Maz’ya’s $\Phi$-inequalities on domains

This paper contains development of the theory of Maz’ya’s $\Phi$-inequalities given in [11] and [12]. These inequalities might be thought of as corrections of the invalid endpoint Hardy–Littlewood–Sobolev inequality

$${\|\operatorname{I}_{\alpha}[f]\|_{L_{p}(\mathbb{R}^{d})}\lesssim\|f\|_{L_{1}(\mathbb{R}^{d})},\qquad p=\frac{d}{d-\alpha}.}$$

(1.1)

The symbol $\operatorname{I}_{\alpha}$ denotes the Riesz potential of order $\alpha$:

$${\operatorname{I}_{\alpha}[f]=c_{d,\alpha}\int\limits_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}\,dy,\qquad f\in C_{0}^{\infty}(\mathbb{R}^{d}).}$$

(1.2)

For the classical Hardy–Littlewood–Sobolev inequality and its applications, see the original paper [7] or modern presentation in Subsection $\mathrm{VIII}.4.2$ of [9]. The counterexample to (1.1) is provided by a sequence of functions $f$ approximating a delta measure. Linear corrections of the Hardy–Littlewood–Sobolev inequality at the endpoint are sometimes called Bourgain–Brezis inequalities. Usually, one adds translation and dilation invariant constraints for the function $f$ that exclude the delta measures. We refer the reader to the pioneering paper [1], to more recent studies [3], [10], [14], and to the surveys [8] and [15].

In [4], Vladimir Maz’ya suggested a modification of the Hardy–Littlewood–Sobolev inequality for $p=1$ that allowed substitution of a delta measure (see also Problem $5.1$ in the problem book [5] as well). He conjectured that the inequality

$${\Big{|}\int\limits_{\mathbb{R}^{d}}\Phi(\nabla u(x))\,dx\Big{|}\lesssim\|\Delta u\|_{L_{1}(\mathbb{R}^{d})}^{d/(d-1)}}$$

(1.3)

holds true for all smooth compactly supported functions $u$ with a uniform constant, provided the positively $d/(d-1)$ homogeneous function $\Phi\colon\mathbb{R}^{d}\to\mathbb{R}$ satisfies the cancellation condition

$${\int\limits_{S^{d-1}}\Phi(\zeta)\,d\sigma(\zeta)=0.}$$

(1.4)

By $\sigma$ we denote the Hausdorff measure on the unit sphere. The necessity of this condition follows from substitution of $u$ that mimics the fundamental solution of the Laplace equation, i.e. $\Delta u$ mimics a delta measure (see Section 2 below for details on such arguments). By a positively $q$-homogeneous function we mean a function $\Psi$ defined on a Euclidean space such that $\Psi(\lambda y)=\lambda^{q}\Psi(y)$ for any $y$ and any $\lambda\in\mathbb{R}_{+}$. Note that the inequality (1.3) is non-linear; there is no hint to convexity in it as well. The cases where $p=2$ and $\Phi$ is a quadratic form were considered in [6], with sharp constants established. Maz’ya’s conjecture was proved in [11]. We cite the main result of that paper.

Theorem 1.1 implies Maz’ya’s conjecture (1.3) via the representation

$${\nabla u(x)=c_{d}\int\limits_{\mathbb{R}^{d}}\frac{(x-y)\Delta u(y)}{|x-y|^{d}}\,dy,\qquad u\in C_{0}^{\infty}(\mathbb{R}^{d}),}$$

(1.7)

here $c_{d}$ is a certain constant. The classical Sobolev inequalities are valid for functions on domains. Since the companion Bourgain–Brezis inequalities were adjusted to this setting in [2] and [3], it is also desirable to find analogs of Maz’ya’s $\Phi$ inequalities for functions on domains. Here the main results are. In these theorems, we assume that $K\colon\mathbb{R}^{d}\to\mathbb{R}^{\ell}$ and $\Phi\colon\mathbb{R}^{\ell}\to\mathbb{R}$ are Lipschitz on the unit sphere and positively $(\alpha-d)$ and $p$ homogeneous, respectively. We always have the homogeneity relation $p=d/(d-\alpha)$. The number $\beta$ belongs to $(0,1)$.

