theorem 2.1. Then the $F$ -log-canonical threshold of $J$ at $x\in U$ is equal to:

(2.3)

\operatorname{lct}_{F}(J;x)=\min_{i:\,x\in\pi(E_{i})}\frac{b_{i}+1}{a_{i}}.

Remark 2.4.

Note that while the data $\left\{E_{i}\right\}_{i\in T}$ and $\left\{(a_{i},b_{i})\right\}_{i\in T}$ depends on the log-principalization $\pi$ , Definition 1.5 is intrinsic, thus the right-hand side of (2.3) does not depend on $\pi$ .

3. The one-dimensional case

3.1. A formula for $\epsilon_{\star}$

In this section we provide a formula for $\epsilon_{\star}$ in the one-dimensional case (Theorem 1.2). The formula will be phrased in terms of the $F$ -log-canonical threshold, where $F$ is any local field of characteristic zero.

When $F$ is non-Archimedean, we denote by $\mathcal{O}_{F}$ its ring of integers, by $k_{F}$ its residue field, and by $q_{F}$ the number of elements in $k_{F}$ . Write $\varpi_{F}\in\mathcal{O}_{F}$ for a fixed uniformizer (i.e., a generator of the maximal ideal of the ring of integers) of $F$ , and let $\left|\,\cdot\,\right|_{F}$ be as in $\mathsection$ 1.4, so that $\left|\varpi_{F}\right|_{F}=q_{F}^{-1}$ . We write $\mu_{F}$ for the Haar measure on $F$ , normalized such that $\mu_{F}(\mathcal{O}_{F})=1$ when $F$ is non-Archimedean, and such that $\mu_{F}$ is the Lebesgue measure when $F$ is Archimedean. We write $\mu_{\mathcal{O}_{F}}:=\mu_{F}|_{\mathcal{O}_{F}}$ . We write $dx$ instead of $\mu_{F}$ when we integrate a function $g(x)$ with respect to $\mu_{F}$ . We denote by $\left\|(a_{1},...,a_{n})\right\|_{F}:=\max_{i}\left|a_{i}\right|_{F}$ the maximum norm on $F^{n}$ .

For an analytic map $\phi:X\rightarrow F$ , and $x\in X$ , we set $\phi_{x}(z):=\phi(z)-\phi(x)$ .

To prove Theorem 1.2, we reduce to the monomial case using Hironaka’s resolution of singularities ( $\mathsection$ 2), and prove the monomial case in Lemma 3.2. However, we first note that the upper bound in (1.7) can be proved by elementary arguments, as follows:

Lemma 3.1.

Let $X$ be an analytic $F$ -manifold, and let $\phi:X\to F$ be a locally dominant analytic map. Then for every $x\in X$ with $\operatorname{lct}_{F}(\phi_{x};x)<1$ , one has:

\epsilon_{\star}(\phi;x)\leq\frac{\operatorname{lct}_{F}(\phi_{x};x)}{1-\operatorname{lct}_{F}(\phi_{x};x)}.

Proof.

We need to show

1-\frac{1}{1+\epsilon_{\star}(\phi;x)}\leq\operatorname{lct}_{F}(\phi_{x};x).

Let $\epsilon<\epsilon_{\star}(\phi;x)$ . Then there exists a neighborhood $U$ of $x$ such that $\phi_{*}\sigma\in L^{1+\epsilon}$ for every $\sigma\in\mathcal{M}_{c,\infty}(U)$ . Write $\phi_{*}\sigma=g(t)\mu_{F}$ . Let $B(a,\delta):=\left\{t:\left|t-a\right|_{F}\leq\delta\right\}$ , and note that $\mu_{F}(B(a,\delta))\sim\delta$ . By Jensen’s inequality we have:

(\phi_{*}\sigma)(B(a,\delta))=\frac{1}{\delta}\int_{B(a,\delta)}\delta g(t)dt\leq\left(\frac{1}{\delta}\int_{B(a,\delta)}\left(\delta g(t)\right)^{1+\epsilon}dt\right)^{\frac{1}{1+\epsilon}}\lesssim\delta^{1-\frac{1}{1+\epsilon}},

i.e. we have the distributional estimate $\sigma\left(\left\{z:\left|\phi(z)-\phi(x)\right|_{F}\leq\delta\right\}\right)\lesssim\delta^{1-\frac{1}{1+\epsilon}}$ . Using Fubini’s theorem, we obtain:

	$\displaystyle\int_{U}\left\|\phi_{x}(z)\right\|_{F}^{-s}d\sigma$	$\displaystyle=\int_{U}\left(\int_{0}^{\infty}1_{\left\{(t,z)\,:\left\|\phi_{x}(z)\right\|_{F}^{-s}\geq t\right\}}(t,z)dt\right)d\sigma$
		$\displaystyle=\int_{0}^{\infty}\sigma\left\{z:\left\|\phi(z)-\phi(x)\right\|_{F}^{-s}\geq t\right\}dt$
		$\displaystyle=\int_{0}^{\infty}\sigma\left\{z:\left\|\phi(z)-\phi(x)\right\|_{F}\leq t^{-1/s}\right\}dt$
		$\displaystyle\lesssim 1+\int_{1}^{\infty}t^{-\frac{1}{s}\left(1-\frac{1}{1+\epsilon}\right)}dt<\infty,$

whenever $s<1-\frac{1}{1+\epsilon}$ . This implies that $\operatorname{lct}_{F}(\phi_{x};x)\geq 1-\frac{1}{1+\epsilon_{\star}(\phi;x)}$ . ∎

We now return to the main narrative. Specializing to the setting of Hironaka’s theorem, we consider the monomial case.

Lemma 3.2.

Let $f:F^{n}\rightarrow F$ be a monomial map $f(x_{1},...,x_{n})=x_{1}^{a_{1}}\cdot...\cdot x_{n}^{a_{n}}$ , let $g:F^{n}\rightarrow\mathbb{\mathbb{R}}_{\geq 0}$ be a continuous function and let $\mu=g(x)\left|x_{1}\right|_{F}^{b_{1}}\cdot...\cdot\left|x_{n}\right|_{F}^{b_{n}}\mu_{F}^{n}$ , for $a_{1},...,a_{n}\in\mathbb{Z}_{\geq 1}$ and $b_{1},...,b_{n}\in\mathbb{Z}_{\geq 0}$ . Then:

\epsilon_{\star}(f_{*}\mu)\,\begin{cases}=\infty&\text{if\, }\min_{i}\frac{b_{i}+1}{a_{i}}\geq 1,\\ \geq\frac{\min_{i}\frac{b_{i}+1}{a_{i}}}{1-\min_{i}\frac{b_{i}+1}{a_{i}}}&\text{otherwise}.\end{cases}

Furthermore, if $g(0)\neq 0$ , the second bound is in fact an equality.

Proof.

Without loss of generality we may assume $\min_{i}\frac{b_{i}+1}{a_{i}}=\frac{b_{n}+1}{a_{n}}$ . We first consider the special case when $\frac{b_{n}+1}{a_{n}}$ is the unique minimum. We write $f$ as a composition $f=\operatorname{pr}\circ\psi$ , where $\psi:F^{n}\rightarrow F^{n}$ is given by $\psi(x_{1},...,x_{n})=(x_{1},...,x_{n-1},x_{1}^{a_{1}}\cdot...\cdot x_{n}^{a_{n}})$ and $\operatorname{pr}$ is the projection to the last coordinate. Write $x=(\overline{x},x_{n})$ , where $\overline{x}=(x_{1},...,x_{n-1})$ , and similarly $y=(\overline{y},y_{n})$ . Note that $\operatorname{Jac}_{(x_{1},...,x_{n})}(\psi)=a_{n}x_{1}^{a_{1}}\cdot...\cdot x_{n-1}^{a_{n-1}}x_{n}^{a_{n}-1}$ and that if $\psi(\overline{x},x_{n})=(\overline{y},y_{n})$ , then $\overline{x}=\overline{y}$ and $x_{n}^{a_{n}}=y_{n}y_{1}^{-a_{1}}\cdot...\cdot y_{n-1}^{-a_{n-1}}$ . Hence the Radon–Nikodym density of $\psi_{*}\mu$ is equal to

	$\displaystyle\frac{d(\psi_{*}\mu)}{d(\mu_{F}^{n})}(\overline{y},y_{n})$	$\displaystyle=\sum_{t:\psi(\overline{y},t)=y}\frac{\left(\prod_{j=1}^{n-1}\left\|y_{j}\right\|_{F}^{b_{j}}\right)\left\|t\right\|_{F}^{b_{n}}g(\overline{y},t)}{\left\|\operatorname{Jac}_{(y_{1},...,y_{n-1},t)}(\psi)\right\|_{F}}$
		$\displaystyle=\sum_{t:\psi(\overline{y},t)=y}g(\overline{y},t)\left\|a_{n}\right\|_{F}^{-1}\cdot\left(\prod_{j=1}^{n-1}\left\|y_{j}\right\|_{F}^{b_{j}-a_{j}}\right)\left\|t\right\|_{F}^{b_{n}-a_{n}+1}$
		$\displaystyle=\left(\sum_{t:\psi(\overline{y},t)=y}g(\overline{y},t)\right)\cdot\left\|a_{n}\right\|_{F}^{-1}\left(\prod_{j=1}^{n-1}\left\|y_{j}\right\|_{F}^{b_{j}-a_{j}-\frac{(b_{n}-a_{n}+1)a_{j}}{a_{n}}}\right)\left\|y_{n}\right\|_{F}^{\frac{b_{n}-a_{n}+1}{a_{n}}}.$

Since $\frac{b_{n}+1}{a_{n}}<\frac{b_{j}+1}{a_{j}}$ we get for every $1\leq j\leq n-1$ ,

a_{n}(b_{j}-a_{j}+1)>(b_{n}-a_{n}+1)a_{j},

and hence:

b_{j}-a_{j}-\frac{(b_{n}-a_{n}+1)a_{j}}{a_{n}}>-1.

In particular, integrating over the first $n-1$ coordinates, we get that $\frac{d(f_{*}\mu)}{d\mu_{F}}(s)\lesssim\left|s\right|_{F}^{\frac{b_{n}-a_{n}+1}{a_{n}}}$ . If $\frac{b_{n}+1}{a_{n}}\geq 1$ then $\epsilon_{\star}(f_{*}\mu)=\infty$ as required. Thus we may assume that $\frac{b_{n}+1}{a_{n}}<1$ . Then $f_{*}\mu\in\mathcal{M}_{c,1+\epsilon}(F)$ , whenever $\frac{b_{n}-a_{n}+1}{a_{n}}(1+\epsilon)>-1$ , i.e. whenever

\epsilon<\frac{b_{n}+1}{a_{n}-b_{n}-1}=\frac{\frac{b_{n}+1}{a_{n}}}{1-\frac{b_{n}+1}{a_{n}}}=\frac{\min_{i}\frac{b_{i}+1}{a_{i}}}{1-\min_{i}\frac{b_{i}+1}{a_{i}}}.

If $g(0)\neq 0,$ we also have $\frac{d(f_{*}\mu)}{d\mu_{F}}(s)\gtrsim\left|s\right|_{F}^{\frac{b_{n}-a_{n}+1}{a_{n}}}$ , whence the inequality in the statement of the lemma is in fact an equality.

It is left to deal with the case when $\min_{i}\frac{b_{i}+1}{a_{i}}$ is not uniquely achieved. For the lower bound, take

\widetilde{\mu}=\left|x_{1}\right|_{F}^{b_{1}}...\left|x_{n-1}\right|_{F}^{b_{n-1}}\left|x_{n}\right|_{F}^{b_{n}-\delta}g(x)\mu_{F}^{n},

for an arbitrarily small $\delta>0$ . Since $\widetilde{\mu}\geq\mu$ inside a small neighborhood of $0$ , we deduce that $f_{*}\mu\in\mathcal{M}_{c,1+\epsilon}(F)$ for $\epsilon<\frac{\frac{b_{n}-\delta+1}{a_{n}}}{1-\frac{b_{n}-\delta+1}{a_{n}}}$ . Since $\delta$ can be taken arbitrarily small we are done. Similarly, for the upper bound we take

\widetilde{\mu}=\left|x_{1}\right|_{F}^{b_{1}+\delta}\cdot...\cdot\left|x_{n-1}\right|_{F}^{b_{n-1}+\delta}\left|x_{n}\right|_{F}^{b_{n}}g(x)\mu_{F}^{n}\leq\mu,

with $g(0)\neq 0$ and deduce that $\epsilon_{\star}(f_{*}\mu)\leq\frac{\frac{b_{n}+1}{a_{n}}}{1-\frac{b_{n}+1}{a_{n}}}$ . ∎

Proof of Theorem 1.2.

Let $\phi:X\to F$ be a locally dominant analytic map, and let $x_{0}\in X$ . Replacing $\phi$ with $\phi_{x_{0}}=\phi-\phi(x_{0})$ , we may assume that $\phi(x_{0})=0$ . We may further assume that $X\subseteq F^{n}$ is open, and apply Theorem 2.1, to get a log-principalization $\pi:\widetilde{X}\rightarrow X$ , such that, locally on a chart around a point in $\pi^{-1}(x_{0})$ , $\phi_{x_{0}}\circ\pi(\widetilde{x}_{1},...,\widetilde{x}_{n})=C\widetilde{x}_{1}^{a_{1}}\cdot...\cdot\widetilde{x}_{n}^{a_{n}}$ , and

\operatorname{Jac}_{\widetilde{x}}(\pi)=v(\widetilde{x})\cdot\widetilde{x}_{1}^{b_{n}}\cdot...\cdot\widetilde{x}_{n}^{b_{n}}.

Let $\sigma\in\mathcal{M}_{c,\infty}(X)$ , with $\sigma=g(x)\mu_{F}^{n}$ , and $g>0$ in a neighborhood of $x_{0}$ . Then $\sigma=\pi_{*}\mu$ , where

\mu=(g\circ\pi)(\widetilde{x}_{1},...,\widetilde{x}_{n})\left|v(\widetilde{x})\right|_{F}\left|\widetilde{x}_{1}\right|_{F}^{b_{1}}...\left|\widetilde{x}_{n}\right|_{F}^{b_{n}}\mu_{F}^{n}.

Since $\phi_{*}\sigma=(\phi\circ\pi)_{*}\mu$ , the theorem now follows from Lemmas 3.2 and 2.3. ∎

3.2. Relation to Fourier decay and other invariants

Let $\phi:X\rightarrow Y$ be an $F$ -analytic map between $F$ -analytic manifolds $X$ and $Y$ . We have seen that each of the exponents $\epsilon_{\star}(\phi;x)$ and $\operatorname{lct}_{F}(\phi_{x};x)$ provides a different quantification for the singularities of $\phi$ near $x\in X$ . When $Y=F^{m}$ , one can further consider other invariants involving the Fourier transform of pushforward measures.

In $\mathsection$ 1.1, we have defined $k_{\star}(\phi;x)$ as the minimal number of self-convolutions after which the pushforward densities of smooth measures supported near $x$ become bounded. Note that by the Plancherel theorem, for each $\nu\in\mathcal{M}_{c,1}(F^{m})$ we have $\nu^{*k}\in\mathcal{M}_{c,2}(F^{m})$ if and only if $\mathcal{F}(\nu)\in L^{2k}(F^{m})$ , whence

\nu^{*k}\in\mathcal{M}_{c,\infty}(F^{m})\Longrightarrow\mathcal{F}(\nu)\in L^{2k}(F^{m})\Longrightarrow\nu^{*2k}\in\mathcal{M}_{c,\infty}(F^{m}).

Thus, the exponent $k_{\star}(\phi;x)$ is, in general, roughly comparable to the $L^{p}$ -class of $\mathcal{F}(\phi_{*}\mu)$ rather than to the $L^{p}$ -class of the pushforward measure $\phi_{*}\mu$ itself. Instead of the $L^{p}$ -class, we now focus on an invariant quantifying the Fourier decay of $\phi_{*}\mu$ on the power-law scale:

\delta_{\star}(\phi;x):=\sup_{U\ni x}\inf_{\mu\in\mathcal{M}_{c,\infty}(U)}\delta_{\star}(\phi_{*}\mu),

where

(3.1)

\delta_{\star}(\nu):=\sup\{\delta\geq 0\,:\,\left|\mathcal{F}(\nu)(y)\right|\lesssim\left\|y\right\|_{F}^{-\delta}\}.

The study of the invariant $\delta_{\star}(\mu)$ , and variations of it, goes back at least to the 1920’s, when the classical van der Corput lemma was introduced, relating lower bounds on the derivative of a smooth function $f:\mathbb{\mathbb{R}}\rightarrow\mathbb{\mathbb{R}}$ , to bounds as in (3.1), see [Ste93, Proposition 2], and [CCW99]. This invariant was also studied extensively in Igusa’s work [Igu78] in the case $\dim Y=1$ ; it is much less understood in high dimensions.

Remark 3.3.

Note that $\epsilon_{\star}(\phi;x)$ and $\operatorname{lct}_{F}(\phi_{x};x)$ are preserved under analytic changes of coordinates around $x$ . On the other hand, $\delta_{\star}(\phi;x)$ and $k_{\star}(\phi;x)$ might depend on the choice of coordinate system. For example, the map $\phi(x,y)=(x,x^{2}(1+y^{1000}))$ satisfies $\epsilon_{\star}(\phi)=\frac{1}{999}$ (by Theorem 1.1), while $\delta_{\star}(\phi)=\frac{1}{2}$ and $k_{\star}(\phi)\leq 4$ . By applying the change of coordinates $\psi(x,y)=(x,y-x^{2})$ , we get $\psi\circ\phi(x,y)=(x,x^{2}y^{1000})$ , and still have $\epsilon_{\star}(\psi\circ\phi)=\frac{1}{999}$ , while $\delta_{\star}(\psi\circ\phi)=\frac{1}{1000}$ and $k_{\star}(\psi\circ\phi)\geq 1000$ .

