A stationary set method
for estimating oscillatory integrals

The argument in this subsection was shared to us by Wright, who further attributed it to Stein in the real setting and to Hua (see [Hua65, page 15]) in the $p$-adic setting. A similar form of it was used for instance in [Wri20]. Their argument also works for general valued fields, Archimedean or non-Archimedian. Here we only present this argument in the real setting. It shows that the upper bound (1.1) is tight in an average sense (up to a log factor). Let us be more precise. Denote

and

We will prove that

where $Q$ is a bounded semialgebraic function of complexity $\leq k$, $\delta:=L_{Q}\leq 1$, and the implicit constants are allowed to depend on $d$ and $k$.

The first inequality in (1.4) follows directly from Theorem 1.1. We only need to prove the second inequality. Let $g:[0,1]^{d}\to\mathbb{R}$ be a measurable function. Let $\psi:\mathbb{R}\to\mathbb{R}$ be a Schwartz function satisfying $\mathbf{1}_{[0,1]}(x)\leq\psi(x)$ for every $x\in\mathbb{R}$ whose Fourier transform is compactly supported. We first show that

To see this, we write

By the definition of $\psi$, this is bounded by

The bound (1.5) now follows from the triangle inequality.

Recall that $\delta=L_{Q}$. Assume that $\widehat{\psi}(s)$ is supported on $[-C,C]$. For every $j$ with $\delta\lesssim 2^{-j}\leq 1$, it holds that

where in the second last inequality we applied (1.5). This, combined with the trivial bound

implies the second inequality in (1.4).

A well-known method of estimating oscillatory integrals is through van der Corput estimates, which assume a lower bound for $D^{\beta}Q$ for some multi-index $\beta\in\mathbb{N}^{d}$. This method is very efficient in dimension one, see for instance Stein [Ste93], or Arkhipov, Chubarikov and Karatsuba [AKC79, Section 1.1] for a more quantitative, but perhaps lesser-known, formulation. In [CCW99], Carbery, Christ, and Wright established van der Corput estimates in dimensions greater than one with very mild assumptions on the relevant phase functions. We also refer to Wright [Wri11a], [Wri11b] and [Wri20] for estimating multi-dimensional oscillatory integrals in settings other than $\mathbb{R}^{d}$ (for instance the $p$-adic setting).

The results in the literature that are perhaps closest to ours were obtained by Bruna, Nagel and Wainger [BNW88] (see also Nagel, Seeger and Wainger [NSW93] for a related result). To describe this result, let $S\subset\mathbb{R}^{n+1}$ be a smooth, compact hypersurface that is convex and of finite type. Let $d\sigma$ be the surface measure on $S$. For $x\in S$, let $\nu_{x}$ denote the outward unit normal to $S$ at $x$. Bruna, Nagel, and Wainger [BNW88, Theorem B] proved that, for large $\lambda\gg 1$, we have

The left-hand side of (1.10) is the Fourier transform of the surface-carried measure $s$. The neighborhood of $x$ on the right-hand side of (1.10) plays a similar role to our stationary sets in Theorem 1.1.

In a slightly different direction, Varchenko [Var76] obtained asymptotic estimates for oscillatory integrals

where $\lambda\in\mathbb{R}$ is a large parameter, $\varphi$ is an analytic phase function, and $\chi$ is a function supported near one point, say the origin. The asymptotic expansion involves terms in the Newton polyhedron of the phase function. Instead of asking a “global” condition of the form $D^{\beta}\varphi\geq 1$ as in [CCW99], Varchenko assumed the support of $\chi$ to be small enough; he used the resolution of singularities for analytic functions, and his results are more “local”. For further work along this line, we refer to Karpushkin [Kar80] and [Kar83], Phong, Stein and Sturm [PSS99] and Greenblatt [Gre10], and the references therein. Operator-valued oscillatory integrals in a similar spirit have also been extensively studied. It is beyond our capability to give a review of even a small part of it, and therefore we refer interested readers to more classical works including Phong and Stein [PS97] and Seeger, Sogge and Stein [SSS91], and more recent works including Ikromov, Kempe and Müller [IKM10] and Ikromov and Müller [IM16].