We note that the new cancellation condition (1.9) implies (1.6). The main difference of the $\Phi$-inequalities from classical estimates in the spirit of Sobolev is that there is no monotonicity of the estimated quantity with respect to domain. In other words, it is completely unclear how to bound $\int_{\Omega}\Phi(K*f)$ with $\int_{\mathbb{R}^{d}}\Phi(K*f)$; seemingly, there are no such bounds. In the classical case, however, $\int_{\Omega}|f|^{p}\leq\int_{\mathbb{R}^{d}}|f|^{p}$.

We have also considered the case where $\Omega$ is a half-space.

The requirement that the boundary of $\Omega$ is $C^{1,\beta}$ smooth in Theorem 1.2 means the following. In a neighborhood of any boundary point, $\Omega$ coincides with the subgraph of a differentiable function whose gradient is $\beta$-Hölder. In particular, if we choose the coordinates in the way that in a neighborhood of the origin

$${\Omega=\{{x\in\mathbb{R}^{d}}\mid{x_{d}\geq h(x_{1},x_{2},\ldots,x_{d-1})}\},\quad h(0)=0,\text{ and }\nabla h(0)=0,}$$

(1.11)

then

$${|h(y)|\leq C|y|^{1+\beta}}$$

(1.12)

for sufficiently small $y\in\mathbb{R}^{d-1}$ and an absolute constant $C=C(\Omega)$.

The assumption that $f$ has zero mean appearing in the Theorems 1.1 and 1.3 is necessary, since without them $K*f$ decays as $|x|^{\alpha-d}$ at infinity, and there is no hope for the bound in question unless we regularize the integral on the left hand side of the inequality; this requirement might be thought of as a condition of compact support for functions in Sobolev inequalities. The proofs of the two theorems above rely on the circle of ideas developed in [11] and [12] and form, in a sense, a natural continuation of that papers. We also present corollaries in the spirit of the original Maz’ya’s setting (1.3).

We have used the notation $\bar{\Omega}$ to denote the closure of $\Omega$. The boundary $\partial\Omega$ is equipped with the Hausdorff measure of dimension $d-1$. The second summand on the right hand side of (1.14) is unavoidable as it can be seen from considering harmonic functions $u$.

While Corollary 1.2 follows from Theorem 1.3 almost immediately, the derivation of Corollary 1.1 from Theorem 1.2 requires some efforts. The strategy is to extend $u$ to a compactly supported function on $\mathbb{R}^{d}$ in such a way that the $L_{1}$-norm of the Laplacian is controlled by the right hand side of (1.14) and then apply Theorem 1.2 via (1.7). A similar strategy was used in [3] in the context of classical Bourgain–Brezis inequalities. Here we need a specific extension theorem, see Proposition 5.1.

In the forthcoming Section 2, we prove the necessity of cancellation conditions in Theorems 1.2 and 1.3. Section 3 is devoted to reduction of our inequalities to Theorem 3.1 below; the latter theorem might be informally thought of as the Besov space version of Maz’ya’s inequalities. Section 4 contains the proof of the latter theorem. The concluding Section 5 finishes the proofs of Theorems 1.2 and 1.3 and provides derivation Corollaries 1.1 and 1.2.

I wish to thank Vladimir Maz’ya for asking me the question that motivated the paper (the question was whether Corollary 1.1 holds true). I am also grateful to Alexander Nazarov, Mikhail Novikov, Bogdan Raiţă, and Alexander Tyulenev for discussions concerning extension of smooth functions (results around Proposition 5.1).

We first verify the necessity of (1.9) in Theorem 1.2 and then explain the modifications needed to justify necessity in Theorem 1.3.

Let $\xi\in S^{d-1}$. Choose a point $z_{\xi}\in\partial\Omega$ such that the inward pointing normal vector to $\partial\Omega$ at $z_{\xi}$ equals $\xi$. For example, we may choose $z_{\xi}$ as follows: let it be the point $z\in\operatorname{cl}\Omega$ such that $\langle{z},{\xi}\rangle$ is the smallest among all $z\in\Omega$. By rotating and shifting $\Omega$, we may, without loss of generality, assume $\xi=(0,0,\ldots,0,1)$ and $z_{\xi}$ is the origin.