We next discuss the relations between the different exponents. In the one-dimensional case, it turns out that all of the exponents above are essentially equivalent (whenever $\operatorname{lct}_{F}(\phi;x)\leq 1$ ). In order to explain this, we need to discuss the structure of pushforward measures by analytic maps.

3.2.1. Asymptotic expansions of pushforward measures and their Fourier transform

Let $F$ be a local field, $f:U\rightarrow F$ be a locally dominant analytic map with $U\subseteq F^{n}$ an open set. Let $\mu\in\mathcal{M}_{c}^{\infty}(U)$ , and consider the pushforward measure $f_{*}\mu$ . Fix a non-trivial additive character $\Psi$ of $F$ . We may identify between $F$ and $F^{\vee}$ by $t\longleftrightarrow\Psi_{t}$ where $\Psi_{t}(y)=\Psi(ty)$ . The Fourier transform $\mathcal{F}(f_{*}\mu)$ can then be written as

\mathcal{F}(f_{*}\mu)(t)=\int_{F}\Psi(ty)df_{*}\mu(y)=\int_{U}\Psi(t\cdot f(x))d\mu(x).

To $\mu$ and $f$ , one can further associates Igusa’s local zeta function

Z_{\mu,f}(s):=\int_{U}\left|f(x)\right|_{F}^{s}d\mu(x),\,\,\,\,\,\,s\in\mathbb{C},\,\,\,\,\operatorname{Re}(s)>0.

Igusa’s local zeta function admits a meromorphic continuation to the complex plane (see [BG69, Ati70, Ber72] for the Archimedean case, and [Igu74, Igu78] for the non-Archimedean case). The poles of $Z_{\mu,f}(s)$ (as well as certain twisted versions of it), and the Laurent expansions around them, controls the asymptotic expansions for $f_{*}\mu$ as $\left|y\right|_{F}\rightarrow 0$ and for $\mathcal{F}(f_{*}\mu)$ when $\left|t\right|_{F}\rightarrow\infty$ , via the theory of Mellin transform (see [Igu78, Theorems 4.2, 4.3 and 5.3]). We next describe the asymptotic expansions of $f_{*}\mu$ and $\mathcal{F}(f_{*}\mu)$ .

Definition 3.4 (Asymptotic expansion, see [Igu78, Section I.2]).

Let $x_{\infty}$ be $0$ or $\infty$ . A sequence $\{\varphi_{k}\}_{k\in\mathbb{N}}$ of complex-valued functions on an open subset $U$ of $F\in\{\mathbb{\mathbb{R}},\mathbb{C}\}$ , with $\varphi_{k}(x)\neq 0$ in a punctured neighborhood of $x_{\infty}$ , is said to constitute an asymptotic scale, if for every $k$ ,

{\displaystyle\lim\limits_{x\to x_{\infty}}\frac{\varphi_{k+1}(x)}{\varphi_{k}(x)}=0}.

A function $f:U\rightarrow\mathbb{C}$ is said to have an asymptotic expansion at $x_{\infty}$ , if there exists a sequence $\{a_{n}\}_{n\in\mathbb{N}}$ of complex numbers such that for every $k\geq 0$ , there exists $C>0$ such that for all $x$ close enough to $x_{\infty}$ :

\left|f(x)-\sum\limits_{i=0}^{k}a_{i}\varphi_{i}(x)\right|\leq C\cdot\varphi_{k+1}(x).

In this case, we write

(3.2)

f(x)\approx\sum\limits_{k=0}^{\infty}a_{k}\varphi_{k}(x)\text{ as }x\rightarrow x_{\infty}.

Example 3.5.

Given a monotone increasing sequence $\Lambda=(-1<\lambda_{0}<\lambda_{1}<...<\lambda_{n}<...)$ of real numbers, with no finite accumulation points, and given a sequence $\{m_{n}\}_{n\in\mathbb{N}}$ of positive integers, set $\varphi_{0},\varphi_{1},\varphi_{2},...$ to be the sequence:

x^{\lambda_{0}}\log(x)^{m_{0}-1},x^{\lambda_{0}}\log(x)^{m_{0}-2},...,x^{\lambda_{0}},x^{\lambda_{1}}\log(x)^{m_{1}-1},...,x^{\lambda_{1}},...

for $x>0$ . Then $\{\varphi_{k}\}_{k\in\mathbb{N}}$ is an asymptotic scale at $x_{\infty}=0$ .

We now describe the asymptotic expansions of pushforward measures and their Fourier transforms. We fix a local field $F$ of characteristic $0$ and an analytic $F$ -manifold $X$ . If $F$ is non-Archimedean we further fix a uniformizer $\varpi_{F}\in\mathcal{O}_{F}$ . We set $F_{1}^{\times}:=\{x\in F:\left|x\right|_{F}=1\}$ and denote by $\mathbb{\mathrm{ac}}:F^{\times}\rightarrow F_{1}^{\times}$ the angular component map

\mathbb{\mathrm{ac}}(t)=\begin{cases}t\varpi_{F}^{-\mathbb{\mathrm{val}}(t)}&\text{if }F\text{ is non-Archimedean}\\ \frac{t}{\left|t\right|}&\text{if }F\text{ is Archimedean}.\end{cases}

Theorem 3.6 ([Jea70, Mal74, Igu78], see also [VZnG17, Section 4] and [Den91, Theorem 1.3.2 and Corollary 1.4.5]).

Let $f:X\rightarrow F$ be a locally dominant analytic map, let $\mu\in\mathcal{M}_{c}^{\infty}(X)$ , and write $f_{*}\mu=g(y)d\mu_{F}$ . Suppose that $0$ is the only critical value of $f$ . Then there exist:

•

a sequence $\Lambda=\{\lambda_{k}\}_{k\geq 0}$ , consisting of strictly increasing positive real numbers with $\lim\limits_{k\to\infty}\lambda_{k}=\infty$ , if $F\in\{\mathbb{\mathbb{R}},\mathbb{C}\}$ , or a finite set of complex numbers $\lambda_{k}$ , with $\operatorname{Re}\lambda_{k}>0$ , and $\operatorname{Im}\lambda_{k}\in\frac{2\pi}{N\ln(q_{F})}\mathbb{N}$ for some $N\in\mathbb{N}$ , if $F$ is non-Archimedean;
•

a sequence $\{m_{k}\}_{k\geq 0}$ of positive integers;
•

smooth functions $a_{k,m,\mu},b_{k,m,\mu}$ on $F_{1}^{\times}$ ,

such that:

(1)

For $F\in\{\mathbb{\mathbb{R}},\mathbb{C}\}$ , $g(y)$ admits an asymptotic expansion¹¹1The asymptotic expansion is also termwise differentiable, and uniform in the angular component; we refer to [Igu78, p. 19-24] for the precise meaning of those notions. of the form

(3.3)

g(y)\approx\sum_{k\geq 0}\sum\limits_{m=1}^{m_{k}}a_{k,m,\mu}(\mathbb{\mathrm{ac}}(y))\cdot\left|y\right|_{F}^{\lambda_{k}-1}(\log\left|y\right|_{F})^{m-1},\left|y\right|_{F}\rightarrow 0,

and $\mathcal{F}(f_{*}\mu)$ admits an asymptotic expansion of the form,

(3.4)

\mathcal{F}(f_{*}\mu)(t)\approx\sum_{k\geq 0}\sum\limits_{m=1}^{m_{k}}b_{k,m,\mu}(\mathbb{\mathrm{ac}}(t))\cdot\left|t\right|_{F}^{-\lambda_{k}}(\log\left|t\right|_{F})^{m-1},\left|t\right|_{F}\rightarrow\infty.

(2)

For $F$ non-Archimedean, $g(y)$ and $\mathcal{F}(f_{*}\mu)$ admits an expansion as in (3.3) and (3.4), where (( $\approx$ )) is replaced with (( $=$ )), and both sums are finite.
(3)

For each $\lambda_{k}$ , the functions $(b_{k,m,\mu})_{m=1}^{m_{k}}$ are determined by the functions $(a_{k,m,\mu})_{m=1}^{m_{k}}$ . If $\operatorname{Re}\lambda_{k}<1$ , the map taking the latter to the former is one-to-one, and, moreover, the leading function $b_{k,m_{k},\mu}$ is not identically zero (provided that $m_{k}$ is defined so that $a_{k,m_{k},\mu}$ is not identically $0$ ).

Remark 3.7.

(1)

The maps $(a_{k,m,\mu})_{m=1}^{m_{k}}\mapsto(b_{k,m,\mu})_{m=1}^{m_{k}}$ of item (3) can be explicitly described; see [VZnG17, Proposition 4.6] and [Igu78, Section 2.2].
(2)

Note that Igusa’s theory (and in particular Theorem 3.6) was originally developed for polynomial maps but works for analytic maps as well. Indeed, the proof uses resolution of singularities to reduce to the case of pushforward of measures with monomial density by monomial maps. The same reduction can be made for analytic maps via an analytic version of resolution of singularities (as stated in Theorem 2.1). For a generalization of Theorem 3.6 to the case of meromorphic maps, see [VZnG17, Section 5].

We record the following immediate corollary of Equation (3.3) in Theorem 3.6:

Corollary 3.8.

Let $f:X\to F$ be a locally dominant analytic map, let $\mu\in\mathcal{M}_{c}^{\infty}(X)$ , and suppose that $\epsilon_{\star}(f_{*}\mu)<\infty$ . Then $f_{*}\mu\notin\mathcal{M}_{c,1+\epsilon_{\star}(f_{*}\mu)}(F)$ , i.e. the supremum in the definition of $\epsilon_{\star}$ is not achieved.

Theorem 3.6 also implies the following corollary, relating $\epsilon_{\star}$ to $\delta_{\star}$ :

Corollary 3.9.

Let $\phi:X\rightarrow F$ be a locally dominant analytic map such that $0$ is the only critical value. Then for each $\mu\in\mathcal{M}_{c}^{\infty}(X)$

(3.5)

\epsilon_{\star}(\phi_{*}\mu)=\begin{cases}\infty&\text{if }\delta_{\star}(\phi_{*}\mu)\geq 1,\\ \frac{\delta_{\star}(\phi_{*}\mu)}{1-\delta_{\star}(\phi_{*}\mu)}&\text{if }\delta_{\star}(\phi_{*}\mu)<1.\end{cases}

Proof.

Write $\phi_{*}\mu=g(y)d\mu_{F}$ . If $F=\mathbb{\mathbb{R}}$ , then by Theorem 3.6, $g(y)$ can be expanded near $y=0$ , so that the leading term is $\left|y\right|_{F}^{\lambda_{k_{0}}-1}(\log\left|y\right|_{F})^{m_{k_{0}}-1}$ for some $\lambda_{k_{0}}\in\mathbb{\mathbb{R}}_{>0}$ and $m_{k_{0}}\in\mathbb{N}$ . Similarly, $\mathcal{F}(\phi_{*}\mu)$ can be expanded near $t=\infty$ . If $\lambda_{k_{0}}<1$ , then by Item (3) of Theorem 3.6, $\left|y\right|_{F}^{-\lambda_{k_{0}}}(\log\left|y\right|_{F})^{m_{k_{0}}-1}$ is the leading term of $\mathcal{F}(\phi_{*}\mu)$ , and thus $\delta_{\star}(\phi_{*}\mu)=\lambda_{k_{0}}$ . If $\lambda_{k_{0}}=1$ , then Item (3) implies that $\delta_{\star}(\phi_{*}\mu)\geq 1$ .

Since $0$ is the only critical value of $\phi$ , $g(y)$ is bounded outside any neighborhood of $0$ , so $g(y)^{1+\epsilon}$ is integrable if and only if it is integrable in a small ball $B$ around $0$ , and this holds if and only if $(\lambda_{k_{0}}-1)(1+\epsilon)>-1$ , i.e. either $\lambda_{k_{0}}\geq 1$ and then $\epsilon_{\star}(\phi_{*}\mu)=\infty$ , or $\epsilon<\frac{\lambda_{k_{0}}}{1-\lambda_{k_{0}}}=\frac{\delta_{\star}(\phi_{*}\mu)}{1-\delta_{\star}(\phi_{*}\mu)}$ .

The case when $F$ is non-Archimedean should be done with care, since there might be multiple terms in (3.3) with the same real part, and some cancellations may occur (see Example 3.10 below). Let $\lambda:=\underset{k}{\min}\operatorname{Re}\lambda_{k}$ and suppose $\lambda<1$ . Then by Theorem 3.6, $g(y)$ can be written as $g(y)=g_{\mathrm{lead}}(y)+g_{\mathrm{error}}(y)$ , where

g_{\mathrm{lead}}(y):=\sum_{m}\sum_{j=0}^{N-1}\widetilde{a}_{j,m,\mu}(\mathbb{\mathrm{ac}}(y))\cdot\left|y\right|_{F}^{\lambda+\frac{2\pi ij}{N\ln(q_{F})}-1}(\log\left|y\right|_{F})^{m-1},

$\widetilde{a}_{j,m,\mu}(\mathbb{\mathrm{ac}}(y))$ is equal to $a_{k_{j},m,\mu}(\mathbb{\mathrm{ac}}(y))$ if $\lambda_{k_{j}}=\lambda+\frac{2\pi ij}{N\ln(q_{F})}\in\Lambda$ and $0$ otherwise, and furthermore,

g_{\mathrm{error}}(y):=\sum_{k:\operatorname{Re}\lambda_{k}>\lambda}\sum\limits_{m=1}^{m_{k}}a_{k,m,\mu}(\mathbb{\mathrm{ac}}(y))\cdot\left|y\right|_{F}^{\lambda_{k}-1}(\log\left|y\right|_{F})^{m-1}.

We first show there is an arithmetic progression $I_{a,N,l}:=\{a+bN\}_{l<b\in\mathbb{N}}$ for some $a,l\in\mathbb{N}$ , and $y_{0}\in F_{1}^{\times}$ , such that

(3.6)

\left|g_{\mathrm{lead}}(y)\right|>C_{F}\left|y\right|_{F}^{\lambda-1},

for all $y\in S_{a,N,l,d,y_{0}}:=\left\{z\in\mathcal{O}_{F}:\mathbb{\mathrm{val}}(\mathbb{\mathrm{ac}}(z)-y_{0})\geq d,\,\mathbb{\mathrm{val}}(z)\in I_{a,N,l}\right\}$ , for some constant $C_{F}$ independent of $y$ . It is enough to show (3.6) for the terms in $g_{\mathrm{lead}}(y)$ where $m=m_{0}$ is maximal such that for some $0\leq j\leq N-1$ , $\widetilde{a}_{j,m,\mu}(\mathbb{\mathrm{ac}}(y))$ is not identically zero. Note that

\left|y\right|_{F}^{\frac{2\pi ij}{N\ln(q_{F})}}=q_{F}^{-\mathbb{\mathrm{val}}(y)\frac{2\pi ij}{N\ln(q_{F})}}=e^{-\ln(q_{F})\mathbb{\mathrm{val}}(y)\frac{2\pi ij}{N\ln(q_{F})}}=e^{-\mathbb{\mathrm{val}}(y)\frac{2\pi ij}{N}}.

Let $y_{0}\in F_{1}^{\times}$ be such that $\widetilde{a}_{j,m_{0},\mu}(y_{0})\neq 0$ for some $0\leq j\leq N-1$ . Choose $d\in\mathbb{N}$ such that each of $\widetilde{a}_{0,m_{0},\mu}(z),\dots,\widetilde{a}_{0,m_{0},\mu}(z)$ is constant on the ball $\mathbb{\mathrm{val}}(z-y_{0})\geq d$ . Note that the functions $f_{0},...,f_{N-1}:\mathbb{Z}/N\mathbb{Z}\rightarrow\mathbb{C}$ , $f_{j}(t)=e^{-t\frac{2\pi ij}{N}}$ are the irreducible characters of $\mathbb{Z}/N\mathbb{Z}$ and hence they are linearly independent. In particular, there exists $a\in\mathbb{N}$ , such that

\left|\sum_{j=0}^{N-1}\widetilde{a}_{j,m_{0},\mu}(y_{0})e^{-t\frac{2\pi ij}{N}}\right|=C_{F}>0,

for all $t\in I_{a,N,0}$ . Taking $y\in S_{a,N,l,d,y_{0}}$ for $l\gg 1$ , we deduce (3.6) as required.

Finally, by Item (3) of Theorem 3.6 and by an argument similar to the one above, it follows that $\delta_{\star}(\phi_{*}\mu)=\lambda$ . Hence, (3.6) implies that $\int_{S_{a,N,l,d,y_{0}}}g(y)^{1+\epsilon}d\mu_{F}$ diverges for every $\epsilon\geq\frac{\lambda}{1-\lambda}=\frac{\delta_{\star}(\phi_{*}\mu)}{1-\delta_{\star}(\phi_{*}\mu)}$ . On the other hand, a similar argument as in the case $F=\mathbb{\mathbb{R}}$ shows that $\int_{F}g(y)^{1+\epsilon}d\mu_{F}$ converges for $\epsilon<\frac{\lambda}{1-\lambda}$ , so the corollary follows. ∎

In the proof of Corollary 3.9 we have seen there might be some cancellations between the terms in (3.3) with the same real part, and that these cancellations are insignificant for infinitely many values of $\mathbb{\mathrm{val}}(y)$ . Here is a simple example which illustrates this phenomenon.

