A note on strong-form stability for the Sobolev inequality

TL;DR

This paper proves a strong quantitative version of the Sobolev inequality in Euclidean space, linking the inequality gap to the gradient difference from extremal functions for functions in the Sobolev space.

Contribution

It establishes a strong form of the quantitative Sobolev inequality, providing a precise control of the gradient difference in terms of the inequality gap.

Findings

The gap in the Sobolev inequality controls the gradient difference from extremal functions.

The result applies to functions in the homogeneous Sobolev space rac{1}{p}(rac{1}{n})

The paper extends the understanding of stability in Sobolev inequalities.

Abstract

In this note, we establish a strong form of the quantitive Sobolev inequality in Euclidean space for $p \in (1,n)$. Given any function $u \in \dot W^{1,p}(\mathbb{R}^n)$, the gap in the Sobolev inequality controls $\| \nabla u -\nabla v\|_{p}$, where $v$ is an extremal function for the Sobolev inequality.

Equations

$$\Vert \nabla u{\Vert }_p\ge S\Vert u{\Vert }_{p^{\ast }}.$$

$$\bar{v}(x)={\left(1+|x{|}^{p^{\prime }}\right)}^{(p-n)/p},$$

$${\mathcal{M}}=\Big{\{}v\ \big{|}\ v(x)=c\,\bar{v}\left(\lambda(x-y)\right)\text{ for some }c\in{\mathbb{R}},\,\lambda\in{\mathbb{R}}_{+},\,y\in{\mathbb{R}}^{n}\Big{\}}$$

$$A(u)$$

$$\delta (u)$$

$$A(u{)}^2\le C\delta (u).$$

$${\left(\frac{\Vert \nabla u-\nabla v{\Vert }_p}{\Vert u{\Vert }_{p^{\ast }}}\right)}^{\alpha }\le C_1\delta (u)+C_2\frac{\Vert u-v{\Vert }_{p^{\ast }}}{\Vert u{\Vert }_{p^{\ast }}}.$$

$$A(u{)}^{\beta }\le C\delta (u).$$

$$\beta =\{\begin{array}{ll}{p}^{\prime }{\left(p^{\ast }\left(3+4p-\frac{3p+1}{n}\right)\right)}^2& \text{ if }p\in (1,2)\\ p{\left(p^{\ast }\left(3+4p-\frac{3p+1}{n}\right)\right)}^2& \text{ if }p\in [2,n).\end{array}$$

$${\Vert \frac{F+G}{2}\Vert }_p^{p^{\prime }}+{\Vert \frac{F-G}{2}\Vert }_p^{p^{\prime }}$$

$${\Vert \frac{F+G}{2}\Vert }_p^p+{\Vert \frac{F-G}{2}\Vert }_p^p$$

$${\Vert \frac{\nabla u-\nabla v}{2}\Vert }_p^{p^{\prime }}$$

$$\Vert \nabla v{\Vert }_p^p$$

$${\Vert \nabla u+\nabla v\Vert }_p^{p^{\prime }}$$

$${\Vert \frac{\nabla u-\nabla v}{2}\Vert }_p^{p^{\prime }}$$

$${\Vert \frac{u+v}{2}\Vert }_{p^{\ast }}^{p^{\prime }}$$

$${\left(\Vert u{\Vert }_{p^{\ast }}-{\Vert \frac{u-v}{2}\Vert }_{p^{\ast }}\right)}^{p^{\prime }}$$

$${\Vert \frac{\nabla u-\nabla v}{2}\Vert }_p^p$$

$$\le \Vert \nabla u{\Vert }_p^p-S^p{\Vert \frac{u+v}{2}\Vert }_{p^{\ast }}^p$$

$$\le \Vert \nabla u{\Vert }_p^p-S^p\Vert u{\Vert }_{p^{\ast }}^p+p\Vert u{\Vert }_{p^{\ast }}^{p-1}{\Vert \frac{u-v}{2}\Vert }_{p^{\ast }}.$$

$$\lambda (u)=\inf \left\{\frac{\Vert u-v{\Vert }_{p^{\ast }}}{\Vert u{\Vert }_{p^{\ast }}}:v\in \mathcal{M},\Vert v{\Vert }_{p^{\ast }}=\Vert u{\Vert }_{p^{\ast }}\right\}$$

$$\lambda (u{)}^{\beta }\le C\frac{\Vert \nabla u{\Vert }_p-S_{p,n}\Vert u{\Vert }_{p^{\ast }}}{\Vert u{\Vert }_{p^{\ast }}},$$

$$\frac{\Vert \nabla u{\Vert }_p-S\Vert u{\Vert }_{p^{\ast }}}{\Vert u{\Vert }_{p^{\ast }}}\le \delta (u).$$