As an application of our stationary set estimate, we obtain the sharp convergence exponent of Tarry’s problem in dimension two for an arbitrary degree. Let $k\geq 2$ and $\xi=(\xi_{1},\dots,\xi_{d})\in\mathbb{R}^{d}$ with $d\geq 1$. Set

For a multi-index $\beta=(\beta_{1},\dots,\beta_{d})\in\mathbb{N}_{0}^{d}$, denote $\xi^{\beta}=\xi_{1}^{\beta_{1}}\dots\xi_{d}^{\beta_{d}}$. Let $\phi_{d,k}(\xi)$ denote the vector $(\xi^{\beta})_{1\leq|\beta|\leq k}\in\mathbb{R}^{N}$. Let $S_{d,k}$ denote the $d$-dimensional manifold

which is sometimes referred to as a Parsell-Vinogradov manifold. For $W\subset[0,1]^{d}$, define the Fourier extension operator associated to $W$ to be

The convergence exponent of Tarry’s problem is defined to be

Theorem 1.1 will serve as a main ingredient in the proof of the following theorem, in which we obtain the sharp convergence exponent of Tarry’s problem in dimension two.

The quantity $\|E_{[0,1]}^{(1,k)}1\|_{L^{p}(\mathbb{R}^{N})}$ appears in the leading coefficient of the asymptotic expansion of Vinogradov’s mean value, as was first shown by Hua for large exponents $p$ [Hua47, Chapter 10.3], and in the full range of $p$ in [Woo19, Corollary 1.3]. The problem of determining the convergence exponent (1.15) goes back to Hua’s works [Hua52, Hua59]. In our notation, Hua proved that $p_{1,k}\leq k^{2}/2+k$, and posed the problem of determining $p_{1,k}$. This problem was resolved by Arkhipov, Chubarikov and Karatsuba [AKC79] (see also [ACK04, Theorem 1.3] in their book), where they proved that

for every $k$. For the connection to the original Tarry problem, we refer to [Woo19, Section 13].

In the same paper [AKC79], the authors also studied Tarry’s problem in higher dimensions, and obtained that

for every $d,k\geq 2$. At the end of their paper, they also stated as an open problem to find the exact value of $p_{d,k}$ in higher dimensions. Later, Ikromov [Ikr97] obtained the lower bound

In the quadratic case, that is, the case $k=2$, the problem was resolved by Mockenhaupt [Moc96], who proved that $p_{d,2}=2(d+1)$ for every $d\geq 2$.

For more partial results and for results on other systems of monomials, we refer to Chakhkiev [Cha03], Ikromov and Safarov [IS13], Safarov [Saf19], and Dzhabbarov [Dzh19, Dzh20].

As a consequence of Theorem 1.2, we obtain the following Fourier restriction estimates.

When $k\equiv 3\mod 4$, it is reasonable to believe that (1.21) also holds for every $p>p_{2,k}$. Here our result covers the range $p>p_{2,k}+1$.

Recall that Theorem 1.2 states that (1.21) holds with $f=1$ whenever $p>p_{2,k}$. That such an estimate implies (1.21) for a general $f\in L^{\infty}$ for an even $p$ is standard, and is exactly how Mockenhaupt [Moc96] proved (1.21) for every $p>p_{d,2}$ and every $d\geq 2$. In the case of Mockenhaupt, the exponent $p_{d,2}=2(d+1)$ is always even, which allows him to obtain sharp estimates in $p$. In our case, our exponent $p_{2,k}$ may be odd, and by doing this reduction we may be off the sharp exponent of $p$ by 1.

Because of its applications to areas like partial differential equations, analytic number theory, and combinatorics, the Fourier restriction theory has received significant amount of attentions in the past few decades. A number of methods, including the bilinear method (see for instance Tao, Vargas and Vega [TVV98], Wolff [Wol01] and Tao [Tao03]), the multilinear method (see for instance Bennett, Carbery and Tao [BCT06] and Guth [Gut10]), the broad-narrow analysis of Bourgain and Guth [BG11] and the polynomial method (see for instance Guth [Gut16]), have been introduced to study the Fourier restriction phenomenon. Perhaps it is not exaggerating to say that each of the above mentioned papers generated an area of active research. Our method in the current paper seems to be disjoint from the above mentioned ones, and it would be very interesting to see how they can be connected.

Let us first remark on the methods of proof of Theorem 1.2, and compare with those in the literature.

First of all, in the quadratic case $k=2$ in [Moc96], in order to bound the $L^{p}$ norm of $E_{[0,1]^{d}}^{(d,2)}1$, Mockenhaupt computed the function $E_{[0,1]^{d}}^{(d,2)}1$ directly. To be more precise, he first smoothed out the function $\mathbf{1}_{[0,1]^{d}}$ to $e^{-|\xi|^{2}/2}$, took the advantage of having a quadratic phase function and computed a Fourier transform directly (roughly speaking via completing squares). It seems difficult to generalize this approach to surfaces of cubic degree or higher. The result of Mockenhaupt [Moc96] has recently been recovered in [GOZZK20] via the broad-narrow analysis of Bourgain and Guth and decoupling inequalities for quadratic forms, and the approach there does not work either for degrees $k\geq 3$.