Example 3.10.

Let $d\in\mathbb{N}$ and let $p>d$ be a prime. Let $\phi:\mathbb{Q}_{p}\rightarrow\mathbb{Q}_{p}$ be the map $\phi(x)=x^{d}$ . Write $\phi_{*}\mu_{\mathbb{\mathbb{Z}}_{p}}=g(y)\mu_{\mathbb{\mathbb{Z}}_{p}}$ . Then for almost all $y\in\mathbb{\mathbb{Z}}_{p}$ we have

g(y)=\#\phi^{-1}(y)\cdot\left|y\right|_{p}^{-1+\frac{1}{d}}=a(\mathbb{\mathrm{ac}}(y))1_{\{z\in\mathbb{\mathbb{Z}}_{p}:d|\mathbb{\mathrm{val}}(z)\}}(y)\cdot\left|y\right|_{p}^{-1+\frac{1}{d}}.

where $a(z):=\#\{x\in\mathbb{\mathbb{Z}}_{p}:x^{d}=z\}$ . Note that by Schur orthogonality, $\sum_{j=0}^{d-1}e^{-t\frac{2\pi ij}{d}}=d$ if $d|t$ and is $0$ if $d\nmid t\in\mathbb{N}$ . In particular

1_{\{z\in\mathbb{\mathbb{Z}}_{p}:d|\mathbb{\mathrm{val}}(z)\}}(y)=\frac{1}{d}\sum_{j=0}^{d-1}e^{-\mathbb{\mathrm{val}}(y)\frac{2\pi ij}{d}}=\frac{1}{d}\sum_{j=0}^{d-1}\left|y\right|_{p}^{\frac{2\pi ij}{d\ln(q_{F})}},

and therefore, the expansion of $g(y)$ as in (3.3) is

g(y)=\sum_{j=0}^{d-1}\frac{a(\mathbb{\mathrm{ac}}(y))}{d}\left|y\right|_{p}^{-1+\frac{1}{d}+\frac{2\pi ij}{d\ln(q_{F})}}.

3.2.2. Relations between the invariants

We now show that $\epsilon_{\star}(\phi;x),\delta_{\star}(\phi;x)$ and $k_{\star}(\phi;x)$ are essentially determined by the log-canonical threshold $\operatorname{lct}_{F}(\phi_{x};x)$ whenever $\operatorname{lct}_{F}(\phi;x)\leq 1$ . In particular, we show that

\epsilon_{\star}(\phi;x),\delta_{\star}(\phi;x),\frac{1}{k_{\star}(\phi;x)}\text{ and }\operatorname{lct}_{F}(\phi_{x};x)

are asymptotically equivalent as $\operatorname{lct}_{F}(\phi_{x};x)\rightarrow 0$ .

Theorem 1.2 already shows that $\epsilon_{\star}(\phi;x)=\frac{\operatorname{lct}_{F}(\phi_{x};x)}{1-\operatorname{lct}_{F}(\phi_{x};x)}$ if $\operatorname{lct}_{F}(\phi_{x};x)<1$ . We further have the following:

Proposition 3.11.

Let $\phi:X\rightarrow F$ be a dominant $F$ -analytic map. Then:

(1)

If $\operatorname{lct}_{F}(\phi_{x};x)<1$ , then $\delta_{\star}(\phi;x)=\operatorname{lct}_{F}(\phi_{x};x)$ . In particular,

\operatorname{lct}_{F}(\phi_{x}*\phi_{y};(x,y))\,\begin{cases}=\operatorname{lct}_{F}(\phi_{x};x)+\operatorname{lct}_{F}(\phi_{y};y)~{},&\operatorname{lct}_{F}(\phi_{x};x)+\operatorname{lct}_{F}(\phi_{y};y)<1,\\ \geq 1~{},&\text{otherwise}.\end{cases}

(2)

We have:

\left\lceil\frac{1}{\operatorname{lct}_{F}(\phi_{x};x)}\right\rceil\leq k_{\star}(\phi;x)\leq\left\lfloor\frac{1}{\operatorname{lct}_{F}(\phi_{x};x)}\right\rfloor+1.

Proof.

Let us first prove Item (1). Let $x\in X$ . Replacing $\phi$ with $\phi_{x}$ , we may assume that $\phi(x)=0$ . We may choose an open neighborhood $U$ such that $0$ is the only critical value of $\phi|_{U}$ , $\overline{U}$ is compact and such that for each $\mu\in\mathcal{M}_{c}^{\infty}(U)$ , and each $s<\operatorname{lct}_{F}(\phi;x)$ , one has $\int\left|\phi(x)\right|_{F}^{-s}d\mu(s)<\infty$ . Taking any $\mu$ which does not vanish at $x$ , we get by Theorem 1.2 and Lemma 3.2 that

\epsilon_{\star}(\phi_{*}\mu)=\frac{\operatorname{lct}_{F}(\phi_{x};x)}{1-\operatorname{lct}_{F}(\phi_{x};x)}=\epsilon_{\star}(\phi;x)<\infty.

Corollary 3.9 implies that

\delta_{\star}(\phi_{*}\mu)=\frac{\epsilon_{\star}(\phi_{*}\mu)}{1+\epsilon_{\star}(\phi_{*}\mu)}=\frac{\epsilon_{\star}(\phi;x)}{1+\epsilon_{\star}(\phi;x)}=\operatorname{lct}_{F}(\phi_{x};x).

Since the equalities above hold for $\mu$ of arbitrarily small support around $x$ , we get $\delta_{\star}(\phi;x)=\delta_{\star}(\phi_{*}\mu)=\operatorname{lct}_{F}(\phi_{x};x)$ as required. Since Fourier transform translates convolution into product, we have

\delta_{\star}(\phi*\phi;(x,y))=\delta_{\star}(\phi;x)+\delta_{\star}(\phi;y),

which implies the second part of Item (1) (see also [Den91, Section 5.1]).

We now turn to Item (2). Set $k_{0}:=\left\lceil\frac{1}{\operatorname{lct}_{F}(\phi_{x};x)}\right\rceil$ . For a positive integer $k<k_{0}$ we get by Item (1), that $\operatorname{lct}_{F}(\phi_{x}^{*k};(x,...,x))<1$ , so that $\epsilon_{\star}(\phi_{x}^{*k};(x,...,x))<\infty$ . This implies the lower bound $k_{\star}(\phi;x)\geq k_{0}$ . The upper bound follows from (1.3) and Theorem 1.2. ∎

Remark 3.12.

Item (1) of Proposition 3.11 can be seen as an $F$ -analytic interpretation of a theorem by Thom–Sebastiani [ST71] (see e.g. [MSS18] for the case $F=\mathbb{C}$ ).

The combination of Corollaries 3.8 and 3.9 implies Theorem 1.3, as follows.

Proof of Theorem 1.3.

Let $\mu_{j}\in\mathcal{M}_{c}^{\infty}(F^{n_{j}})$ , and suppose that

(3.7)

\nu_{1}*\nu_{2}\in\mathcal{M}_{1+\epsilon}(F),\quad\epsilon>0,

where $\nu_{j}=(\phi_{j})_{*}\mu_{j}$ . Assume by contradiction that

\frac{\epsilon_{\star}(\nu_{1})}{1+\epsilon_{\star}(\nu_{1})}+\frac{\epsilon_{\star}(\nu_{2})}{1+\epsilon_{\star}(\nu_{2})}\leq\frac{\epsilon}{1+\epsilon},

For each $j\in\{1,2\}$ , choose a finite cover $\bigcup_{i}U_{i,j}$ of $\operatorname{Supp}(\mu_{j})$ by open balls in $F^{n_{j}}$ , such that for each $i$ , $\phi_{j}|_{U_{i,j}}$ has at most one critical value $z_{i,j}$ of $\phi_{j}$ . We can write $\mu_{j}=\sum_{i}\mu_{i,j}$ with $\mu_{i,j}$ supported inside $U_{i,j}$ . Taking $\phi_{i,j}=\phi_{j}|_{U_{i,j}}-z_{i,j}$ , we can find $i_{1}$ and $i_{2}$ such that

\epsilon_{\star}(\nu_{j})=\epsilon_{\star}\left((\phi_{i_{j},j})_{*}(\mu_{i_{j},j})\right),\quad j=1,2.

Now, Corollary 3.9 implies that

\delta_{\star}\left((\phi_{i_{j},j})_{*}(\mu_{i_{j},j})\right)=\frac{\epsilon_{\star}\left((\phi_{i_{j},j})_{*}(\mu_{i_{j},j})\right)}{1+\epsilon_{\star}\left((\phi_{i_{j},j})_{*}(\mu_{i_{j},j})\right)},

whence

\delta_{\star}\left((\phi_{i_{1},1})_{*}(\mu_{i_{1},1})*(\phi_{i_{2},2})_{*}(\mu_{i_{2},2})\right)\leq\frac{\epsilon}{1+\epsilon}.

Using Corollary 3.9 once again, we obtain that

\epsilon_{\star}(\nu_{1}*\nu_{2})\leq\epsilon_{\star}\left((\phi_{i_{1},1})_{*}(\mu_{i_{1},1})*(\phi_{i_{2},2})_{*}(\mu_{i_{2},2})\right)\leq\epsilon.

In view of Corollary 3.8, this contradicts (3.7). ∎

3.3. Consistency of the various bounds in the one-dimensional case

The following proposition, which can be seen as a variant of the Łojasiewicz gradient inequality (see e.g. [Lo65, p.92], [BM88, Proposition 6.8]), shows that the formula for $\epsilon_{\star}(\phi;x)$ given in Theorem 1.2 in the one-dimensional case is consistent with the lower and upper bounds in Theorems 1.1 and 1.5. The proof of Proposition 3.13 is similar to [Fee19, Theorem 1], but applied to any local field.

Note that for $\phi:F^{n}\rightarrow F$ , we have $\mathcal{J}_{\phi}=\langle\frac{\partial\phi}{\partial x_{1}},...,\frac{\partial\phi}{\partial x_{n}}\rangle$ , which we denote by $\langle\nabla\phi\rangle$ .

Proposition 3.13.

Let $\phi:F^{n}\rightarrow F$ be an analytic map. Then for every $x\in F^{n}$ , we have

\frac{\operatorname{lct}_{F}(\langle\nabla\phi\rangle;x)}{1-\operatorname{lct}_{F}(\langle\nabla\phi\rangle;x)}\geq\frac{\operatorname{lct}_{F}(\phi_{x};x)}{1-\operatorname{lct}_{F}(\phi_{x};x)}\geq\operatorname{lct}_{F}(\langle\nabla\phi\rangle;x)\geq\operatorname{lct}_{F}(\phi_{x};x),

where the middle (resp. left) inequality holds whenever $\operatorname{lct}_{F}(\phi_{x};x)<1$ (resp. $\operatorname{lct}_{F}(\langle\nabla\phi\rangle;x)<1$ ).

Proof.

The middle inequality follows from Theorems 1.1 and 1.2, and the left inequality follows from the right inequality, so it is left to show that $\operatorname{lct}_{F}(\langle\nabla\phi\rangle;x_{0})\geq\operatorname{lct}_{F}(\phi_{x_{0}};x_{0})$ for a fixed $x_{0}\in F^{n}$ .

Recall that $\left\|(a_{1},...,a_{n})\right\|_{F}:=\max_{i}\left|a_{i}\right|_{F}$ is the maximum norm on $F^{n}$ . We would like to relate between $\int_{B}\left\|\nabla\phi\right\|_{F}^{-s}dx$ and $\int_{B}\left|\phi_{x_{0}}\right|_{F}^{-s}dx$ , where $B$ is a small ball around $x_{0}$ , for $s>0$ small enough. Let $\pi:\widetilde{X}\rightarrow F^{n}$ be a resolution of singularities of $\phi_{x_{0}}$ . Then we have

(3.8)

\int_{B}\left\|\nabla\phi\right\|_{F}^{-s}dx=\int_{\pi^{-1}(B)}\left\|\nabla\phi|_{\pi(\widetilde{x})}\right\|_{F}^{-s}\left|\operatorname{Jac}_{\widetilde{x}}(\pi)\right|_{F}d\widetilde{x}.

Since $\pi^{-1}(B)$ is compact, by working locally over finitely many pieces, and using Theorem 2.1, we may replace $\pi^{-1}(B)$ by a compact neighborhood $L\subseteq F^{n}$ of $0$ , $\pi(0)=x_{0}$ , and further assume that for $\widetilde{x}=(\widetilde{x}_{1},...,\widetilde{x}_{n})$ :

(3.9)

\widetilde{\phi}(\widetilde{x}):=\phi\circ\pi(\widetilde{x})=C\widetilde{x}_{1}^{a_{1}}\cdots\widetilde{x}_{n}^{a_{n}}\text{ and }\operatorname{Jac}_{\widetilde{x}}(\pi)=v(\widetilde{x})\widetilde{x}_{1}^{b_{1}}\cdots\widetilde{x}_{n}^{b_{n}},

for some analytic unit $v(\widetilde{x})$ and a constant $C\in F$ . Note that on $L$ all of the entries of $d_{\widetilde{x}}\pi$ are smaller, in absolute value, than some constant $\widetilde{C}$ , so that the operator norm $\left\|d_{\widetilde{x}}\pi\right\|_{\mathrm{op}}:=\underset{v\in F^{n}:\left\|v\right\|_{F}=1}{\sup}\left\|d_{\widetilde{x}}\pi\cdot v\right\|_{F}$ of $d_{\widetilde{x}}\pi$ is bounded by a constant. Since $\nabla\widetilde{\phi}|_{\widetilde{x}}=\nabla\phi|_{\pi(\widetilde{x})}\cdot d_{\widetilde{x}}\pi$ , for $s>0$ we get:

	$\displaystyle\text{$\int_{L}\left\\|\nabla\phi\|_{\pi(\widetilde{x})}\right\\|_{F}^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}$}d\widetilde{x}$	$\displaystyle\leq\int_{L}\left\\|d_{\widetilde{x}}\pi\right\\|_{\mathrm{op}}^{s}\left\\|\nabla\widetilde{\phi}\|_{\widetilde{x}}\right\\|_{F}^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}d\widetilde{x}\lesssim\int_{L}\left\\|\nabla\widetilde{\phi}\|_{\widetilde{x}}\right\\|_{F}^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}d\widetilde{x}$
		$\displaystyle\lesssim\int_{L}\left(\max_{j}\left\|\widetilde{x}_{1}\right\|_{F}^{a_{1}}\cdot...\cdot\left\|\widetilde{x}_{j}\right\|_{F}^{a_{j}-1}\cdot...\cdot\left\|\widetilde{x}_{n}\right\|_{F}^{a_{n}}\right)^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}d\widetilde{x}$
		$\displaystyle\lesssim\int_{L}\left(\left\|\widetilde{x}_{1}\right\|_{F}^{a_{1}}...\left\|\widetilde{x}_{n}\right\|_{F}^{a_{n}}\right)^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}d\widetilde{x}\lesssim\int_{\pi(L)}\left\|\phi_{x_{0}}\right\|_{F}^{-s}dx,$

which concludes the proof. ∎

Remark 3.14.

Here is an alternative approach to Proposition 3.13, as suggested by the referee. For simplicity suppose $F=\mathbb{C}$ . We may assume that $x=0\in\mathbb{C}^{n}$ . Recall from [HS06, Definition 1.1.1] that the integral closure $\overline{I}$ of an ideal $I$ in a commutative ring $R$ , is the set of elements $r\in R$ for which there are $a_{j}\in I^{j}$ such that $\sum_{j=0}^{n}a_{j}r^{n-j}=0$ for some $n\in\mathbb{N}$ . We now take $R=\mathbb{C}\{x_{1},...,x_{n}\}$ to be the ring of convergent power series, $I:=\langle\nabla\phi\rangle$ and $J:=\langle\phi_{0}\rangle$ . By [HS06, Corollary 7.1.4], we have $J\subseteq\overline{I}$ and in particular $\operatorname{lct}_{\mathbb{C}}(J;0)\leq\operatorname{lct}_{\mathbb{C}}(\overline{I};0)$ (see [Mus12, Property 1.15]). However, by [Mus12, Property 1.12] (see also [dFM09, Proposition 2.4]) we have $\operatorname{lct}_{\mathbb{C}}(I;0)=\operatorname{lct}_{\mathbb{C}}(\overline{I};0)$ , which implies that $\operatorname{lct}_{\mathbb{C}}(J;0)\leq\operatorname{lct}_{\mathbb{C}}(I;0)$ as required. This argument likely generalizes to any local field $F$ of characteristic $0$ , i.e. the full generality of Proposition 3.13.

3.4. Some examples in higher dimension

We next discuss the higher dimensional case (i.e. $\dim Y>1$ ). Here we will see that the connection between the four invariants $\epsilon_{\star}(\phi;x)$ , $\delta_{\star}(\phi;x)$ , $k_{\star}(\phi;x)$ and $\operatorname{lct}_{F}(\phi_{x};x)$ is not as tight as in the one-dimensional case. We first provide a simpler description of $\delta_{\star}(\phi;x)$ .

Lemma 3.15.

Let $F$ be a local field of characteristic $0$ and $\phi:F^{n}\rightarrow F^{m}$ be an analytic map. Then

\delta_{\star}(\phi;x)=\inf_{\ell}\{\delta_{\star}(\ell\circ\phi;x)\},

where $\ell$ runs over all non-zero linear functionals $\ell:F^{m}\rightarrow F$ .