$$a^q-b^q\ge a-b.$$

$$|a+b{|}^{p^{\prime }}+|a-b{|}^{p^{\prime }}$$

$${\left(\sum |a_i{|}^s\right)}^{1/s}+{\left(\sum |b_i{|}^s\right)}^{1/s}$$

$$\begin{array}{rl}|F-G{|}^{p^{\prime }}+|F+G{|}^{p^{\prime }}& \le {\left(\sum (|F_i+G_i{|}^{p^{\prime }}+|F_i-G_i{|}^{p^{\prime }}{)}^{2/p^{\prime }}\right)}^{p^{\prime }/2}\\ & \le 2{\left(\sum (|F_i{|}^p+|G_i{|}^p{)}^{2/p}\right)}^{p^{\prime }/2}.\end{array}$$

$${\left(\sum (|a_i{|}^p+|b_i{|}^p{)}^{2/p}\right)}^{1/2}\le {\left({\left(\sum |a_i{|}^2\right)}^{p/2}+{\left(\sum |b_i{|}^2\right)}^{p/2}\right)}^{1/p}.$$

$$|F+G{|}^{p^{\prime }}+|F-G{|}^{p^{\prime }}\le 2{\left(|F{|}^p+|G{|}^p\right)}^{p^{\prime }/p}.$$

$$\Vert h_1{\Vert }_{L^s({\mathbb{R}}^n)}+\Vert h_2{\Vert }_{L^s({\mathbb{R}}^n)}$$

$$\Vert F+G{\Vert }_p^{p^{\prime }}+\Vert F-G{\Vert }_p^{p^{\prime }}$$

$$\le 2{\left(\int |F{|}^p+|G{|}^p\right)}^{p^{\prime }/p}.$$

Full text

1. Introduction

Sobolev inequalities, broadly speaking, establish integrability or regularity properties of a function in terms of the integrability of its gradient. A fundamental example is the classical Sobolev inequality on Euclidean space, which states the following. Given $n\geq 2$ and $p\in(1,n)$, there exists a constant $S=S(n,p)$ such that

[TABLE]

for any function $u\in\dot{W}^{1,p}({\mathbb{R}}^{n})$. Here, $p^{*}=np/(n-p)$, and $\dot{W}^{1,p}({\mathbb{R}}^{n})$ is the space of functions such that $u\in L^{p^{*}}({\mathbb{R}}^{n})$ and $|{\nabla}u|\in L^{p}({\mathbb{R}}^{n})$. Let us take $S$ to be the largest possible constant for which (1.1) holds. Aubin [Aub76] and Talenti [Tal76] determined that equality is achieved in (1.1) for the function

[TABLE]

as well as its translations, dilations, and constant multiples. Here and in the sequel, we let $p^{\prime}=p/(p-1)$ denote the Hölder conjugate of $p$. In fact, these functions are the only such extremal functions for (1.1), and we will let

[TABLE]

denote this $(n+2)$-dimensional space of extremal functions.

Brezis and Lieb raised the question of quantitative stability for the Sobolev inequality in [BL85], asking whether the deviation of a given function from attaining equality in (1.1) controls its distance to the family of extremal functions ${\mathcal{M}}$. The strongest notion of distance that one expects to control is the $L^{p}$ norm between gradients. With this in mind, let us define the asymmetry of a function $u\in\dot{W}^{1,p}({\mathbb{R}}^{n})$ by

[TABLE]

Note that $A(u)$ is invariant under the symmetries of the Sobolev inequality (translations, dilations, and constant multiples) and is equal to zero if and only if $u\in{\mathcal{M}}$. To quantify the deviation from equality in (1.1), we define the deficit of a function $u\in\dot{W}^{1,p}({\mathbb{R}}^{n})$ to be

[TABLE]

Like the asymmetry, the deficit is a non-negative functional that is invariant under translations, dilations, and constant multiples, and is equal to zero if and only if $u\in{\mathcal{M}}$.

By way of a concentration compactness argument as in [Lio85], one readily establishes the qualitative stability of (1.1). That is, if $\{u_{i}\}$ is a sequence of functions with $\delta(u_{i})\to 0$, then $A(u_{i})\to 0$. The first quantitative result was established in the case $p=2$ in [BE91], where Bianchi and Egnell showed that there is a dimensional constant $C$ such that

[TABLE]