Let $f_{0}$ be a smooth function with unit integral supported in the unit ball. Consider its dilations $f_{n}$:

$${f_{n}(x)=n^{d}f(nx),\qquad x\in\mathbb{R}^{d},\ n\in\mathbb{N}.}$$

(2.1)

We will be using the homogeneity relation

$${K*f_{n}(x)=n^{d-\alpha}K*f(nx),\qquad x\in\mathbb{R}^{d},}$$

(2.2)

and the asymptotic formula

$${K*f(x)=K(x)+O(|x|^{\alpha-d-1}),\qquad x\to\infty.}$$

(2.3)

This formula follows from the homogeneity and smoothness assumptions on the kernel $K$. Recall Lemma $6.5$ in [11]:

$${\big{|}\Phi(a+b)-\Phi(a)\big{|}\lesssim|a|^{p-1}|b|,\qquad\text{provided}\quad 2|b|\leq|a|,\quad a,b\in\mathbb{R}^{\ell}.}$$

(2.4)

Then, by (2.3) and (2.4),

$${\Phi(K*f(x))=\Phi(K(x))+O\Big{(}|K(x)|^{p-1}|x|^{\alpha-d-1}\Big{)}=\Phi(K(x))+O\Big{(}|x|^{-d-1}\Big{)},\qquad x\to\infty.}$$

(2.5)

We plug the function $x\mapsto f_{n}(x_{1},x_{2},\ldots,x_{d-1},x_{d}-2/n)$ into (1.8); note that such a function is supported in $\Omega$ provided $n$ is sufficiently large. Then, by (2.2), the left hand side of (1.8) equals

$${\Big{|}\int\limits_{n\Omega}\Phi(K*f(x_{1},x_{2},\ldots,x_{d-1},x_{d}-2))\,dx\Big{|}.}$$

(2.6)

Let $B_{r}(x)$ be the open Euclidean ball of radius $r$ centered at $x$. Note that $K*f$ is a continuous function. Therefore, by the asymptotic formula (2.5), the integral (2.6) differs by a uniformly (w.r.t. $n$) bounded quantity from

$${\Big{|}\int\limits_{n\Omega\setminus B_{4}(0)}\Phi(K(x))\,dx\Big{|}.}$$

(2.7)

Now let

$${\operatorname{I}=\int\limits_{\genfrac{}{}{0.0pt}{-2}{\zeta\in S^{d-1}}{\zeta_{d}>0}}\Phi(K(\zeta))\,d\sigma(\zeta);\qquad\operatorname{M}=\int\limits_{\zeta\in S^{d-1}}\Big{|}\Phi(K(\zeta))\Big{|}\,d\sigma(\zeta).}$$

(2.8)

We wish to show $\operatorname{I}=0$, this will prove the necessity of the first identity in (1.9) (the necessity of the second is obtained by plugging a similar function generated by $f$ whose integral is $-1$). We rewrite (2.7) using a change of variables and homogeneity:

$${\int\limits_{n\Omega\setminus B_{4}(0)}\Phi(K(x))\,dx=\int\limits_{4}^{\infty}r^{d-1}\!\!\!\int\limits_{S^{d-1}\cap\frac{n}{r}\Omega}\Phi(K(r\zeta))\,d\sigma(\zeta)\,dr=\int\limits_{4}^{\infty}r^{-1}\!\!\!\int\limits_{S^{d-1}\cap\frac{n}{r}\Omega}\Phi(K(\zeta))\,d\sigma(\zeta)\,dr.}$$

(2.9)

Note that, in fact, the domain of integration for the outer integral is bounded by $n\operatorname{diam}\Omega$, we will use this fact slightly later. Let

$${\Psi(\rho)=\!\!\!\!\int\limits_{S^{d-1}\cap\rho^{-1}\Omega}\!\!\!\Phi(K(\zeta))\,d\sigma(\zeta),\qquad\rho\in(0,\infty).}$$

(2.10)

This function satisfies the bounds for all $\rho\in(0,\infty)$; the second formula follows from (1.12). Let $\gamma\in(0,1)$ be a parameter. Then, (2.9) equals

$${\int\limits_{4}^{\infty}r^{-1}\Psi(r/n)\,dr=\int\limits_{4}^{n^{\gamma}}r^{-1}\Psi(r/n)\,dr+\int\limits_{n^{\gamma}}^{O(n)}r^{-1}\Psi(r/n)\,dr.}$$