Proof.

For each $\mu\in\mathcal{M}_{c}^{\infty}(F^{n})$ , we have

\left|\mathcal{F}(\phi_{*}\mu)(z)\right|\lesssim\left\|z\right\|_{F}^{-\delta}\Longleftrightarrow\left|\mathcal{F}(\phi_{*}\mu)(ta_{1},...,ta_{m})\right|\lesssim\left|t\right|_{F}^{-\delta},

for each $a=(a_{1},...,a_{m})$ with $\left\|a\right\|_{F}=1$ and $t\in F$ . Setting $\ell_{a}(y_{1},...,y_{m})=\sum_{i=1}^{m}a_{i}y_{i}$ , we have,

\mathcal{F}(\phi_{*}\mu)(ta_{1},...,ta_{m})=\int_{F^{m}}\Psi(t\ell_{a}(y))\cdot d\phi_{*}\mu=\int_{F}\Psi(tz)\cdot d(\ell_{a}\circ\phi)_{*}\mu=\mathcal{F}\left((\ell_{a}\circ\phi)_{*}\mu\right)(t),

which concludes the proof of the lemma. ∎

The following examples demonstrate that the connection between the four invariants can get loose as the dimension of $Y$ grows.

Example 3.16.

Consider the map $\phi:\mathbb{C}^{m}\rightarrow\mathbb{C}^{m}$ , defined by $\phi(x_{1},...,x_{m})=(x_{1}^{d},x_{1}^{d}x_{2},...,x_{1}^{d}x_{m})$ . Then:

(1)

$\epsilon_{\star}(\phi)=\frac{1}{dm-1}$ .
(2)

$\operatorname{lct}_{\mathbb{C}}(J)=\frac{1}{d}$ , where $J=\langle x_{1}^{d},x_{1}^{d}x_{2},...,x_{1}^{d}x_{m}\rangle=\langle x_{1}^{d}\rangle$ .
(3)

$dm\leq k_{\star}(\phi)\leq dm+1$ .
(4)

$\delta_{\star}(\phi)=\frac{1}{d}$ .

Item (1) follows from Theorem 1.1. Young’s inequality (see 1.3) shows that $k_{\star}(\phi)\leq dm+1$ , on the other hand we have $\operatorname{lct}_{\mathbb{C}}(J)=\frac{1}{d}$ which implies that at least $dm$ convolutions are needed to obtain the (FRS) property (see [GH24, Lemmas 3.23 and 3.26]), hence $k_{\star}(\phi)\geq dm$ . Note that for any $a_{1},..,a_{m}\in\mathbb{C}$ with $\left\|a\right\|_{\mathbb{C}}=1$ we have $\operatorname{lct}_{\mathbb{C}}(a_{1}x_{1}^{d}+a_{2}x_{1}^{d}x_{2}+...+a_{m}x_{1}^{d}x_{m})=\frac{1}{d}$ so $\delta_{\star}(\phi)=\frac{1}{d}$ , by Lemma 3.15.

Remark 3.17.

One can replace $\phi$ in Example 3.16, with

\phi(x_{1},...,x_{m})=(x_{1},x_{1}^{d}x_{2},...,x_{1}^{d}x_{m}).

Here, we get $\epsilon_{\star}(\phi)=\frac{1}{dm-d}$ , which is similar to Example 3.16 when $m$ is large, while $\operatorname{lct}_{\mathbb{C}}(J)=1$ , where $J=\langle x_{1},x_{1}^{d}x_{2},...,x_{1}^{d}x_{m}\rangle=\langle x_{1}\rangle$ , which is much larger than in Example 3.16.

The following example shows that a reverse Young inequality (Theorem 1.3) does not hold in dimension $m\geq 2$ .

Example 3.18.

Consider $\phi(x,y)=(x,x^{2}(1+y^{1000}))$ . Then $\epsilon_{\star}(\phi)=\frac{1}{999}$ , while $\delta_{\star}(\phi)=\frac{1}{2}$ , and consequently $k_{\star}(\phi)\leq 4$ .

4. Lower bound: proof of Theorem 1.1

In this section we prove Theorem 1.1. We start with the equidimensional case, which we restate as Proposition 4.1 below.

Recall that given a locally dominant analytic map $\phi:X\to Y$ between two $F$ -analytic manifolds, the Jacobian ideal sheaf $\mathcal{J}_{\phi}$ is defined locally as the ideal in the algebra of analytic functions on $X$ generated by the $m\times m$ -minors of the differential $d_{x}(\phi)$ of $\phi$ .

Proposition 4.1.

Let $X,Y$ be $F$ -analytic manifolds, $\dim X=\dim Y=n$ , and let $\phi:X\to Y$ be a locally dominant analytic map. Then for any $x_{0}\in X$ ,

(4.1)

\epsilon_{\star}(\phi;x_{0})=\operatorname{lct}_{F}(\mathcal{J}_{\phi};x_{0}).

Proof.

We may assume that $X,Y$ are compact balls in $F^{n}$ , and $x_{0}=0$ . Since $\dim X=\dim Y=n$ , we have $\mathcal{J}_{\phi}=\langle\operatorname{Jac}_{x}(\phi)\rangle$ , where $\operatorname{Jac}_{x}(\phi)=\det\,(d_{x}\phi)$ . Since $\phi$ is analytic and $X$ is compact, there is an open dense set $U\subseteq X$ and $M\in\mathbb{N}$ such that $\#\left\{\phi^{-1}\phi(x)\right\}\leq M$ and $\operatorname{Jac}_{x}(\phi)\neq 0$ , for every $x\in U$ . We choose a disjoint cover $\bigcup_{i\in\mathbb{N}}U_{i}$ of $U$ by locally closed subsets²²2Recall that a set is locally closed if it is the difference of two open sets. $U_{i}$ such that $\phi|_{U_{i}}$ is a diffeomorphism. Write $\mu:=\mu_{F}^{n}|_{X}$ , $\mu_{i}:=\mu_{F}^{n}|_{U_{i}}$ , and note that $\phi_{*}\mu=g(y)\cdot\mu_{F}^{n}$ and $\phi_{*}\mu_{i}=g_{i}(y)\cdot\mu_{F}^{n}$ , where

g(y)=\sum_{x\in\phi^{-1}(y)}\left|\operatorname{Jac}_{x}(\phi)\right|_{F}^{-1}\text{ and }g_{i}(y)=\left|\operatorname{Jac}_{\phi^{-1}(y)}(\phi)\right|_{F}^{-1}.

We have

(4.2)

\int_{Y}g(y)^{1+s}dy=\int_{\phi(U)}g(y)^{s}d\phi_{*}\mu=\int_{U}g\circ\phi(x)^{s}d\mu\geq\int_{U}\frac{1}{\left|\operatorname{Jac}_{x}(\phi)\right|_{F}^{s}}d\mu.

On the other hand, since $\#\left\{i\in I:y\in\phi(U_{i})\right\}\leq M$ for each $y\in Y$ , using Jensen’s inequality, we have:

	$\displaystyle\int_{Y}g(y)^{1+s}dy$	$\displaystyle\leq\int_{Y}\left(\sum_{i\in I:y\in\phi(U_{i})}g_{i}(y)\right)^{1+s}dy\leq M^{s}\int_{Y}\sum_{i\in I:y\in\phi(U_{i})}g_{i}(y)^{1+s}dy$
(4.3)			$\displaystyle=M^{s}\sum_{i\in I}\int_{Y}g_{i}(y)^{1+s}dy=M^{s}\sum_{i\in I}\int_{U_{i}}\frac{1}{\left\|\operatorname{Jac}_{x}(\phi)\right\|_{F}^{s}}d\mu=M^{s}\int_{U}\frac{1}{\left\|\operatorname{Jac}_{x}(\phi)\right\|_{F}^{s}}d\mu,$

which implies the proposition. ∎

Proof of Theorem 1.1.

Let $\phi:X\to Y$ be a locally dominant analytic map, and let $x_{0}\in X$ . Since the claim is local, we may assume that $X\subseteq F^{n}$ is an open subset, and $Y=F^{m}$ , with $n\geq m$ . Let $B$ be a small ball around $x_{0}$ . For each subset $I\subseteq\{1,...,n\}$ of size $m$ , let $M_{I}$ be the corresponding $m\times m$ -minor of $d_{x}\phi$ , and set

V_{I}:=\left\{x\in B:\max_{I^{\prime}\subseteq\{1,...,n\}}\left|M_{I^{\prime}}(x)\right|_{F}=\left|M_{I}(x)\right|_{F}\right\}.

If $V_{I}\neq\varnothing$ , the map $\phi_{I}(x):=(\phi(x),x_{j_{1}},...,x_{j_{n-m}})$ is locally dominant, where $\operatorname{Jac}_{x}(\phi_{I})=M_{I}(x)$ , and $\{j_{1},...,j_{n-m}\}=\{1,...,n\}\backslash I$ . For each $V_{I}$ of positive measure, let $\mu_{B}:=\mu_{F}^{n}|_{B}$ , $\mu_{I}:=\mu_{F}^{n}|_{B\cap V_{I}}$ and write

\phi_{*}\mu_{B}=g(y)\cdot\mu_{F}^{m},\,\,\,\phi_{*}\mu_{I}=g_{I}(y)\cdot\mu_{F}^{m}\text{ and }\left(\phi_{I}\right)_{*}\mu_{I}=\widetilde{g}_{I}(z)\cdot\mu_{F}^{n}.

Let $\widetilde{B}$ be a large ball in $F^{n-m}$ which contains the projection of $\phi_{I}(B\cap V_{I})$ from $F^{n}$ to the last $n-m$ coordinates $F^{n-m}$ . Since $\phi=\pi_{I}\circ\phi_{I}$ where $\pi_{I}:F^{n}\rightarrow F^{m}$ is a projection to the first $m$ coordinates, we have

g_{I}(y)=\int_{F^{n-m}}\widetilde{g}_{I}(y,z_{m+1},...,z_{n})dz_{m+1}...dz_{n}=\int_{\widetilde{B}}\widetilde{g}_{I}(y,z_{m+1},...,z_{n})dz_{m+1}...dz_{n}.

By Jensen’s inequality, we have

	$\displaystyle\int_{F^{m}}g_{I}(y)^{1+s}dy=$	$\displaystyle\int_{F^{m}}\mu_{F}^{n-m}(\widetilde{B})^{1+s}\left(\frac{1}{\mu_{F}^{n-m}(\widetilde{B})}\int_{\widetilde{B}}\widetilde{g}_{I}(y,z_{m+1},...,z_{n})dz_{m+1}...dz_{n}\right)^{1+s}dy$
	$\displaystyle\leq$	$\displaystyle\mu_{F}^{n-m}(\widetilde{B})^{s}\cdot\int_{F^{m}\times\widetilde{B}}\widetilde{g}_{I}(y,z_{m+1},...,z_{n})^{1+s}dydz_{m+1}...dz_{n}\lesssim\int_{F^{n}}\widetilde{g}_{I}(z)^{1+s}dz,$

for every $s>0$ . Since $\bigcup_{I}V_{I}$ is of full measure in $B$ , and using Proposition 4.1 and (1.5), we have:

	$\displaystyle\int_{F^{m}}g(y)^{1+s}dy$	$\displaystyle\lesssim\sum_{I}\int_{F^{m}}g_{I}(y)^{1+s}dy\lesssim\sum_{I}\int_{F^{n}}\widetilde{g}_{I}(z)^{1+s}dz$
		$\displaystyle\lesssim\sum_{I}\int_{V_{I}}\left\|M_{I}(x)\right\|_{F}^{-s}dx\lesssim\int_{B}\min_{I\subseteq\{1,...,n\}}[\left\|M_{I}(x)\right\|_{F}^{-s}]dx<\infty,$

for every $s<\operatorname{lct}_{F}(\mathcal{J}_{\phi};x_{0})$ , as required. ∎

5. Proof of Theorem 1.5 – an upper bound over $\mathbb{C}$

In this section we prove Theorem 1.5. We use the following easy consequence of the coarea formula (see [Fed69]).

Lemma 5.1.

Let $B\subseteq\mathbb{C}^{n}$ be a compact ball, and let $\phi:B\to\mathbb{C}^{m}$ be a dominant analytic map. Let $\mu_{B}:=\mu_{\mathbb{C}}^{n}|_{B}$ and write $\phi_{*}\mu_{B}=g(y)\cdot\mu_{\mathbb{C}}^{m}$ . Then:

g(y)=\int_{\phi^{-1}(y)}\frac{dH_{2(n-m)}(x)}{\det\left((d_{x}\phi)^{*}(d_{x}\phi)\right)},

where the integral is taken with respect to the $2(n-m)$ -dimensional Hausdorff measure (recall $\mathsection$ 1.4(7)).

Let $\phi:X\rightarrow Y$ be a locally dominant analytic map between complex analytic manifolds, and let $x_{0}\in X$ . Since the claim is local, we may assume that $Y=\mathbb{C}^{m}$ and $X=B$ is a ball in $\mathbb{C}^{n}$ . To show (1.12), it is enough to bound $\epsilon_{\star}(\phi_{*}\mu_{B})$ , where $B$ is of arbitrarily small radius around $x_{0}$ . Denote $G(x):=\det\left((d_{x}\phi)^{*}(d_{x}\phi)\right)$ . By Lemma 5.1, we have:

g(y)=\int_{\phi^{-1}(y)}\frac{dH_{2(m-n)}(x)}{G(x)}

whence

(5.1)

\int_{Y}g(y)^{1+\epsilon}dy=\int_{X}\left[\int_{\phi^{-1}(\phi(x))}\frac{dH_{2(m-n)}(x^{\prime})}{G(x^{\prime})}\right]^{\epsilon}dx.

We apply Theorem 2.1 to the ideal $\mathcal{J}_{\phi}$ . Let $\pi:\widetilde{X}\to B$ be the corresponding resolution. Without loss of generality, we can assume that $\widetilde{X}\subseteq\mathbb{C}^{m}$ is an open subset, so that

\det d_{\widetilde{x}}\pi=v(\widetilde{x})\widetilde{x}_{1}^{b_{1}}\cdots\widetilde{x}_{n}^{b_{n}}~{},\quad v(0)\neq 0,

and each of the $m\times m$ minors $M_{I}$ of $d_{\widetilde{x}}\phi$ satisfies

M_{I}(\pi(\widetilde{x}))=u_{I}(\widetilde{x})\widetilde{x}_{1}^{a_{1}}\cdots\widetilde{x}_{n}^{a_{n}},

where at least one of the functions $u_{I}$ does not vanish at $0$ .

We first perform the change of variables $x=\pi(\widetilde{x})$ in the external integral in (5.1), yielding

\int_{Y}g(y)^{1+\epsilon}dy=\int_{\widetilde{X}}\left|\det d_{\widetilde{x}}\pi\right|_{\mathbb{C}}\left[\int_{\phi^{-1}(\phi(\pi(\widetilde{x})))}\frac{dH_{2(m-n)}(x^{\prime})}{G(x^{\prime})}\right]^{\epsilon}d\widetilde{x},

and then perform the change of variables $x^{\prime}=\pi(t)$ in the internal integral. Since the map $\pi$ is continuously differentiable, and $B$ is compact, the product of the singular values of the restriction of $d_{t}\pi$ to any subspace is bounded from below in absolute value by a number times the absolute value of the determinant of $d_{t}\pi$ . Thus,

(5.2)

\int_{Y}g(y)^{1+\epsilon}dy\gtrsim\int_{\widetilde{X}}\left|\det d_{\widetilde{x}}\pi\right|_{\mathbb{C}}\left[\int_{\widetilde{\phi}^{-1}(\widetilde{\phi}(\widetilde{x}))}\frac{\left|\det d_{t}\pi\right|_{\mathbb{C}}\,dH_{2(m-n)}(t)}{G(\pi(t))}\right]^{\epsilon}d\widetilde{x},

where $\widetilde{\phi}:=\phi\circ\pi$ . By the Cauchy–Binet formula, the right-hand side is equal to

\int_{\widetilde{X}}\left|\det d_{\widetilde{x}}\pi\right|_{\mathbb{C}}\left[\int_{\widetilde{\phi}^{-1}(\widetilde{\phi}(\widetilde{x}))}\frac{\left|v(t)\right|_{\mathbb{C}}\left|t_{1}\right|_{\mathbb{C}}^{b_{1}}\cdot...\cdot\left|t_{n}\right|_{\mathbb{C}}^{b_{n}}\,dH_{2(m-n)}(t)}{\sum_{I}\left|u_{I}(t)\right|_{\mathbb{C}}\left|t_{1}\right|_{\mathbb{C}}^{a_{1}}\cdot...\cdot\left|t_{n}\right|_{\mathbb{C}}^{a_{n}}}\right]^{\epsilon}d\widetilde{x}.