This result, in addition to being optimal in the strength of the distance controlled, is sharp in the sense that the exponent $2$ cannot be replaced by a smaller one. The proof relies strongly on the fact that $W^{1,2}({\mathbb{R}}^{n})$ is a Hilbert space, and in the absence of this structure, the case when $p\neq 2$ has proven much more difficult to treat. Nevertheless, in [CFMP09], Cianchi, Fusco, Maggi, and Pratelli established a quantitative stability result in which the deficit controls the distance of a function to ${\mathcal{M}}$ in terms of the $L^{p^{*}}$ norm; see Theorem 2.1 below for a precise statement. The argument combines symmetrization arguments in the spirit of [FMP08] with a mass transportation argument in one dimension. More recently, in [FN19], Figalli and the author strengthened this result in the case $p\geq 2$ by showing that the deficit of a function controls a power of $A(u)$. The main idea there was to view $W^{1,p}({\mathbb{R}}^{n})$ as a weighted Hilbert space and to establish a spectral gap for the linearized operator in the second variation as in [BE91]. However, bounding the difference between the deficit and the second variation required the use of the main result of [CFMP09].

In this note, we establish a reduction theoreom that, paired with [CFMP09], allows us to deduce a strong-form quantitative stability result in which the deficit of a function controls a power of $A(u)$. For $p\geq 2$, this recovers the main result of [FN19] with a simpler proof, while in the case $p\in(1,2)$, it provides the first known quantitative estimate for (1.1) at the level of gradients.

Theorem 1.1**.**

Fix $n\geq 2$ and $p\in(1,n).$ There exist constants $C_{1}(n,p)$ and $C_{2}(n,p)$ such that the following holds. For any $u\in\dot{W}^{1,p}({\mathbb{R}}^{n})$ and for any $v\in{\mathcal{M}}$ with $\|u\|_{p^{*}}=\|v\|_{p^{*}},$ we have

[TABLE]

Here, $\alpha=p^{\prime}$ if $p\in(1,2)$ and $\alpha=p$ if $p\in[2,n)$.

Pairing Theorem 1.1 with the main result of [CFMP09] (Theorem 2.10 below), we establish the following quantitative estimate.

Corollary 1.2**.**

Fix $n\geq 2$ and $p\in(1,n)$. There exist constants $C=C(n,p)$ and $\beta=\beta(n,p)$ such that the following holds. For any $u\in\dot{W}^{1,p}({\mathbb{R}}^{n})$, we have

[TABLE]

The value of $\beta$ in Corollary 1.2 is given by

[TABLE]

The proof of Theorem 1.1 is elementary and at its core relies on the convexity of the function $t\mapsto t^{p}$. It is inspired by the recent paper [HS], in which Hynd and Seuffert give a qualitative description of extremal functions in (a certain form of) Morrey’s inequality. Interestingly, they are able to establish a quantitative stability result, even without knowing the explicit form of extremal functions.

Quantitive stability for Sobolev-type inequalities has been a topic of interest in recent years. Closely related to the main results here, a strong-form quantitative stability result was shown for the Sobolev inequality (1.1) with $p=1$ in [FMP13], following [FMP07, Cia06]. Quantitative stability results have also been shown for (a different form of) Morrey’s inequality [Cia08], the log-Sobolev inequality [IM14, BGRS14, FIL16], the higher order Sobolev inequality [BWW03, GW10], the fractional Sobolev inequality [CFW13], Gagliardo-Nirenberg-Sobolev inequalities [CF13, DT13, DT16, Seu, Ngu], and Strichartz inequalities [Neg].

More broadly, strong-form stability estimates (in which the gap in a given inequality controls the strongest possible norm, typically involving the oscillation of a set or function) have been studied for various functional and geometric inequalities. For instance, such results have been shown for isoperimetric inequalities in Euclidean space [FJ14], on the sphere [BDF17], and in hyperbolic space [BDS15], as well as for anisotropic [Neu16] and Gaussian [Eld15, BBJ17] isoperimetric inequalities.

Apart from their innate interest from a variational perspective, quantitative stability estimates have found applications in the study of geometric problems [FM11, CS13, KM14] and PDE [CF13, DT16]. Certain applications, such as those in [FMM18, CNT], necessitate strong-form quantitative estimates of the type established here.

Acknowledgments: The author is supported by Grant No. DMS-1638352 at the Institute for Advanced Study.

2. Proofs of Theorem 1.1 and Corollary 1.2

In the proof of Theorem 1.1, we will make use of the following version of Clarkson’s inequalities for vector-valued functions, which state the following. Let $F,G:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ with $|F|,|G|\in L^{p}({\mathbb{R}}^{n})$. Then

[TABLE]

if $p\geq 2$. These inequalities were shown for scalar- and complex-valued functions in [Cla36], and were extended to functions mapping from ${\mathbb{R}}$ to ${\mathbb{R}}^{n}$ in [Boa40]. Though Clarkson’s inequalities have been generalized in a number of directions, we could not locate a reference for the precise form of (2.1) and (2.2), so in Section 3 we prove (2.2) and show how to deduce (2.1) from its scalar-valued analogue.