Next, we turn to the upper bound (1.18) established in [AKC79]. The authors of [AKC79] first derived an accurate pointwise bound for a $d$-dimensional oscillatory integral, and then applied this bound to obtain a good pointwise bound for $|E_{[0,1]^{d}}^{(d,k)}1(x)|$ in terms of $x$. Their pointwise bound is sharp for some, but not all, $x\in\mathbb{R}^{N}$, resulting in an integrability exponent that is not sharp.

In our paper, we propose an entirely different approach, which we call a stationary set method. This method, combined with a careful study of the geometry of stationary sets in our problem (see Subsection 4.2), and certain rigidity properties of stationary sets (see Subsection 4.3), allows us to prove sharp $L^{p}$ integribility of $|E_{[0,1]^{d}}^{(d,k)}1(x)|$ for $d=2$ and every $k\geq 2$.

In the end, we comment on Corollary 1.4. The estimate (1.21) with a sharp range of $p$ when $d=1$ was proven by Drury [Dru85]; indeed, he proved a stronger result with $\|f\|_{\infty}$ replaced by $\|f\|_{q}$ for an optimal range of $q$ as well. As a consequence, Drury [Dru85] recovered the result of Arkhipov, Chubarikov and Karatsuba [AKC79] and gave an alternative proof for (1.17). It is worth mentioning that Drury’s result can be partially recovered via the broad-narrow analysis of Bourgain and Guth [BG11] and the multi-linear restriction estimates of Bennett, Carbery and Tao [BCT06]; to be more precise, via these more recent tools, one can prove (1.21) for an optimal range of $p$, and therefore obtain another proof of (1.17). Moreover, as was shown in [GOZZK20], if one combines the broad-narrow analysis of Bourgain and Guth, with the multi-linear Fourier restriction estimates of Bennett, Bez, Flock and Lee [BBFL18] and the decoupling inequalities for quadratic Parsell-Vinogradov surfaces in Guo and Zhang [GZ19], then one can also recover the result of Mockenhaupt [Moc96] (i.e. the case where $d\geq 2$ and $k=2$). We tried to prove Corollary 1.4 by this approach, but did not succeed. For $d=2$ and $k=3$, we encountered problems in the narrow part of the analysis; while for $k\geq 4$, we encountered significant difficulties in both the broad and the narrow part.

In this subsection, we introduce some terminologies and tools in model theory. They will be useful when we prove Lemma 2.9, which is a crucial ingredient in the proof of Theorem 1.1.

A language $L$ is a triple of tuples

where each $c_{i},i\in I,$ is a constant symbol, each $f_{j}^{(m_{j})},j\in J,$ is a function symbol of arity $m_{j}$, and each $R_{k}^{(n_{k})},k\in K,$ is a relation symbol of arity $n_{k}$.

An $L$-structure $\mathcal{M}$ is a tuple

where $M$ is a set, for each $i\in I$, $c_{i}^{\mathcal{M}}\in M$, for each $j\in J$, $f_{j}^{(m_{j}),\mathcal{M}}$ is a function $M^{m_{j}}\rightarrow M$, and for each $k\in K$, $R_{k}^{(n_{k}),\mathcal{M}}\subset M^{n_{k}}$ is a $n_{k}$-ary relation on $M$. We call

interpretations in $\mathcal{M}$ of the corresponding symbols in $L$. It is a common abuse of notation to denote the structure $\mathcal{M}$ by the underlying set $M$, and we will take this liberty.

An $L$-formula, is obtained using the logic connectives $\vee,\wedge,\neg$ (denoting disjunction, conjunction and negation, respectively), and the quantifiers $\exists,\forall$, from atomic formulas. An atomic formula is a formula either of the form $t_{1}=t_{2}$, where $t_{1},t_{2}$ are terms built out of the function and constant symbols and variables, or of the form $R^{(n)}(t_{1},\ldots,t_{n})$ where $R^{(n)}$ is an $n$-ary relation symbol and $t_{1},\ldots,t_{n}$ are terms. A variable $X$ occurring in an $L$-formula $\phi$ is called a bound variable of the formula, if it appears as $\exists X$ or $\forall X$ in $\phi$. Otherwise, $X$ is called a free variable of $\phi$. If $X_{1},\ldots,X_{n}$ are the free variables of a formula $\phi$, we will denote this by writing $\phi$ as $\phi(X_{1},\ldots,X_{n})$.