We bound the internal integral from below as follows. For each $\widetilde{x}\in\widetilde{X}$ , set

Q(\widetilde{x}):=\left\{t\in\widetilde{X}\,\,:\,\,\sum_{j=1}^{n}\frac{\left|t_{j}-\widetilde{x}_{j}\right|_{\mathbb{C}}}{\left|\widetilde{x}_{j}\right|_{\mathbb{C}}}\leq\frac{1}{4}\right\}

Then

		$\displaystyle\int_{\widetilde{\phi}^{-1}(\widetilde{\phi}(\widetilde{x}))}\frac{\left\|v(t)\right\|_{\mathbb{C}}\left\|t_{1}\right\|_{\mathbb{C}}^{b_{1}}\cdot...\cdot\left\|t_{n}\right\|_{\mathbb{C}}^{b_{n}}\,dH_{2(m-n)}(t)}{\sum_{I}\left\|u_{I}(t)\right\|_{\mathbb{C}}\left\|t_{1}\right\|_{\mathbb{C}}^{a_{1}}\cdot...\cdot\left\|t_{n}\right\|_{\mathbb{C}}^{a_{n}}}$
	$\displaystyle\gtrsim$	$\displaystyle\int_{\widetilde{\phi}^{-1}(\widetilde{\phi}(\widetilde{x}))\cap Q(\widetilde{x})}\left\|t_{1}\right\|_{\mathbb{C}}^{b_{1}-a_{1}}\cdot...\cdot\left\|t_{n}\right\|_{\mathbb{C}}^{b_{n}-a_{n}}\,dH_{2(m-n)}(t)$
(5.3)		$\displaystyle\gtrsim$	$\displaystyle\left\|\widetilde{x}_{1}\right\|_{\mathbb{C}}^{b_{1}-a_{1}}\cdot...\cdot\left\|\widetilde{x}_{n}\right\|_{\mathbb{C}}^{b_{n}-a_{n}}\,H_{2(n-m)}(\widetilde{\phi}^{-1}(\widetilde{\phi}(\widetilde{x}))\cap Q(\widetilde{x})).$

Claim 5.2.

For any $\widetilde{x}\in\widetilde{X}$ ,

H_{2(n-m)}(\widetilde{\phi}^{-1}(\widetilde{\phi}(\widetilde{x}))\cap Q(\widetilde{x}))\gtrsim\min_{|I|=n-m}\prod_{i\in I}\left|\widetilde{x}_{i}\right|_{\mathbb{C}}.

Proof of Claim 5.2.

Let $T:\widetilde{X}\to\widetilde{X}$ be the affine map given by $T(\widetilde{x}^{\prime})_{j}=\widetilde{x}_{j}(1+\widetilde{x}^{\prime}_{j})$ . Then $Q(\widetilde{x})=TB_{\frac{1}{2}}$ , where $B_{\frac{1}{2}}$ is a ball of radius $\frac{1}{2}$ centered at the origin. By a theorem of Lelong [Lel57] (see also [Thi67, LG86]), for any analytic set $M$ of pure dimension $n-m$ , and any $r>0$ , one has

\frac{H_{2(n-m)}(M\cap B_{r})}{H_{2(n-m)}(M_{0}\cap B_{r})}\geq\lim_{\rho\to+0}\frac{H_{2(n-m)}(M\cap B_{\rho})}{H_{2(n-m)}(M_{0}\cap B_{\rho})},

where $M_{0}$ is a linear subspace of the same dimension $n-m$ . The limit on the right-hand side is the Lelong number of $M$ , which is the algebraic multiplicity of $M$ at $\widetilde{x}$ ; it is strictly positive; thus

H_{2(n-m)}(M\cap B_{r})\gtrsim r^{2(n-m)}.

Applying this estimate to $M=T^{-1}(\widetilde{\phi}^{-1}(\widetilde{\phi}(\widetilde{x})))$ and observing that

H_{2(n-m)}(TA)\geq\min_{|I|=n-m}\prod_{i\in I}\left|\widetilde{x}_{i}\right|_{\mathbb{C}}\cdot H_{2(n-m)}(A),

for any Borel set $A\subset\widetilde{X}$ , we obtain the claimed assertion. ∎

We now proceed with the proof of the theorem. Using Claim 5.2, we deduce

\int_{\widetilde{\phi}^{-1}(\widetilde{\phi}(\widetilde{x}))}\frac{\left|\det d_{t}\pi\right|_{\mathbb{C}}\,dH_{2(m-n)}(t)}{G(\pi(t))}\gtrsim\prod_{i=1}^{n}\left|\widetilde{x}_{i}\right|_{\mathbb{C}}^{b_{i}-a_{i}+1},

whence,

\int_{Y}g(y)^{1+\epsilon}dy\gtrsim\int_{\widetilde{X}}\prod_{i=1}^{n}\left|\widetilde{x}_{i}\right|_{\mathbb{C}}^{b_{i}+\epsilon(b_{i}-a_{i}+1)}d\widetilde{x}.

This integral diverges whenever there is an index $i$ such that

b_{i}+\epsilon(b_{i}-a_{i}+1)\leq-1,

i.e.

\epsilon_{\star}(\phi;x)\leq\min_{i}\left\{\frac{b_{i}+1}{a_{i}-b_{i}-1}\,\,:\,\,b_{i}+1<a_{i}\right\}.

On the other hand,

\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};x_{0})=\min_{i}\left\{\frac{b_{i}+1}{a_{i}}\right\}

and under the assumption that this quantity is strictly less than $1$ we have an index $i$ such that

\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};x_{0})=\frac{b_{i}+1}{a_{i}}<1.

For this index,

\frac{b_{i}+1}{a_{i}-b_{i}-1}=\frac{\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};x_{0})}{1-\operatorname{lct}_{\mathbb{C}}(\mathcal{J}_{\phi};x_{0})}.

This concludes the proof of Theorem 1.5.

6. Geometric characterization of $\epsilon_{\star}=\infty$

From now on let $K$ be a number field and let $\mathcal{O}_{K}$ be its ring of integers. Let $\operatorname{Loc}_{0}$ be the collection of all non-Archimedean local fields $F$ which contain $K$ . We use the notation $\operatorname{Loc}_{0,\gg}$ , for the collection of $F\in\operatorname{Loc}_{0}$ with large enough residual characteristic, depending on some given data. Throughout this section, we denote by $B(y,r)$ a ball of radius $r$ centered at $y\in Y(F)$ .

In this section we focus on algebraic morphisms $\varphi:X\rightarrow Y$ between algebraic $K$ -varieties. We would like to characterize morphisms $\varphi$ where $\epsilon_{\star}(\varphi_{F})=\infty$ for certain collections of local fields $F$ , in terms of the singularities of $\varphi$ . In order to effectively do this, it is necessary to consider an “algebraically closed” collection of local fields, such as the following:

•

$\{\mathbb{C}\}$ .
•

$\{F\}_{F\in\operatorname{Loc}_{0,\gg}}$ , or $\{F\}_{F\in\operatorname{Loc}_{0}}$ .

Aizenbud and Avni have shown the following characterization of the (FRS) property:

Theorem 6.1 ([AA16, Theorem 3.4]).

Let $\varphi:X\rightarrow Y$ be a map between smooth $K$ -varieties. Then $\varphi$ is (FRS) if and only if for each $F\in\operatorname{Loc}_{0}$ and every $\mu\in\mathcal{M}_{c}^{\infty}(X(F))$ , one has $\left(\varphi_{F}\right)_{*}(\mu)\in\mathcal{M}_{c,\infty}(Y(F))$ .

The Archimedean counterpart of this theorem was studied in [Rei], where it was shown that given an (FRS) morphism $\varphi$ , then $\varphi_{\mathbb{\mathbb{R}}}$ and $\varphi_{\mathbb{C}}$ are $L^{\infty}$ -morphisms. For the other direction, the non-Archimedean proof of [AA16, Theorem 3.4] (see [AA16, Section 3.7]), can be easily adapted to the complex case, with less complications due to the fact that $\mathbb{C}$ is algebraically closed. We arrive at the following characterization of $L^{\infty}$ -morphisms.

Corollary 6.2.

Let $\varphi:X\rightarrow Y$ be a map between smooth $K$ -varieties. Then the following are equivalent:

(1)

$\varphi$ is (FRS).
(2)

For every local field $F$ containing $K$ , the map $\varphi_{F}$ is an $L^{\infty}$ -morphism.
(3)

For each $F\in\operatorname{Loc}_{0,\gg}$ , the map $\varphi_{F}$ is an $L^{\infty}$ -map.
(4)

The map $\varphi_{\mathbb{C}}$ is an $L^{\infty}$ -map.

Our goal is to characterize the weaker property that $\epsilon_{\star}(\varphi_{F})=\infty$ over $F=\mathbb{C}$ , or over all $F\in\operatorname{Loc}_{0}$ (Theorem 1.12, restated below as Theorem 6.6). Let us first present an example showing the (FRS) condition is too strong for this purpose. Let $\varphi:\mathbb{A}_{\mathbb{Q}}^{2}\rightarrow\mathbb{A}_{\mathbb{Q}}^{1}$ be the map $\varphi(x,y)=xy$ . Then:

	$\displaystyle\varphi_{\mathbb{Q}_{p}}^{-1}(B(0,p^{-k}))\cap\mathbb{Z}_{p}^{2}$	$\displaystyle=\left\{(x,y)\in\mathbb{Z}_{p}^{2}:\mathbb{\mathrm{val}}(x)+\mathbb{\mathrm{val}}(y)\geq k\right\}$
		$\displaystyle=\bigcup_{r\geq k,0\leq l\leq r}\left\{(x,y)\in\mathbb{Z}_{p}^{2}:\mathbb{\mathrm{val}}(x)=l,\mathbb{\mathrm{val}}(y)=r-l\right\}.$

In particular, we have

	$\displaystyle\frac{\varphi_{\mathbb{Q}_{p}*}(\mu_{\mathbb{Z}_{p}}^{2})(B(0,p^{-k}))}{\mu_{\mathcal{O}_{F}}(B(0,p^{-k}))}$	$\displaystyle=p^{k}\cdot\sum_{r=k}^{\infty}\sum_{l=0}^{r}\left(\frac{p-1}{p}\right)^{2}p^{-l}p^{-(r-l)}$
		$\displaystyle=p^{k}\cdot\left(\frac{p-1}{p}\right)^{2}\sum_{r=k}^{\infty}(r+1)p^{-r}\geq\left(\frac{p-1}{p}\right)^{2}(k+1).$

Hence, the measure $\varphi_{\mathbb{Q}_{p}*}(\mu_{\mathbb{Z}_{p}}^{2})$ does not have bounded density. On the other hand, since $\operatorname{lct}_{\mathbb{Q}_{p}}(\varphi_{\mathbb{Q}_{p}};0)=1$ , and by considering the asymptotic expansion of $\varphi_{\mathbb{Q}_{p}*}(\mu_{\mathbb{Z}_{p}}^{2})$ as in Theorem 3.6, one sees:

(1)

$\varphi_{\mathbb{Q}_{p}*}(\mu_{\mathbb{Z}_{p}}^{2})$ explodes logarithmically around $0$ , i.e. the density of $\varphi_{\mathbb{Q}_{p}*}(\mu_{\mathbb{Z}_{p}}^{2})$ behaves like $\mathbb{\mathrm{val}}(t)$ , around $0$ .
(2)

$\epsilon_{\star}(\varphi_{\mathbb{Q}_{p}})=\infty$ for every prime $p$ .

By Corollary 6.2, $\varphi$ cannot be (FRS), and indeed $\left\{xy=0\right\}$ is not normal, so in particular it does not have rational singularities.

In order to prove Theorem 1.12, we recall the notion of jet schemes. Let $X\subseteq\mathbb{A}_{K}^{n}$ be an affine $K$ -scheme whose coordinate ring is

K[x_{1},\dots,x_{n}]/(f_{1},\dots,f_{k}).

Then the $m$ -th jet scheme $J_{m}(X)$ of $X$ is the affine scheme with the following coordinate ring:

K[x_{1},\dots,x_{n},x_{1}^{(1)},\dots,x_{n}^{(1)},\dots,x_{1}^{(m)},\dots,x_{n}^{(m)}]/(\{f_{j}^{(u)}\}_{j=1,u=1}^{k,m}),

where $f_{i}^{(u)}$ is the $u$ -th formal derivative of $f_{i}$ .

Let $\varphi:\mathbb{A}^{n_{1}}\rightarrow\mathbb{A}^{n_{2}}$ be a morphism between affine spaces. Then the $m$ -th jet morphism $J_{m}(\varphi):\mathbb{A}^{n_{1}(m+1)}\rightarrow\mathbb{A}^{n_{2}(m+1)}$ of $\varphi$ is given by formally deriving $\varphi$ , $J_{m}(\varphi)=(\varphi,\varphi^{(1)},\dots,\varphi^{(m)})$ . Similarly, the $m$ -th jet $J_{m}(\varphi)$ of a morphism $\varphi:X\rightarrow Y$ of affine $K$ -schemes, is given by the formal derivative of $\varphi$ . Both $J_{m}(X)$ and $J_{m}(\varphi)$ can be generalized to arbitrary $K$ -schemes and $K$ -morphisms (see [CLNS18, Chapter 3] and [EM09] for more details).

Given a subscheme $Z\subseteq X$ of a smooth variety $X$ , with $Z$ defined by an ideal $J$ , we denote by $\operatorname{lct}(X,Z):=\operatorname{lct}(J)$ the log-canonical threshold of the pair $(X,Z)$ . Mustaţă showed that the log-canonical threshold $\operatorname{lct}(X,Z)$ can be characterized in terms of the growth rate of the dimensions of the jet schemes of $Z$ :

Theorem 6.3 ([Mus02, Corollary 0.2], [CLNS18, Corollary 7.2.4.2]).

Let $X$ be a smooth, geometrically irreducible $K$ -variety, and let $Z\subsetneq X$ be a closed subscheme. Then

\operatorname{lct}(X,Z)=\dim X-\sup_{m\geq 0}\frac{\dim J_{m}(Z)}{m+1}.

Furthermore, the supremum is achieved for $m$ divisible enough. In particular, $\operatorname{lct}(X,Z)$ is a rational number.

Note that $\operatorname{lct}(X,Z)$ depends on $Z$ and $\dim X$ , and neither on the ambient space $X$ not on the embedding of $Z$ in $X$ .

We now introduce the following definitions from [GH24]. For a morphism $\varphi:X\rightarrow Y$ between schemes, we denote by $X_{y,\varphi}$ the scheme theoretic fiber of $\varphi$ over $y\in Y$ .

Definition 6.4.

Let $\varphi:X\rightarrow Y$ be a morphism of smooth, geometrically irreducible $K$ -varieties, and let $\epsilon>0$ .

(1)

$\varphi$ is called $\epsilon$ -flat if for every $x\in X$ we have $\dim X_{\varphi(x),\varphi}\leq\dim X-\epsilon\dim Y$ .
(2)

$\varphi$ is called $\epsilon$ -jet flat if $J_{m}(\varphi):J_{m}(X)\rightarrow J_{m}(Y)$ is $\epsilon$ -flat for every $m\in\mathbb{N}$ .
(3)

$\varphi$ is called jet-flat if it is $1$ -jet flat.

In particular, note that $\varphi$ is flat if and only if it is $1$ -flat.

Note that by Theorem 6.3, $\varphi$ is $\epsilon$ -jet-flat if and only if $\operatorname{lct}(X,X_{\varphi(x),\varphi})\geq\epsilon\dim Y$ for all $x\in X$ . We will need the following lemma to give a jet scheme interpretation to rational and semi-log-canonical singularities (from Theorem 6.3).

Lemma 6.5.

Let $\varphi:X\rightarrow Y$ be a morphism of smooth $K$ -varieties. Then:

(1)

$\varphi$ is (FRS) if and only if $J_{m}(\varphi)$ is flat with locally integral fibers, for every $m\geq 0$ (in particular, $\varphi$ is jet-flat).
(2)

$\varphi$ is jet-flat if and only if $\varphi$ is flat with fibers of semi-log-canonical singularities.

Proof.

Item (1) is proved in [GH24, Corollary 3.12] and essentially follows from a characterization of rational singularities, by Mustaţă [Mus01]. For the proof of Item (2), note that by [GH24, Corollary 2.7], $\varphi$ is jet-flat if and only if $J_{m}(\varphi)$ is flat over $Y\subseteq J_{m}(Y)$ . Since a fiber of a morphism between smooth varieties is flat if and only if its fibers are local complete intersections, the latter condition is equivalent to the condition that every $x\in X$ , and every $m\in\mathbb{N}$ , the scheme $J_{m}(X_{\varphi(x),\varphi})$ is a local complete intersection. By [Ish18, Corollary 10.2.9] and [EI15, Corollary 3.17], this is equivalent to the condition that $X_{\varphi(x),\varphi}$ has semi-log-canonical singularities, for every $x\in X$ . ∎

6.1. Proof of Theorem 1.12

We are now in a position to prove Theorem 1.12. Let us recall its formulation, slightly restated using Lemma 6.5.

Theorem 6.6.

Let $\varphi:X\rightarrow Y$ be a map between smooth $K$ -varieties. Then the following are equivalent:

(1)

$\varphi$ is jet-flat.
(2)

For every local field $F$ containing $K$ , we have $\epsilon_{\star}(\varphi_{F})=\infty$ .
(3)

For every $F\in\operatorname{Loc}_{0,\gg}$ we have $\epsilon_{\star}(\varphi_{F})=\infty$ .
(4)

We have $\epsilon_{\star}(\varphi_{\mathbb{C}})=\infty$ .

The proof of Theorem 6.6 is done by showing both implications $(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(1)$ and $(1)\Rightarrow(2)\Rightarrow(4)\Rightarrow(1)$ . The implications $(2)\Rightarrow(3)$ and $(2)\Rightarrow(4)$ are immediate. We first prove $(3)\Rightarrow(1)$ and $(4)\Rightarrow(1)$ . Then we will prove $(1)\Rightarrow(2)$ in the non-Archimedean case in $\mathsection$ 6.2, and the Archimedean case in $\mathsection$ 6.3.