(2.11)

The absolute value of the second integral is bounded by $\operatorname{M}(1-\gamma)\log n+O(1)$ by (LABEL:eq211), whereas the first equals $\operatorname{I}\gamma\log n+O(1+n^{(\gamma-1)\beta})$ via (LABEL:eq212). If $\operatorname{I}\neq 0$, we may choose $\gamma$ such that the whole integral tends to infinity as $n\to\infty$, which contradicts the initial inequality. We have proved necessity in Theorem 1.2.

To prove necessity in Theorem 1.3, we follow a similar route. Without loss of generality, we may assume $\xi=(0,0,\ldots,1)$ and plug the function

$${x\mapsto f_{n}(x_{1},x_{2},\ldots,x_{d-1},x_{d}-1/n)-f(x_{1},x_{2},\ldots,x_{d-1},x_{d}-1)}$$

(2.12)

into (1.10). We need to subtract the second term to obtain a function with zero mean. Applying the tricks we have used to pass from (1.8) to the uniform boundedness of (2.7) to remove the contribution of the second term to the left hand side of the inequality, we arrive at the integral

$${\Big{|}\int\limits_{\genfrac{}{}{0.0pt}{-2}{|x|<n}{x_{d}>0}}\Phi(K(x))\,dx\Big{|}.}$$

(2.13)

The uniform boundedness (w.r.t $n$) of this integral leads to (1.9) via the same change of variable.

The necessity of (1.6) in Theorem 1.3 is proved by considering the function

$${x\mapsto f_{n}(x_{1},x_{2},\ldots,x_{d-1},x_{d}-1)-f(x_{1},x_{2},\ldots,x_{d-1},x_{d}-1).}$$

(2.14)

We split the kernel into parts:

$${K_{n}(x)=\begin{cases}K(x),\qquad&|x|\in[2^{-n-1},2^{-n});\\ 0,\qquad&\text{otherwise}.\end{cases}}$$

(3.1)

Then, $K=\sum_{n}K_{n}$ and $K_{n}(x)=2^{(d-\alpha)n}K_{0}(2^{n}x)$. We will also use the notation

$${K_{\leq n}(x)=\sum\limits_{k\leq n}K_{k}(x).}$$

(3.2)

The condition (1.6) implies

$${\int\limits_{\mathbb{R}^{d}}\Phi(K_{n}(x))\,dx=\int\limits_{\mathbb{R}^{d}}\Phi(-K_{n}(x))\,dx=0,}$$

(3.3)

while (1.9) yields

$${\int\limits_{\langle{x},{\xi}\rangle\geq 0}\!\!\!\Phi(K_{n}(x))\,dx=\!\!\!\int\limits_{\langle{x},{\xi}\rangle\geq 0}\!\!\!\Phi(-K_{n}(x))\,dx=0.}$$

(3.4)

The following lemma is a simpler variation of Lemma $2.4$ in [11].

The function $\mathcal{M}_{p}\colon\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ originated in [12] in this context and played the pivotal role in [11] (see formulas $(2.17)$ and $(5.4)$ of that paper):

$${\mathcal{M}_{p}(x,y)=\begin{cases}\min(x^{p-1}y,xy^{p-1}),\qquad&p\in(1,2];\\ \frac{1}{2}(x^{p-1}y+xy^{p-1}),\qquad&p\in(2,\infty),\end{cases}\qquad x\geq 0,y\geq 0.}$$

(3.7)

The lemma below is completely similar to Lemma $2.6$ in [11], we omit its proof.

Since $\mathcal{M}_{p}$ is a non-negative function, the estimate

$${\sum\limits_{n\in\mathbb{Z}}\int\limits_{\Omega}\mathcal{M}_{p}\Big{(}|K_{\leq n}*f|,|K_{n+1}*f|\Big{)}\lesssim\|f\|_{L_{1}(\mathbb{R}^{d})}^{p}}$$

(3.9)

follows from Theorem $2.3$ in [11]. A companion estimate cannot be reduced to Theorem $2.2$ in [11] since now we need more cancellation conditions on the kernel $K$.