Given an $L$-formula $\phi(X_{1},\ldots,X_{n})$, an $L$-structure $\mathcal{M}=(M,\ldots)$, and $\bar{a}\in M^{n}$, one can define in the obvious way (using induction on the structure of the formula $\phi$) whether $\phi(\bar{a})$ is true in the structure $M$ (denoted $M\models\phi(\bar{a})$). Thus, every $L$-formula $\phi(X_{1},\ldots,X_{n})$ defines a subset, $\phi(M)$, of $M^{n}$, defined by

Given an $L$-structure $M$, a formula $\phi(Y_{1},\ldots,Y_{m},X_{1},\ldots,X_{n})$ with free variables

and a tuple $\bar{a}=(a_{1},\ldots,a_{m})\in M^{m}$, we call $\phi(\bar{a},X_{1},\ldots,X_{n})$ an $L(M)$-formula ($L$-formula with parameters in $M$) with free variables $X_{1},\ldots,X_{n}$.

An $L$-theory $T$ is a set of of $L$-formulas without free variables (also called $L$-sentences). An $L$-structure $M$ is a model of $T$ (denoted $M\models T$), if $M\models\phi$ for each $\phi\in T$. An $L$-theory $T$ admits quantifier elimination if for every $L$-formula $\phi(X_{1},\ldots,X_{n})$ there exists a quantifier-free $L$-formula $\psi$ such that for every model $M$ of $T$, $M\models(\phi(X_{1},\ldots,X_{n})\leftrightarrow\psi(X_{1},\ldots,X_{n}){\color[rgb]{0,0,1})}$ (here and elsewhere, $\phi\leftrightarrow\psi$ is the standard abbreviation for the formula $(\neg\phi\vee\psi)\wedge(\neg\psi\vee\phi)$).

The theory of the $L$-structure $M$, denoted $\mathrm{Th}(M)$, is the set of $L$-sentences which are true in $M$. Two $L(M)$-formulas $\phi(\bar{a},X_{1},\ldots,X_{n})$ and $\psi(\bar{b},X_{1},\ldots,X_{n})$ are said to be equivalent modulo $\mathrm{Th}(M)$, if $M\models(\forall X_{1})\cdots(\forall X_{n})(\phi(\bar{a},X_{1},\ldots,X_{n})\leftrightarrow\psi(\bar{b},X_{1},\ldots,X_{n}))$. The definable sets of an $L$-structure $M$ are sets of the form

where $\phi$ is an $L(M)$-formula. The definable functions are those whose graphs are definable sets.

For the rest of the article, we will denote by $L=((0,1),(+,\cdot),(\leq))$ the language of ordered fields.

The set $\mathbb{R}$ with the usual interpretations of $0,1,+,\cdot,\leq$ is an $L$-structure. Notice that, by definition, semi-algebraic sets are definable sets in the $L$-structure $\mathbb{R}$. The Tarski-Seidenberg theorem [Tar51] states that the the $L$-theory, $\mathrm{RCOF}$, of real closed ordered fields admits quantifier elimination. Here, $\mathrm{RCOF}$ is the set of $L$-sentences axiomatizing ordered fields, in which every non-negative element is a square and every polynomial having odd degree has a root. By definition, we see that the field $\mathbb{R}$ (with the usual interpretations of $0,1,+,\cdots,\leq$) is a model of $\mathrm{RCOF}$. As a consequence of the Tarski-Seidenberg theorem [Tar51], one immediately obtains that every definable set of the $L$-structure $\mathbb{R}$ is semi-algebraic.

As an illustration of Tarski-Seidenberg theorem, observe that the $L$-formula $\phi$ in (2.3) (which has quantifiers) is equivalent modulo the theory $\mathrm{RCOF}$ to the quantifier-free $L$-formula

As a consequence,

Another simple geometric consequence of the Tarski-Seidenberg theorem is that if $S_{1}$ and $S_{2}$ are a semialgebraic sets in $\mathbb{R}$, then the set $\{x\in\mathbb{R}:\exists y\in S_{1}\text{ s.t. }x+y\in S_{2}\}$ is also semialgebraic.

While the Tarski-Seidenberg theorem is not quantitative as stated above, there are effective versions with complexity estimates. In particular, given any $L$-formula $\phi$, there exists a quantifier-free $L$-formula $\psi$ equivalent to $\phi$ modulo $\mathrm{RCOF}$, such that the number and degrees of the polynomials appearing in $\psi$ is bounded by a function of the number and degrees of the polynomials appearing in $\phi$, as well as the number of quantified and free variables. This is in fact a consequence of the original proof of Tarski [Tar51] of the Tarski-Seidenberg theorem, and we omit detailed explanation here. We remark that the above mentioned function can be made explicit, see for instance [BPR06, Theorem 14.16].

With the preparation above, let us prove a lemma that will play a crucial role in the proof of Theorem 1.1.

With Lemma 2.9 in mind, we now prove Theorem 1.1.