Proposition 6.7.

Let $\varphi:X\rightarrow Y$ be a map between smooth $K$ -varieties. Assume that either $\epsilon_{\star}(\varphi_{\mathbb{C}})=\infty$ or $\epsilon_{\star}(\varphi_{F})=\infty$ for all $F\in\operatorname{Loc}_{0,\gg}$ . Then $\varphi$ is jet-flat.

Proof.

Working locally, and composing $\varphi$ with an étale map $\Phi:Y\rightarrow\mathbb{A}^{m}$ , we may assume that $Y=\mathbb{A}^{m}$ . Let $\psi:\mathbb{A}^{m}\rightarrow\mathbb{A}^{m}$ be any dominant morphism, and let $\mu_{1}\in\mathcal{M}_{c}^{\infty}(X(F))$ and $\mu_{2}\in\mathcal{M}_{c}^{\infty}(F^{m})$ . By Theorem 1.1, for all $F\in\operatorname{Loc}_{\gg}\cup\left\{\mathbb{C}\right\}$ , we have $\psi_{*}(\mu_{2})\in L^{1+\epsilon}$ for some $\epsilon>0$ . Taking $q$ large enough, by Young’s convolution inequality, one has:

(\varphi*\psi)_{*}(\mu_{1}\times\mu_{2})=\varphi_{*}(\mu_{1})*\psi_{*}(\mu_{2})\subseteq L^{q}*L^{1+\epsilon}\subseteq L^{\infty},

where $\varphi*\psi$ is as in Definition 1.7. By Corollary 6.2, we get that $\varphi*\psi$ is (FRS). We now claim that since $\varphi$ is a morphism whose convolution with any dominant morphism produces an (FRS) morphism, $\varphi$ must be jet-flat.

Indeed, assume it is not the case. Then by Theorem 6.3 and Definition 6.4, there exist $y\in\overline{K}^{m}$ and $N>0$ such that the scheme theoretic fiber $X_{y,\varphi}$ of $\varphi$ over $y$ satisfies

\operatorname{lct}(X,X_{y,\varphi})=\dim X-\sup_{k\geq 0}\frac{\dim J_{k}X_{y,\varphi}}{k+1}\leq m(1-\frac{2}{N}).

Moreover, this supremum is achieved for $k$ divisible enough. Thus the map

J_{k}(\varphi):J_{k}(X)\rightarrow J_{k}(Y)

is not $\left(1-\frac{1}{N}\right)$ -flat for $k$ divisible enough. But on the other hand, the map

\psi_{N}(y_{1},...,y_{m})=(y_{1}^{2N},...,y_{m}^{2N})

satisfies that $J_{l}(\psi_{N})$ is not $\frac{1}{N}$ -flat for divisible enough $l$ (since $\operatorname{lct}(y_{i}^{2N})=\frac{1}{2N}$ ). Thus we may find $k_{0}\in\mathbb{N}$ such that $J_{k_{0}}(\varphi)$ is not $\left(1-\frac{1}{N}\right)$ -flat and $J_{k_{0}}(\psi_{N})$ is not $\frac{1}{N}$ -flat . But then $J_{k_{0}}(\varphi*\psi)=J_{k_{0}}(\varphi)*J_{k_{0}}(\psi)$ is not flat (see [GH24, Lemma 3.26]), which is a contradiction by Fact 6.5, as $\varphi*\psi$ is (FRS). ∎

6.2. $(1)\Rightarrow(2)$ : the non-Archimedean case

We now turn to the proof of $(1)\Rightarrow(2)$ , in the non-Archimedean case. We first prove the following variant of [CGH23, Theorem 4.12].

Proposition 6.8 ([CGH23, Theorem 4.12]).

Let $\varphi:X\rightarrow Y$ be a jet-flat map between smooth $K$ -varieties. Then there exists $M\in\mathbb{N}$ , such that for each $F\in\operatorname{Loc}_{0}$ , each $\mu\in\mathcal{M}_{c}^{\infty}(X(F))$ and each non-vanishing $\tau\in\mathcal{M}^{\infty}(Y(F))$ , one can find $C_{F,\mu,\tau}>0$ such that for each $y\in Y(F)$ and $k\in\mathbb{N}$ one has,

G_{F,\mu}(y,k):=\frac{(\varphi_{*}\mu)(B(y,q_{F}^{-k}))}{\tau(B(y,q_{F}^{-k}))}\leq C_{F,\mu,\tau}k^{M}.

Proof.

We may assume that $Y=\mathbb{A}_{K}^{m}$ and that $\tau=\mu_{F}^{m}$ . We may further assume that $X$ is affine, and thus embeds in $\mathbb{A}_{K}^{n}$ . Let $\widetilde{\mu}$ be the canonical measure on $X(F)$ (see [Ser81, Section 3.3], and also [CCL12, Section 1.2]). It is enough to consider measures $\mu\in\mathcal{M}_{c}^{\infty}(X(F))$ which are of the form $\mu_{l}:=\widetilde{\mu}|_{B(0,q_{F}^{l})\cap X(F)}$ .

Write $g_{F}(l,y)$ for the density of $\varphi_{*}\mu_{l}$ with respect to $\mu_{F}^{m}$ , and set $G_{F}(y,l,k):=G_{F,\mu_{l}}(y,k)$ . Then the collections $\{g_{F}(l,y)\}_{F\in\operatorname{Loc}_{0}}$ and $\{G_{F}(y,l,k)\}_{F\in\operatorname{Loc}_{0}}$ are both motivic functions, in the sense of [CGH18, Section 1.2]. By [CGH18, Theorem 2.1.3], there exists a motivic function $H(l,k)=\left\{H_{F}:\mathbb{Z}\times\mathbb{N}\rightarrow\mathbb{\mathbb{R}}\right\}_{F\in\operatorname{Loc}_{0}}$ , which approximates the supremum of $G(y,l,k)$ , that is:

(6.1)

\frac{1}{C_{F}}H_{F}(l,k)\leq\sup_{y\in Y(F)}G_{F}(y,l,k)\leq H_{F}(l,k),

for all $F\in\operatorname{Loc}_{0}$ and $(l,k)\in\mathbb{Z}\times\mathbb{N}$ , where $C_{F}$ depends only on the local field³³3In the statement of [CGH18, Theorem 2.1.3], the approximation (6.1) is stated for $\left|G_{F}(y,l,k)\right|_{\mathbb{C}}$ instead of $G_{F}(y,l,k)$ . Since $G_{F}(y,l,k)$ is a non-negative real-valued motivic function, their argument yields the current statement as well (see the first four lines of the proof on p.146)..

Since $H$ is motivic, and using [CGH18, Proposition 1.4.2], for each $F\in\operatorname{Loc}_{0}$ , we may divide $\mathbb{Z}\times\mathbb{N}$ into a finite disjoint union $\mathbb{Z}\times\mathbb{N}=\bigsqcup_{A_{F}\in\mathcal{A}}A_{F}$ , with $\left|\mathcal{A}\right|<\infty$ independent of $F$ , such that on each part $A_{F}\subseteq\mathbb{Z}\times\mathbb{N}$ , the following hold. There exist finitely many $a_{i}\in\mathbb{N}$ , $b_{i}\in\mathbb{Q}$ independent of $F$ , a finite set $\Lambda_{F}$ (of size depending on $F$ ), and a finite partition of $A_{F}$ into subsets $A_{F}=\bigsqcup_{\xi\in\Lambda_{F}}A_{F,\xi}$ , such that for all $(k,l)\in A_{F,\xi}$ :

H_{F}(l,k)=\sum_{i=1}^{L}c_{i}(\xi,l,F)\cdot k^{a_{i}}q_{F}^{b_{i}k},

for some constants $c_{i}(\xi,l,F)$ depending on $l,F$ and $\xi$ . Moreover, for fixed $F\in\operatorname{Loc}_{0}$ and $l\in\mathbb{N}$ , the set $A_{F,\xi,l}:=\left\{k\in\mathbb{N}:(k,l)\in A_{F,\xi}\right\}$ is either finite or a fixed congruence class modulo some $e\in\mathbb{Z}_{\geq 1}$ .

To prove the proposition, it is enough to show that for each $F\in\operatorname{Loc}_{0}$ and $l\in\mathbb{Z}$ , we have $H_{F}(l,k)\leq C_{F,l}k^{M}$ on each $A_{F,\xi,l}$ , for some constant $C_{F,l}$ depending on $F,l$ . It is enough to prove this for $A_{F,\xi,l}$ infinite, as otherwise we have

H_{F}(l,k)\leq C_{F,l}:=\sum_{k\in A_{F,\xi,l}}\sum_{i=1}^{L}\left|c_{i}(\xi,l,F)\right|\cdot k^{a_{i}}q_{F}^{b_{i}k}.

Now suppose $A_{F,\xi,l}$ is infinite. By rearranging the constants $c_{i}(\xi,l,F)$ , we may assume that the pairs $(a_{i},b_{i})$ are disjoint, and that $(b_{i},a_{i})>(b_{i+1},a_{i+1})$ in lexicographic order, that is, either $b_{i}>b_{i+1}$ or $b_{i}=b_{i+1}$ and $a_{i}>a_{i+1}$ . Note that if $b_{1}\leq 0$ , then we are done, since for each $F\in\operatorname{Loc}_{0}$ and each $k\in A_{F,\xi,l}$ :

(6.2)

\sup_{y\in Y(F)}G_{F}(y,l,k)\leq H_{F}(l,k)\leq\left(\sum_{i=1}^{L}\left|c_{i}(\xi,l,F)\right|\right)k^{M},

where $M=\max\{a_{i}\}$ . Assume towards contradiction that $b_{1}>0$ , and $c_{1}(\xi,l,F)\neq 0$ . Then for all large enough $k\in A_{F,\xi,l}$ , one has

(6.3)

\sup_{y\in Y(F)}G_{F}(y,l,k)\geq C_{F}^{-1}H_{F}(l,k)\geq q_{F}^{\frac{1}{2}b_{1}k}.

Now let $\psi_{R}:\mathbb{A}_{K}^{m}\rightarrow\mathbb{A}_{K}^{m}$ be the map $\psi_{R}(x_{1},...,x_{m})=(x_{1}^{R},...,x_{m}^{R})$ , for $R:=\left\lceil 4m/b_{1}\right\rceil$ . Then by [GH24, Corollary 3.18], $\varphi*\psi_{R}:X\times\mathbb{A}_{K}^{m}\rightarrow\mathbb{A}_{K}^{m}$ is (FRS). By Corollary 6.2, we have

(6.4)

(\varphi*\psi_{R})_{*}(\mu_{l}\times\mu_{\mathcal{O}_{F}}^{m})\in\mathcal{M}_{c,\infty}(F^{m}).

On the other hand, note that

(\varphi*\psi_{R})^{-1}(B(y,q_{F}^{-k}))\supseteq\varphi^{-1}(B(y,q_{F}^{-k}))\times\psi_{R}^{-1}(B(0,q_{F}^{-k})).

Further note that

\psi_{R}^{-1}(B(0,q_{F}^{-k}))=B\left(0,q_{F}^{-\left\lceil\frac{k}{R}\right\rceil}\right).

Thus, we have

	$\displaystyle\frac{(\varphi\psi_{R})_{}(\mu_{l}\times\mu_{\mathcal{O}_{F}}^{m})(B(y,q_{F}^{-k}))}{\mu_{F}^{m}(B(y,q_{F}^{-k}))}$	$\displaystyle=\frac{\mu_{l}\times\mu_{\mathcal{O}_{F}}^{m}((\varphi*\psi_{R})^{-1}(B(y,q_{F}^{-k}))}{\mu_{F}^{m}(B(y,q_{F}^{-k}))}$
		$\displaystyle\geq\frac{\mu_{l}\left(\varphi^{-1}(B(y,q_{F}^{-k}))\right)}{\mu_{F}^{m}(B(y,q_{F}^{-k}))}\cdot\mu_{\mathcal{O}_{F}}^{m}\left(\psi_{R}^{-1}(B(0,q_{F}^{-k}))\right)$
		$\displaystyle=\frac{\varphi_{*}\mu_{l}(B(y,q_{F}^{-k}))}{\mu_{F}^{m}(B(y,q_{F}^{-k}))}\cdot\mu_{\mathcal{O}_{F}}^{m}\left(B\left(0,q_{F}^{-\left\lceil\frac{k}{R}\right\rceil}\right)\right)$
		$\displaystyle\geq G_{F}(y,l,k)\cdot q_{F}^{-\left\lceil\frac{k}{R}\right\rceil m}.$

By (6.3), for each $k$ large enough, we may find $y_{0}\in Y(F)$ , such that

\frac{(\varphi*\psi_{R})_{*}(\mu_{l}\times\mu_{\mathcal{O}_{F}}^{m})(B(y_{0},q_{F}^{-k}))}{\mu_{F}^{m}(B(y_{0},q_{F}^{-k}))}\geq q_{F}^{-\left\lceil\frac{k}{R}\right\rceil m}q_{F}^{\frac{1}{2}b_{1}k}\geq q_{F}^{\frac{b_{1}}{8}k},

which contradicts (6.4). Thus $b_{1}\leq 0$ and we are done by (6.2). ∎

We are now ready to prove $(1)\Rightarrow(2)$ of Theorem 6.6.

Proof of $(1)\Rightarrow(2)$ of Theorem 6.6, non-Archimedean case.

Let $\varphi:X\to Y$ be a jet-flat morphism. We may assume $Y=\mathbb{A}_{K}^{m}$ . Let $\mu\in\mathcal{M}_{c}^{\infty}(X(F))$ and write $g_{F}$ for the density of $\varphi_{*}\mu$ with respect to $\mu_{F}^{m}$ . Let $Y^{\mathrm{sm},\varphi}$ be the set of $y\in Y$ such that $\varphi$ is smooth over $y$ . For every $F\in\operatorname{Loc}_{0}$ , the map $\varphi_{F}$ is smooth over $Y^{\mathrm{sm},\varphi}(F)$ , and therefore $g_{F}(y)$ is locally constant on $Y^{\mathrm{sm},\varphi}(F)$ .

By [CGH18, Corollary 1.4.3], the constancy radius of $g_{F}(y)$ can be taken to be definable, i.e. there exists a definable function $\alpha:Y^{\mathrm{sm},\varphi}\to\mathbb{N}$ such that $g_{F}(y)$ is constant around every ball $B(y,q_{F}^{-\alpha_{F}(y)})$ . In particular, for every $y\in Y^{\mathrm{sm},\varphi}(F)$ we have $g_{F}(y)=G_{F,\mu}(y,\alpha_{F}(y))$ . In addition, by Proposition 6.8, we have $G_{F,\mu}(y,k)\leq C_{F,\mu}k^{M}$ , for $F\in\operatorname{Loc}_{0}$ . We arrive at the following:

	$\displaystyle\int_{F^{m}}\left\|g_{F}(y)\right\|^{s}dy$	$\displaystyle=\int_{Y^{\mathrm{sm},\varphi}(F)}\left\|G_{F,\mu}(y,\alpha_{F}(y))\right\|^{s}dy\leq C_{F,\mu}^{s}\int_{F^{m}}\alpha_{F}(y)^{Ms}dy$
		$\displaystyle=C_{F,\mu}^{s}\sum_{t\in\mathbb{N}}t^{Ms}\cdot\mu_{F}^{m}(\{y\in F^{m}:\alpha_{F}(y)=t\})$
		$\displaystyle\leq C^{\prime}(F)+C\sum_{t\in\mathbb{N}}t^{Ms}q_{F}^{-\lambda t}<\infty,$

where the last inequality follows by [CGH18, Theorem 3.1.1], since $\lim\limits_{t\to\infty}\mu_{F}^{m}(\{y\in F^{m}:\alpha_{F}(y)=t\})=0$ and thus $\mu_{F}^{m}(\{y\in F^{m}:\alpha_{F}(y)=t\})\leq q_{F}^{-\lambda t}$ for some $\lambda>0$ and every $t$ large enough. ∎

In [CGH23, Theorem 4.12], Cluckers and the first two authors showed that if $\varphi:X\to Y$ is a jet-flat morphism, which is defined over $\mathbb{Z}$ , and one chooses $\mu=\mu_{X(\mathbb{\mathbb{Z}}_{p})}$ and $\tau=\mu_{Y(\mathbb{\mathbb{Z}}_{p})}$ to be the canonical measures on $X(\mathbb{\mathbb{Z}}_{p})$ and $Y(\mathbb{\mathbb{Z}}_{p})$ (see [CGH23, Lemma 4.2]), then the constant $C_{\mathbb{Q}_{p},\mu,\tau}$ in Proposition 6.8 can be taken to be independent of $p$ (i.e. $C_{\mathbb{Q}_{p},\mu,\tau}=C$ ). [CGH23, Theorem 4.12], together with the ideas of the proof of $(1)\Rightarrow(2)$ of Theorem 6.6, allows us to give bounds on the $L^{s}$ norms of $\frac{\varphi_{*}\mu_{X(\mathbb{\mathbb{Z}}_{p})}}{\mu_{Y(\mathbb{\mathbb{Z}}_{p})}}$ , which are independent of $p$ :

Proposition 6.9.