Let $\{Q_{k,j}\}$ be the grid of dyadic cubes:

$${Q_{k,j}=\prod_{i=1}^{d}\big{[}2^{-k}j_{i},2^{-k}(j_{i}+1)\big{)},\qquad j=(j_{1},j_{2},\ldots,j_{d})\in\mathbb{Z}^{d},\ k\in\mathbb{Z}.}$$

(4.1)

If $Q$ is a cube, $c(Q)$ denotes its center, $\ell(Q)$ its sidelength, and $\lambda Q$ is the cube with center $c(Q)$ and sidelength $\lambda\ell(Q)$, $\lambda>0$.

We formulate an analog of Theorem $3.1$ in [11].

Theorem 4.1 is derived from Lemmas 4.2 and 4.3 in the same way as Theorem $3.1$ in [11] is deduced from Lemma $3.1$ of that paper. We will provide a slight simplification of the argument.

Thus, the proof of Theorem 3.1 is naturally split into three estimates:

(4.18)

The estimate (LABEL:eq410) was proved in [11], see formula $(4.11)$ of that paper. The proof of (LABEL:eq412) is relatively simple:

$${\sum\limits_{n\geq 0}2^{-\beta n}\!\!\!\sum\limits_{\genfrac{}{}{0.0pt}{-2}{j\in\mathbb{Z}^{d}}{3Q_{n,j}\in\mathfrak{B}}}\!\!\!\|f\|_{L_{1}(3Q_{n,j})}^{p}\lesssim\sum\limits_{n\geq 0}2^{-\beta n}\|f\|_{L_{1}(\mathbb{R}^{d})}^{p}\lesssim\|f\|_{L_{1}(\mathbb{R}^{d})}^{p}.}$$

(4.19)

It remains to justify (4.18). Let $Q$ be a cube. By $\mathbb{D}_{m}(Q)$ we mean the collection of all dyadic subcubes of generation $m$ of the cube $Q$ ($Q$ itself is of generation $0$). Consider the quantity

$${\mathrm{E}^{\boldsymbol{b}}_{Q,m}[f]=\sum\limits_{\genfrac{}{}{0.0pt}{-2}{Q^{\prime}\in\mathfrak{B}}{Q^{\prime}\in\mathbb{D}_{m}(Q)}}\Big{(}\int\limits_{Q^{\prime}}|f(x)|\,dx\Big{)}^{p},}$$

(4.20)

which is the boundary modification of the quantity

$${\mathrm{E}_{Q,m}[f]=\sum\limits_{Q^{\prime}\in\mathbb{D}_{m}(Q)}\Big{(}\int\limits_{Q^{\prime}}|f(x)|\,dx\Big{)}^{p},}$$

(4.21)

which played an important role in [11]. Note that $\mathrm{E}^{\boldsymbol{b}}_{Q^{\prime},1}[f]\leq\mathrm{E}^{\boldsymbol{b}}_{Q^{\prime},0}[f]$ for any cube $Q^{\prime}$; here we use that a parent of a boundary cube is also a boundary cube. In particular, $\mathrm{E}^{\boldsymbol{b}}_{Q,m+1}[f]\leq\mathrm{E}^{\boldsymbol{b}}_{Q,m}[f]$ for any $m\geq 0$. Therefore,

$${\sum\limits_{m\geq 0}\big{(}\mathrm{E}^{\boldsymbol{b}}_{Q,m}[f]-\mathrm{E}^{\boldsymbol{b}}_{Q,m+1}[f]\big{)}\leq\|f\|_{L_{1}(Q)}^{p},}$$

(4.22)

and all the summands in this sum are non-negative.

This lemma is similar to Lemma $4.2$ in [11], the proof is also similar. We provide it for completeness. Lemma $6.6$ from [11] says

$${\Big{(}\sum\limits_{j=1}^{n}z_{j}\Big{)}^{p}-\sum\limits_{j=1}^{n}z_{j}^{p}\gtrsim\Big{(}\sum\limits_{j=1}^{n}z_{j}\Big{)}^{p-1}\min\limits_{i}\sum\limits_{j\neq i}z_{j}}$$

(4.24)

for any collection of non-negative numbers $z_{j}$. We will need a slight modification.