Proof of Theorem 1.1.

We first consider the case $p\in(1,2)$. Applying (2.1) with $F={\nabla}u$ and $G={\nabla}v$, we find that

[TABLE]

Next, the Sobolev inequality (1.1) implies that

[TABLE]

In (2.4) we have used the assumption that $\|u\|_{p*}=\|v\|_{p^{*}}$. Together (2.3), (2.4), and (2.5) imply that

[TABLE]

Then, convexity of the function $t\mapsto t^{p^{\prime}}$ implies that

[TABLE]

Together (2.8) and (2.9) imply (2.7). Finally, combining (2.6) and (2.7) and dividing through by $\|u\|_{p^{*}}^{p^{\prime}}$ establishes the proof of (1.2) with $C_{1}=2^{p^{\prime}}$ and $C_{2}=p^{\prime}2^{p^{\prime}-1}$.

Next, the proof for the case $p\geq 2$ is completely analogous. Indeed, applying Clarkson’s inequality (2.2) followed by the Sobolev inequality (1.1), and then (2.7) (with $p$ replacing $p^{\prime}$), we find that

[TABLE]

Dividing by $\|u\|_{p^{*}}^{p}$ establishes (1.2) with $C_{1}=2^{p}$ and $C_{2}=p2^{p-1}$. ∎

Now, let us recall the main result from [CFMP09]. The notion of $L^{p^{*}}$ asymmetry considered there is

[TABLE]

Theorem 2.1** (Cianchi, Fusco, Maggi, Pratelli).**

Fix $n\geq 2$ and $p\in(1,n)$. There exists a constant $C=C(n,p)$ such that the following holds. For any $u\in\dot{W}^{1,p}({\mathbb{R}}^{n})$,

[TABLE]

Here $\beta=\left(p^{*}\left(3+4p-\frac{3p+1}{n}\right)\right)^{2}$.

We now prove Corollary 1.2 by combining Theorems 1.1 and 2.1.

Proof of Corollary 1.2.

The only point to check is that

[TABLE]

To see this, note that for any $q\geq 1$, the function $t\mapsto t^{q}-t$ is increasing for $t\geq 1.$ In particular, if $a\geq b\geq 1$, we have

[TABLE]

Let $a=\|{\nabla}u\|_{p}/S\|u\|_{p^{*}}$ and $b=1$. Then applying (2.12) with $q=p^{\prime}$ for $p\in(1,2)$ and $q=p$ for $p\in[2,n)$ establishes (2.11). With this in hand, Corollary 1.2 follows immediately from (1.2) and (2.10). ∎

3. Clarkson’s inequalities for vector valued functions on ${\mathbb{R}}^{n}$

For $p\in(1,2)$, Clarkson [Cla36] established the following inequality for, in particular, real numbers $a$ and $b$:

[TABLE]

Let us see how to deduce (2.1) from (3.1). We make use of the reverse Minkowski inequality: if $s\in(0,1)$, then for $(a_{1},\dots,a_{n})\subset{\mathbb{R}}^{n}$ and $(b_{1},\dots,b_{n})\subset{\mathbb{R}}^{n}$ we have

[TABLE]

This inequality follows from the concavity of the function $t\mapsto t^{s}.$ We take $s=2/{p^{\prime}}$ and let $a_{i}=|F_{i}+G_{i}|^{p^{\prime}}$ and $b_{i}=|F_{i}-G_{i}|^{p^{\prime}}$ for $i=1,\dots,n$. Here $F_{i}$ denotes the $i$th component of $F$ in some fixed basis. Then, applying (3.2) followed by (3.1), we find that

[TABLE]

On the left-hand side, we have used $|F|$ to denote the Euclidean norm. Next, applying the usual form of Minkowski’s inequality with $r=2/p$ to $(|a_{i}|^{p})$ and $(|b_{i}|^{p})$, we find

[TABLE]

Pairing this with (3.3), we find that

[TABLE]

Finally, we make use of the integral form of (3.2): for $s\in(0,1)$, we have

[TABLE]

We apply (3.5) with $s=p/{p^{\prime}}$ and with $h_{1}=|F+G|^{{p^{\prime}}}$ and $h_{2}=|F-G|^{{p^{\prime}}}$, and then apply (3.4), in order to find that

[TABLE]

This establishes (2.1). The corresponding inequality (2.2) for $p\geq 2$ is straightforward. Note that $a^{p/2}+b^{p/2}\leq(a+b)^{p/2}$ for $p\geq 2.$ Applying this property and then expanding the squares, we have

[TABLE]

Finally, convexity of the function $t\mapsto t^{p/2}$ implies that

[TABLE]

We combine (3.8) and (3.9) and integrate to conclude the proof of (2.2).