Let $\varphi:X\to Y$ be a dominant morphism between finite type $\mathbb{Z}$ -schemes $X$ and $Y$ , with $X_{\mathbb{Q}},Y_{\mathbb{Q}}$ smooth and geometrically irreducible. For any prime $p$ , let $g_{p}$ be the density of $\varphi_{*}\mu_{X(\mathbb{\mathbb{Z}}_{p})}$ with respect to $\mu_{Y(\mathbb{\mathbb{Z}}_{p})}$ . Then the following are equivalent:

(1)

$\varphi_{\mathbb{Q}}:X_{\mathbb{Q}}\to Y_{\mathbb{Q}}$ is jet-flat.
(2)

For every $s>1$ , there exists $C(s)>0$ such that for every prime $p$ ,

$\left\|g_{p}\right\|_{s}:=\int_{Y(\mathbb{\mathbb{Z}}_{p})}\left|g_{p}(y)\right|^{s}d\mu_{Y(\mathbb{\mathbb{Z}}_{p})}<C(s).$

Proof.

The implication $(2)\Rightarrow(1)$ follows from Proposition 6.7. By $(1)\Rightarrow(2)$ of Theorem 6.6, it is enough to prove $(1)\Rightarrow(2)$ for $p\gg_{s}1$ . Suppose that $\varphi_{\mathbb{Q}}$ is jet-flat. We may assume that $Y=\mathbb{A}^{m}$ . Indeed, working locally, and since $Y_{\mathbb{Q}}$ is smooth, we may assume there exists a morphism $\psi:Y\rightarrow\mathbb{A}^{m}$ , such that $\psi_{\mathbb{Q}}$ is an étale map. If $\widetilde{g}_{p}$ is the density of $\psi_{*}\varphi_{*}\mu_{X(\mathbb{\mathbb{Z}}_{p})}$ with respect to $\mu_{p}^{m}$ , and $N$ is an upper bound on the size of the geometric fibers of $\psi$ , then

\left|\frac{\psi_{*}\mu_{Y(\mathbb{\mathbb{Z}}_{p})}}{\mu_{p}^{m}}\right|<N,

for $p$ large enough. In particular, we have:

\int_{Y(\mathbb{\mathbb{Z}}_{p})}\left|g_{p}(y)\right|^{s}d\mu_{Y(\mathbb{\mathbb{Z}}_{p})}\leq\int_{Y(\mathbb{\mathbb{Z}}_{p})}\left|\widetilde{g}_{p}\circ\psi(y))\right|^{s}d\mu_{Y(\mathbb{\mathbb{Z}}_{p})}=\int_{\mathbb{Z}_{p}^{m}}\left|\widetilde{g}_{p}(\widetilde{y})\right|^{s}d(\psi_{*}\mu_{Y(\mathbb{\mathbb{Z}}_{p})})\leq N\int_{\mathbb{Z}_{p}^{m}}\left|\widetilde{g}_{p}(\widetilde{y})\right|^{s}d\mu_{p}^{m}.

As in the proof of $(1)\Rightarrow(2)$ of Theorem 6.6 above, by [CGH18, Corollary 1.4.3], there exists a definable function $\alpha:(\mathbb{A}^{m})^{\mathrm{sm},\varphi}\to\mathbb{N}$ such that $g_{p}(y)=G_{p}(y,\alpha_{\mathbb{Q}_{p}}(y))$ , where $G_{p}(y,k):=G_{\mathbb{Q}_{p},\mu_{X(\mathbb{\mathbb{Z}}_{p})}}(y,k)$ . Hence, applying [CGH23, Theorem 4.12] we can find $C,M\in\mathbb{N}$ such that for $p\gg 1$ :

	$\displaystyle\int_{\mathbb{Z}_{p}^{m}}\left\|g_{p}(y)\right\|^{s}dy$	$\displaystyle=\int_{(\mathbb{A}^{m})^{\mathrm{sm},\varphi}(\mathbb{Q}_{p})\cap\mathbb{Z}_{p}^{m}}\left\|G_{p}(y,\alpha_{\mathbb{Q}_{p}}(y))\right\|^{s}dy\leq C^{s}\int_{(\mathbb{A}^{m})^{\mathrm{sm},\varphi}(\mathbb{Q}_{p})\cap\mathbb{Z}_{p}^{m}}\alpha_{\mathbb{Q}_{p}}(y)^{Ms}dy$
		$\displaystyle=C^{s}\sum_{t\in\mathbb{N}}t^{Ms}\cdot\mu_{p}^{m}(\{y\in\mathbb{Z}_{p}^{m}:\alpha_{\mathbb{Q}_{p}}(y)=t\}).$

By [CGH18, Theorem 3.1.1], there exists $L\in\mathbb{N}$ and $\lambda>0$ such that $\mu_{p}^{m}(\{y\in\mathbb{Z}_{p}^{m}:\alpha_{\mathbb{Q}_{p}}(y)=t\})\leq p^{-\lambda t}$ for every $t>L$ and every prime $p$ . We therefore get the desired claim as:

\int_{\mathbb{Z}_{p}^{m}}\left|g_{p}(y)\right|^{s}dy\leq C^{s}\sum_{t=0}^{L}t^{Ms}+C^{s}\sum_{t>L}t^{Ms}p^{-\lambda t}<C^{s}\left((L+1)\cdot L^{Ms}+\sum_{t>L}t^{Ms}2^{-\lambda t}\right)<C(s).\qed

6.3. $(1)\Rightarrow(2)$ : the Archimedean case

In this subsection we prove $(1)\Rightarrow(2)$ in the cases $F=\mathbb{\mathbb{R}}$ and $F=\mathbb{C}$ . Let $\varphi:X\rightarrow Y$ be a jet-flat morphism between smooth algebraic varieties, defined over $F$ . Using restriction of scalars, we may assume that $F=\mathbb{\mathbb{R}}$ . We would like to show that for each $\mu\in\mathcal{M}_{c}^{\infty}(X(\mathbb{\mathbb{R}}))$ , we have $\varphi_{*}\mu\in\mathcal{M}^{q}(Y(\mathbb{\mathbb{R}}))$ for all $1\leq q<\infty$ . We start with the following proposition.

Proposition 6.10.

Let $\varphi:X\rightarrow Y$ be a jet-flat map between smooth $\mathbb{\mathbb{R}}$ -varieties. Then for every $\mu\in\mathcal{M}_{c}^{\infty}(X(\mathbb{\mathbb{R}}))$ , every non-vanishing $\tau\in\mathcal{M}^{\infty}(Y(\mathbb{\mathbb{R}}))$ , and every $p\in\mathbb{N}$ , one can find $C_{\mu,\tau,p}>0$ and $M_{\mu,p}>0$ such that for each $y\in Y(\mathbb{\mathbb{R}})$ and $0<r<\frac{1}{2}$ one has

\int_{Y(\mathbb{\mathbb{R}})}\left(\frac{\left(\varphi_{*}\mu\right)(B(y,r))}{\tau(B(y,r))}\right)^{p}d\tau(y)<C_{\mu,\tau,p}\left|\log(r)\right|^{M_{\mu,p}}.

Remark 6.11.

Proposition 6.10 is weaker than its non-Archimedean counterpart (Proposition 6.8). The main obstacle is that we do not know of an Archimedean analogue to [CGH18, Theorem 2.1.3], that is, whether one can approximate the supremum of constructible functions by constructible functions as in (6.1). It is conjectured by Raf Cluckers that such a statement should be true under suitable assumptions (see [AM, Conjecture 6.9]). We thank the anonymous referee for bringing this to our attention. We further believe that the current proposition should hold for $M_{\mu,p}=C\cdot p$ for a sufficiently large $C$ independent of $\mu$ .

Analogously to the non-Archimedean case, we introduce the following notion of constructible functions:

Definition 6.12 ([CM11, Section 1.1], see also [LR97]).

(1)

A restricted analytic function is a function $f:\mathbb{\mathbb{R}}^{n}\rightarrow\mathbb{\mathbb{R}}$ such that $f|_{[-1,1]^{n}}$ is analytic and $f|_{\mathbb{\mathbb{R}}^{n}\backslash[-1,1]^{n}}=0$ .
(2)

A subset $A\subseteq\mathbb{\mathbb{R}}^{n}$ is subanalytic if it is definable in $\mathbb{R}_{\mathrm{an}}$ – the extension of the ordered real field by all restricted analytic functions. A function $f:A\rightarrow B$ is subanalytic if its graph $\Gamma_{f}\subseteq\mathbb{\mathbb{R}}^{n}\times\mathbb{\mathbb{R}}^{m}$ is subanalytic.
(3)

A function $h:A\rightarrow\mathbb{\mathbb{R}}$ is called constructible if there exist subanalytic functions $f_{i}:A\rightarrow\mathbb{\mathbb{R}}$ and $f_{ij}:A\rightarrow\mathbb{\mathbb{R}}_{>0}$ , such that:

(6.5) $h(x)=\sum_{i=1}^{N}f_{i}(x)\cdot\prod_{j=1}^{N_{i}}\log(f_{ij}(x)).$

We denote the class of constructible functions on $A$ by $\mathcal{C}(A)$ .
(4)

Given an analytic manifold $Z$ , a measure $\mu\in\mathcal{M}(Z)$ is called constructible if locally it is of the form $f\cdot\left|\omega\right|$ , where $\omega$ is a regular top-form, and $f$ is constructible. We denote the class of constructible measures by $\mathcal{CM}(Z)$ . Similarly, we write $\mathcal{CM}_{c,q}(Z)$ , $\mathcal{CM}^{\infty}(Z)$ and $\mathcal{CM}_{c}^{\infty}(Z)$ .

Note that $X(\mathbb{\mathbb{R}})$ is defined by polynomials, so it is definable in the real field, and in particular subanalytic. Since $\mathcal{C}(X(\mathbb{\mathbb{R}}))$ contains indicators of balls, we may assume that $\mu\in\mathcal{CM}_{c,\infty}(X(\mathbb{\mathbb{R}}))$ when proving Proposition 6.10 and Theorem 6.6.

Proof of Proposition 6.10.

We may assume that $Y=\mathbb{A}_{\mathbb{\mathbb{R}}}^{m}$ , $\tau=\mu_{\mathbb{\mathbb{R}}}^{m}$ and $\mu\in\mathcal{CM}_{c,\infty}(X(\mathbb{\mathbb{R}}))$ . Write $g(y)$ for the density of $\varphi_{*}\mu$ with respect to $\mu_{\mathbb{\mathbb{R}}}^{m}$ . Set

(6.6)

G(y,r)=\frac{(\phi_{*}\mu)(B(y,r))}{r^{m}}=\frac{1}{r^{m}}\int_{B(y,r)}g(y^{\prime})dy^{\prime}.

For each $p\in\mathbb{N}$ , let

G_{p}(r):=\int_{\mathbb{\mathbb{R}}^{m}}G(y,r)^{p}dy.

By [CM11, Theorem 1.3], the functions $G:\mathbb{\mathbb{R}}^{m}\times\mathbb{\mathbb{R}}_{>0}\rightarrow\mathbb{\mathbb{R}}$ and $G_{p}(r):\mathbb{\mathbb{R}}_{>0}\rightarrow\mathbb{\mathbb{R}}$ are constructible. Writing $G_{p}$ as in (6.5), and using a preparation theorem for constructible functions [CM12, Corollary 3.5], there exist $\delta>0$ and $\theta\in\mathbb{\mathbb{R}}$ , such that $\theta=0$ or $\theta\notin[0,\delta]$ , and for each $r\in(0,\delta)$ one can write:

(6.7)

G_{p}(r)=\sum_{i=1}^{M}d_{i}\cdot S_{i}(r)\cdot\left|r-\theta\right|^{\alpha_{i}}\log(\left|r-\theta\right|)^{l_{i}},

where $d_{i}\in\mathbb{\mathbb{R}}$ , $l_{i},M\in\mathbb{N}$ , $\alpha_{i}\in\mathbb{Q}$ , and where $S_{i}$ are certain subanalytic units, called strong functions (see [CM11, Definition 2.3]). We may assume that $\theta=0$ as otherwise, $G_{p}(r)$ is bounded on $(0,\delta)$ and we are done. In addition, [CM12, Corollary 3.5] also ensures that for each $1\leq i\leq M$ , either $S_{i}(r)=1$ for each $r\in(0,\delta)$ , or $\alpha_{i}>-1$ . This additional property is achieved by writing each strong function $S_{i}(r)$ as a converging infinite sum $S_{i}(r)=\sum_{j=0}^{\infty}c_{ij}\cdot r^{j/p}$ for some $p\in\mathbb{Q}_{\geq 0}$ , and split it into a finite sum $\sum_{j=0}^{s}c_{ij}\cdot r^{j/p}$ and an infinite sum $\widetilde{S_{i}}(r):=\sum_{j>s}c_{ij}\cdot r^{j/p}$ . By taking $s$ large enough, and rearranging the terms in (6.7), Cluckers and Miller ensured that $\alpha_{i}>-1$ whenever $S_{i}(r)\neq 1$ . Following the same argument, and taking $s$ even larger, one can guarantee that $\alpha_{i}>N$ for some fixed $N\in\mathbb{N}$ , as large as we wish. Hence, we may assume that $G_{p}(r)$ has the following form:

(6.8)

G_{p}(r)=\sum_{i=1}^{M^{\prime}}d_{i}r^{\alpha_{i}}\log(r)^{l_{i}}+\sum_{i=M^{\prime}+1}^{M}d_{i}S_{i}(r)r^{\alpha_{i}}\log(r)^{l_{i}},

where $\alpha_{i}>N$ for $M^{\prime}+1\leq i\leq M$ , and $N\in\mathbb{N}$ large as we like. For $1\leq i\leq M^{\prime}$ , we may further assume that $(\alpha_{i},l_{i})$ are mutually different and lexicographically ordered, i.e. either $\alpha_{i}<\alpha_{i+1}$ , or $\alpha_{i}=\alpha_{i+1}$ and $l_{i}>l_{i+1}$ . In particular, by taking $\delta$ small enough, we have for $0<r<\delta$ :

(6.9)

\frac{1}{2}d_{1}r^{\alpha_{1}}\left|\log(r)\right|^{l_{1}}<\left|G_{p}(r)\right|<2d_{1}r^{\alpha_{1}}\left|\log(r)\right|^{l_{1}}.

We claim that $\alpha_{1}\geq 0$ . Assume not, then we have for $r$ small enough:

(6.10)

\left\|G(\cdot\,,r)\right\|_{\infty}\geq\left\|G(\cdot\,,r)\right\|_{p}=G_{p}(r)^{\frac{1}{p}}\gtrsim r^{\frac{\alpha_{1}}{p}}\left|\log(r)\right|^{\frac{l_{1}}{p}}\gtrsim r^{\frac{\alpha_{1}}{p}}.

We now use an argument analogous to the one in Proposition 6.8. Take $\psi_{R}:\mathbb{A}^{m}\rightarrow\mathbb{A}^{m}$ to be the map

\psi_{R}(x_{1},...,x_{m})=(x_{1}^{R},...,x_{m}^{R}),

for $R:=\left\lceil 2mp/\left|\alpha_{1}\right|\right\rceil$ and let $\eta\in C_{c}^{\infty}(\mathbb{\mathbb{R}}^{m})$ be a bump function which is equal to one on the unit ball in $\mathbb{\mathbb{R}}^{m}$ . Then $\varphi*\psi_{R}:X\times\mathbb{A}^{m}\rightarrow\mathbb{A}^{m}$ is (FRS), and thus by Corollary 6.2

(6.11)

(\varphi*\psi_{R})_{*}(\mu\times\eta)\in\mathcal{M}_{c,\infty}(\mathbb{\mathbb{R}}^{m}).

Repeating precisely the same argument as in Proposition 6.8, and using (6.10), we may find $\{y_{r}\}_{r}$ such that:

\frac{(\varphi*\psi_{R})_{*}(\mu\times\eta)(B(y_{r},2r))}{\mu_{\mathbb{\mathbb{R}}}^{m}(B(y_{r},2r))}\gtrsim G(y_{r},r)\cdot\psi_{R*}\eta(B(0,r))\gtrsim G(y_{r},r)r^{\frac{m}{R}}\gtrsim r^{\frac{\alpha_{1}}{4p}},

which leads to a contradiction. Hence $\alpha_{1}\geq 0$ , therefore on $(0,\delta)$ for $\delta$ small enough we have

G_{p}(r)\lesssim\left|\log(r)\right|^{l_{1}}.

For $r>\delta$ , we have $G(y,r)\lesssim\delta^{-m}$ and thus $G_{p}(r)\lesssim\delta^{-pm}<\infty$ . This concludes the proof. ∎

In order to prove Theorem 6.6, we need to control the oscillations of constructible functions.

Definition 6.13.

Let $f:\mathbb{\mathbb{R}}^{n}\rightarrow\mathbb{\mathbb{R}}$ be a subanalytic function. Define $\alpha_{f}:\mathbb{\mathbb{R}}^{n}\times\mathbb{\mathbb{R}}_{>0}\rightarrow\mathbb{\mathbb{R}}_{\geq 0}$ by

(6.12)

\alpha_{f}(y,r):=\min\left(1,\sup\left\{t\in\mathbb{\mathbb{R}}_{\geq 0}:\forall y^{\prime}\in B(y,t),\,\left|f(y)-f(y^{\prime})\right|<r\right\}\right).

Note that $\alpha_{f}$ is subanalytic, and for any $r>0$ we have $\alpha_{f}(y,r)>0$ for almost every $y$ . The next lemma extends this construction to the ring of constructible functions.

Lemma 6.14.

Let $g\in\mathcal{C}(\mathbb{\mathbb{R}}^{n})$ . Then there exists a subanalytic function $\alpha_{g}:\mathbb{\mathbb{R}}^{n}\times\mathbb{\mathbb{R}}_{>0}\rightarrow\mathbb{\mathbb{R}}_{\geq 0}$ such that for any $r>0$ we have $\alpha_{g}(y,r)>0$ for almost all $y\in\mathbb{\mathbb{R}}^{n}$ , and

(6.13)

\left|g(y)-g(y^{\prime})\right|\leq r\text{ for all }y^{\prime}\in B(y,\alpha_{g}(y,r)).

Proof.

If $g=\sum_{i=1}^{M}g_{i}$ for $g_{i}\in\mathcal{C}(\mathbb{\mathbb{R}}^{n})$ , and suppose we already constructed $\alpha_{g_{i}}$ , for each $i$ . Then we may set $\alpha_{g}(y,r)=\min_{i}\,\alpha_{g_{i}}(y,\frac{r}{M})$ . Hence, by (6.5), we may assume that

(6.14)

g(y)=f(y)\cdot\prod_{j=1}^{N}\log(f_{j}(y)),

for some subanalytic $f,f_{1},...,f_{N}\in\mathcal{C}(\mathbb{\mathbb{R}}^{n})$ . By setting $\alpha_{g}(y,r)=\alpha_{g}(y,1)$ for $r>1$ , we may assume $r\leq 1$ .

There is an open subanalytic subset $U\subseteq\mathbb{\mathbb{R}}^{n}$ with complement $U^{c}$ of measure $0$ , such that $f|_{U}$ and all $f_{j}|_{U}$ are continuous (see e.g. [DvdD88, Theorem 3.2.11]). Set

M(y):=\max_{j}\left(\max\left\{\left|f(y)\right|+1,\,2f_{j}(y),\,2/f_{j}(y)\right\}\right),

and denote $h(y):=\frac{M(y)^{-N}}{2N}$ . Let $\alpha_{f},\alpha{}_{f_{j}}$ be as in (6.12) and set

\alpha_{g}(y,r)=\frac{1}{2}\min\left\{\alpha_{f}\left(y,rh(y)\right),\alpha_{f_{1}}\left(y,rf_{1}(y)h(y)\right),\dots,\alpha_{f_{N}}\left(y,rf_{N}(y)h(y)\right)\right\}.

Note that $h$ , $\alpha_{f_{i}},\alpha_{f_{ij}}$ are subanalytic, and thus also $\alpha_{g}$ is subanalytic. Moreover, by the continuity of $f|_{U},f_{j}|_{U}$ we get that $\alpha_{f}(y,\,\cdot\,),\alpha{}_{f_{j}}(y,\,\cdot\,)>0$ and thus also $\alpha_{g}(y,\,\cdot\,)>0$ for all $y\in U$ . Note that for any real numbers $a_{1},...,a_{N},b_{1},...,b_{N}\in[-L,L]$ we have

	$\displaystyle\left\|\prod_{i=1}^{N}a_{i}-\prod_{i=1}^{N}b_{i}\right\|$	$\displaystyle=\left\|\sum_{j=1}^{N}\left(\prod_{i=1}^{N-j+1}a_{i}\prod_{i=N-j+2}^{N}b_{i}-\prod_{i=1}^{N-j}a_{i}\prod_{i=N-j+1}^{N}b_{i}\right)\right\|$
(6.15)			$\displaystyle\leq\sum_{j=1}^{N}\left\|\prod_{i=1}^{N-j}a_{i}\prod_{i=N-j+2}^{N}b_{i}\right\|\left\|a_{N-j+1}-b_{N-j+1}\right\|\leq L^{N-1}\sum_{j=1}^{N}\left\|a_{N-j+1}-b_{N-j+1}\right\|.$

By (6.12), for every $y^{\prime}\in B(y,\alpha_{g}(y,r))$ and $r<1$ we have $\left|f(y^{\prime})-f(y)\right|\leq rh(y)\leq\frac{1}{2}$ , and

\left|f(y^{\prime})\right|\leq\left|f(y^{\prime})-f(y)\right|+\left|f(y)\right|\leq M(y).

Similarly, we have:

\left|\log(f_{j}(y^{\prime}))-\log(f_{j}(y))\right|\leq\log(1+rh(y))\leq rh(y)\leq\frac{1}{2},

and

\left|\log(f_{j}(y^{\prime}))\right|\leq\frac{1}{2}+\left|\log(f_{j}(y))\right|\leq\frac{1}{2}+\max\left(f_{j}(y),\frac{1}{f_{j}(y)}\right)\leq M(y).

By (6.15), for every $y^{\prime}\in B(y,\alpha_{g}(y,r))$ we have:

	$\displaystyle\left\|g(y)-g(y^{\prime})\right\|$	$\displaystyle=\left\|f(y)\cdot\prod_{j=1}^{N}\log(f_{j}(y))-f(y^{\prime})\cdot\prod_{j=1}^{N}\log(f_{j}(y^{\prime}))\right\|$
		$\displaystyle\leq M(y)^{N}\left(\left\|f(y)-f(y^{\prime})\right\|+\sum_{j=1}^{N}\left\|\log\left(\frac{f_{j}(y)}{f_{j}(y^{\prime})}\right)\right\|\right)\leq rh(y)(N+1)M(y)^{N}\leq r.\qed$

We can now finish the proof of Theorem 6.6.

Proof of the Archimedean part of $(1)\Rightarrow(2)$ of Theorem 6.6.

Let $\mu\in\mathcal{CM}_{c,\infty}(X(\mathbb{\mathbb{R}}))$ and write $g(y)\in\mathcal{C}(\mathbb{\mathbb{R}}^{m})$ for the density of $\varphi_{*}\mu$ with respect to $\mu_{\mathbb{\mathbb{R}}}^{m}$ . Let $\alpha_{g}$ be as in Lemma 6.14, and set

S(2):=\left\{y\in\mathbb{\mathbb{R}}^{m}:\frac{1}{2}\leq\alpha_{g}(y,\frac{1}{2})\text{ and }g(y)\neq 0\right\}.

Then for each $r\in\mathbb{\mathbb{R}}_{\geq 3}$ define the following subanalytic set:

S(r):=\left\{y\in\mathbb{\mathbb{R}}^{m}:\frac{1}{r}\leq\alpha_{g}(y,\frac{1}{2})<\frac{1}{r-1}\text{ and }g(y)\neq 0\right\}.

We fix $L\in\mathbb{N}$ large enough. Setting

(6.16)

G(y,r)=\frac{(\phi_{*}\mu)(B(y,r))}{r^{m}}=\frac{1}{r^{m}}\int_{B(y,r)}g(y^{\prime})dy^{\prime}

and using Lemma 6.14, Proposition 6.10 and Hölder’s inequality, we have:

	$\displaystyle\int_{S(r)}g(y)^{p}dy$	$\displaystyle\leq\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))+\int_{S(r)\cap\{g(y)>1\}}g(y)^{p}dy\lesssim 1+\int_{\mathbb{\mathbb{R}}^{m}}1_{S(r)}(y)\cdot G(y,\frac{1}{r})^{p}dy$
		$\displaystyle\lesssim 1+\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))^{\frac{1}{1+\frac{1}{L}}}\left(\int_{\mathbb{\mathbb{R}}^{m}}G(y,\frac{1}{r})^{(L+1)p}dy\right)^{\frac{1}{L+1}}$
(6.17)			$\displaystyle\lesssim 1+\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))^{\frac{1}{1+\frac{1}{L}}}\left\|\log\left(\frac{1}{r}\right)\right\|^{\frac{M_{\mu,(L+1)p}}{L+1}}=1+\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))^{\frac{1}{1+\frac{1}{L}}}\left\|\log(r)\right\|^{\frac{M_{\mu,(L+1)p}}{L+1}}.$

Since $\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))$ is a constructible function, using a similar argument as in the Proposition 6.10, and writing $\mu_{\mathbb{\mathbb{R}}}^{m}(S(1/r))$ as in (6.8) and (6.9), we get $\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))\sim r^{\beta}\left|\log(r)\right|^{\gamma}$ , as $r\rightarrow\infty$ , for $\beta\in\mathbb{Q}$ and $\gamma\in\mathbb{N}$ . But since $\sum_{r=2}^{\infty}\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))<\infty$ , we must have $\beta<-1$ . In particular, taking $L$ large enough, we get that $\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))^{\frac{1}{1+\frac{1}{L}}}\sim r^{-1-\delta}\left|\log(r)\right|^{\gamma^{\prime}}$ for some $\delta>0$ . Thus, the following holds for any $\gamma^{\prime\prime}\in\mathbb{\mathbb{R}}$ :

(6.18)

\sum_{r=2}^{\infty}\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))^{\frac{1}{1+\frac{1}{L}}}\left|\log(r)\right|^{\gamma^{\prime\prime}}<\infty.

By (6.17) and (6.18), we get

\int_{\mathbb{\mathbb{R}}^{m}}g(y)^{p}dy=\sum_{r=2}^{\infty}\int_{S(r)}g(y)^{p}dy\lesssim 1+\sum_{r=3}^{\infty}\mu_{\mathbb{\mathbb{R}}}^{m}(S(r))^{\frac{1}{1+\frac{1}{L}}}\left|\log(r)\right|^{\frac{M_{\mu,(L+1)p}}{L+1}}<\infty.\qed

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	$\displaystyle\int_{U}\left\|\phi_{x}(z)\right\|_{F}^{-s}d\sigma$	$\displaystyle=\int_{U}\left(\int_{0}^{\infty}1_{\left\{(t,z)\,:\left\|\phi_{x}(z)\right\|_{F}^{-s}\geq t\right\}}(t,z)dt\right)d\sigma$
		$\displaystyle=\int_{0}^{\infty}\sigma\left\{z:\left\|\phi(z)-\phi(x)\right\|_{F}^{-s}\geq t\right\}dt$
		$\displaystyle=\int_{0}^{\infty}\sigma\left\{z:\left\|\phi(z)-\phi(x)\right\|_{F}\leq t^{-1/s}\right\}dt$
		$\displaystyle\lesssim 1+\int_{1}^{\infty}t^{-\frac{1}{s}\left(1-\frac{1}{1+\epsilon}\right)}dt<\infty,$

	$\displaystyle\frac{d(\psi_{*}\mu)}{d(\mu_{F}^{n})}(\overline{y},y_{n})$	$\displaystyle=\sum_{t:\psi(\overline{y},t)=y}\frac{\left(\prod_{j=1}^{n-1}\left\|y_{j}\right\|_{F}^{b_{j}}\right)\left\|t\right\|_{F}^{b_{n}}g(\overline{y},t)}{\left\|\operatorname{Jac}_{(y_{1},...,y_{n-1},t)}(\psi)\right\|_{F}}$
		$\displaystyle=\sum_{t:\psi(\overline{y},t)=y}g(\overline{y},t)\left\|a_{n}\right\|_{F}^{-1}\cdot\left(\prod_{j=1}^{n-1}\left\|y_{j}\right\|_{F}^{b_{j}-a_{j}}\right)\left\|t\right\|_{F}^{b_{n}-a_{n}+1}$
		$\displaystyle=\left(\sum_{t:\psi(\overline{y},t)=y}g(\overline{y},t)\right)\cdot\left\|a_{n}\right\|_{F}^{-1}\left(\prod_{j=1}^{n-1}\left\|y_{j}\right\|_{F}^{b_{j}-a_{j}-\frac{(b_{n}-a_{n}+1)a_{j}}{a_{n}}}\right)\left\|y_{n}\right\|_{F}^{\frac{b_{n}-a_{n}+1}{a_{n}}}.$

	$\displaystyle\text{$\int_{L}\left\\|\nabla\phi\|_{\pi(\widetilde{x})}\right\\|_{F}^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}$}d\widetilde{x}$	$\displaystyle\leq\int_{L}\left\\|d_{\widetilde{x}}\pi\right\\|_{\mathrm{op}}^{s}\left\\|\nabla\widetilde{\phi}\|_{\widetilde{x}}\right\\|_{F}^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}d\widetilde{x}\lesssim\int_{L}\left\\|\nabla\widetilde{\phi}\|_{\widetilde{x}}\right\\|_{F}^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}d\widetilde{x}$
		$\displaystyle\lesssim\int_{L}\left(\max_{j}\left\|\widetilde{x}_{1}\right\|_{F}^{a_{1}}\cdot...\cdot\left\|\widetilde{x}_{j}\right\|_{F}^{a_{j}-1}\cdot...\cdot\left\|\widetilde{x}_{n}\right\|_{F}^{a_{n}}\right)^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}d\widetilde{x}$
		$\displaystyle\lesssim\int_{L}\left(\left\|\widetilde{x}_{1}\right\|_{F}^{a_{1}}...\left\|\widetilde{x}_{n}\right\|_{F}^{a_{n}}\right)^{-s}\left\|\operatorname{Jac}_{\widetilde{x}}(\pi)\right\|_{F}d\widetilde{x}\lesssim\int_{\pi(L)}\left\|\phi_{x_{0}}\right\|_{F}^{-s}dx,$

	$\displaystyle\left\|\prod_{i=1}^{N}a_{i}-\prod_{i=1}^{N}b_{i}\right\|$	$\displaystyle=\left\|\sum_{j=1}^{N}\left(\prod_{i=1}^{N-j+1}a_{i}\prod_{i=N-j+2}^{N}b_{i}-\prod_{i=1}^{N-j}a_{i}\prod_{i=N-j+1}^{N}b_{i}\right)\right\|$
(6.15)			$\displaystyle\leq\sum_{j=1}^{N}\left\|\prod_{i=1}^{N-j}a_{i}\prod_{i=N-j+2}^{N}b_{i}\right\|\left\|a_{N-j+1}-b_{N-j+1}\right\|\leq L^{N-1}\sum_{j=1}^{N}\left\|a_{N-j+1}-b_{N-j+1}\right\|.$

Integrability of pushforward measures by analytic maps

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

1.1. Application: regularization by convolution

1.2. Main results

1.2.1. A lower bound on ϵ⋆\epsilon_{\star}

Theorem 1.1.

1.2.2. A formula in the one-dimensional case and a reverse Young inequality

Theorem 1.2.

Theorem 1.3 (Reverse Young inequality).

Remark 1.4.

1.2.3. Upper bounds on ϵ⋆\epsilon_{\star}

Theorem 1.5.

Remark 1.6.

1.2.4. Applications to convolutions of algebraic morphisms

Definition 1.7.

Definition 1.8 ([AA16, Definition II]).

Corollary 1.9.

Remark 1.10.

1.2.5. An algebraic characterization of ϵ⋆=∞\epsilon_{\star}=\infty

Example 1.11.

Theorem 1.12.

1.3. Future directions and further applications

1.3.1. ϵ⋆\epsilon_{\star} as an invariant of singularities

Definition 1.13.

Question 1.14.

1.3.2. ϵ⋆\epsilon_{\star} as an invariant of words

Question 1.15.

Question 1.16.

1.3.3. ϵ⋆\epsilon_{\star} as an invariant of representations

1.4. Conventions

Acknowledgement.

2. Preliminaries: embedded resolution of singularities

Theorem 2.1 (See [VZnG08, Theorem 2.3], [DvdD88, Theorem 2.2], [BM89] and [Wlo09]).

Remark 2.2.

Lemma 2.3 (See e.g. [Mus12, Theorem 1.2] and [VZnG08, Theorem 2.7]).

Remark 2.4.

3. The one-dimensional case

3.1. A formula for ϵ⋆\epsilon_{\star}

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Proof of Theorem 1.2.

3.2. Relation to Fourier decay and other invariants

Remark 3.3.

3.2.1. Asymptotic expansions of pushforward measures and their Fourier transform

Definition 3.4 (Asymptotic expansion, see [Igu78, Section I.2]).

Example 3.5.

Theorem 3.6 ([Jea70, Mal74, Igu78], see also [VZnG17, Section 4] and [Den91, Theorem 1.3.2 and Corollary 1.4.5]).

Remark 3.7.

Corollary 3.8.

Corollary 3.9.

Proof.

Example 3.10.

3.2.2. Relations between the invariants

Proposition 3.11.

Proof.

Remark 3.12.

Proof of Theorem 1.3.

3.3. Consistency of the various bounds in the one-dimensional case

Proposition 3.13.

Proof.

Remark 3.14.

3.4. Some examples in higher dimension

Lemma 3.15.

Proof.

Example 3.16.

Remark 3.17.

Example 3.18.

4. Lower bound: proof of Theorem 1.1

Proposition 4.1.

Proof.

Proof of Theorem 1.1.

5. Proof of Theorem 1.5 – an upper bound over ℂ\mathbb{C}

Lemma 5.1.

Claim 5.2.

Proof of Claim 5.2.

1.2.1. A lower bound on $\epsilon_{\star}$

1.2.3. Upper bounds on $\epsilon_{\star}$

1.2.5. An algebraic characterization of $\epsilon_{\star}=\infty$

1.3.1. $\epsilon_{\star}$ as an invariant of singularities

1.3.2. $\epsilon_{\star}$ as an invariant of words

1.3.3. $\epsilon_{\star}$ as an invariant of representations

3.1. A formula for $\epsilon_{\star}$

5. Proof of Theorem 1.5 – an upper bound over $\mathbb{C}$

6. Geometric characterization of $\epsilon_{\star}=\infty$

6.2. $(1)\Rightarrow(2)$ : the non-Archimedean case

Proof of $(1)\Rightarrow(2)$ of Theorem 6.6, non-Archimedean case.

6.3. $(1)\Rightarrow(2)$ : the Archimedean case

Proof of the Archimedean part of $(1)\Rightarrow(2)$ of Theorem 6.6.