Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy

TL;DR

This paper establishes exponential integrability results similar to Moser-Trudinger inequalities for functions with finite non-local, non-convex energy characterized by a specific double integral involving differences of the functions.

Contribution

It introduces new exponential integrability results for functions constrained by a non-local energy integral, extending classical inequalities to a broader function class.

Findings

Proves exponential integrability for functions with finite non-local energy.

Connects non-local energy conditions to classical inequalities.

Provides tools for improved Sobolev, Poincaré, and Hardy inequalities.

Abstract

Let $d \ge 1$, $p \ge d$, and let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\Omega$ such that $$ \mathop{\int_{\Omega} \int_{\Omega}}_{|u(x) - u(y)| > \delta} \frac{1}{|x-y|^{d+p}} \, dx \, dy < + \infty, $$ for some $\delta > 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincar\'e inequalities, and Hardy inequalities.

Equations

$${{\int }_{\Omega }{\int }_{\Omega }}_{|u(x)-u(y)|>\delta }\frac{1}{|x-y{|}^{d+p}}dxdy<+\infty ,$$

$$\underset{n\to \infty }{\lim }{\int }_{\tau }^{\infty }{\rho }_n(r)r^{N-1}dr=0\forall \tau >0,\text{mbox}and\underset{n\to \infty }{\lim }{\int }_0^{+\infty }{\rho }_n(r)r^{N-1}dr=1.$$

$$K_{d,p}:={\int }_{{\mathbb{S}}^{d-1}}|\sigma \cdot e{|}^{p}d\sigma ,$$

$$\underset{n\to \infty }{\mathrm{liminf}}{\int }_{\Omega }{\int }_{\Omega }\frac{|u(x)-u(y){|}^p}{|x-y{|}^p}{\rho }_n(|x-y|)dxdy<+\infty .$$

$$\underset{n\to \infty }{\lim }{\int }_{\Omega }{\int }_{\Omega }\frac{|u(x)-u(y){|}^p}{|x-y{|}^p}{\rho }_n(|x-y|)dxdy=K_{d,p}{\int }_{{\mathbb{R}}^N}|\nabla u(x){|}^{p}dx.$$

$$I_{\delta ,p}(u,O)={{\int }_O{\int }_O}_{|u(x)-u(y)|>\delta }\frac{{\delta }^p}{|x-y{|}^{d+p}}dxdy.$$

$$\underset{\delta \to 0}{\mathrm{liminf}}{I}_{\delta ,p}(u,\Omega )<+\infty .$$

$$\underset{\delta \to 0}{\lim }{I}_{\delta ,p}(u,\Omega )=\frac{1}{p}{K}_{d,p}{\int }_{\Omega }|\nabla u(x){|}^{p}dx,$$

$$I_{\delta ,p}(u,\Omega )\le C_{d,p}{\int }_{{\mathbb{R}}^N}|\nabla u(x){|}^{p}dx\forall u\in W^{1,p}(\Omega ),$$

$$\underset{\delta \to 0}{\mathrm{liminf}}{I}_{\delta ,1}(u,\Omega )\ge K_{d,1}{\int }_{\Omega }|\nabla u|dx$$

$$K_{1,1}{\int }_0^1|\nabla u|dx=\underset{\delta \to 0}{\mathrm{liminf}}{I}_{\delta ,1}(u,(0,1))<\underset{\delta \to 0}{\mathrm{limsup}}{I}_{\delta ,1}(u,(0,1))=+\infty .$$

$$\Big{(}\int_{|u|>\lambda\delta}|u|^{q}\Big{)}^{1/q}\leq C\Big{(}I_{\delta,p}(u,\mathbb{R}^{d})\Big{)}^{1/p},$$

$${\int }_B{\int }_B|u(x)-u(y){|}^{p}dxdy\le C_{d,p}\left(|B{|}^{\frac{d+p}{d}}{I}_{\delta ,p}(u,B)+{\delta }^p|B{|}^2\right).$$

$$\|u\|_{BMO(\Omega)}\leq C_{\Omega}\Big{(}I_{\delta,p}(u,\Omega)+\delta^{d}\Big{)},$$

$$\Vert u{\Vert }_{BMO(\Omega )}:=\underset{\mathrm{ball}B\subset \Omega }{\sup }\frac{1}{|B|}{\int }_B|u-u_B|dx.$$

$$|O|:=\text{mbox}meas(O)\text{mbox}and{u}_O={\text{fint}}_{O}udx\text{mbox}with{\text{fint}}_{O}udx:=\frac{1}{|O|}{\int }_{O}udx.$$

$${\text{fint}}_B{e}^{c|u-u_B|/\Vert u{\Vert }_{BMO(B)}}\le C,$$

$$\underset{\Vert u{\Vert }_{W_0^{1,d}(\Omega )\le 1}}{\sup }{\int }_{\Omega }{e}^{\alpha |u{|}^{d/(d-1)}}\le C,$$

$$\underset{|B{|}^{\frac{p-d}{d}}{\delta }^{-p}{I}_{\delta ,p}(u,B)\le M}{\sup }{\text{fint}}_B{e}^{\alpha {\left(\frac{p}{d}\right)}^{\beta {\delta }^{-1}|u-u_B|}}dx\le C,$$

$$\underset{|B{|}^{\frac{p-d}{d}}{\delta }^{-p}{I}_{\delta ,p}(u,B)\le M_0}{\sup }{\text{fint}}_B{e}^{\alpha {\left(\frac{p}{d}\right)}^{{\delta }^{-1}|u-u_B|}}dx\le C.$$

$$\underset{{\delta }^{-p}{I}_{\delta ,p}(u,\Omega )\le M}{\sup }{\int }_{\Omega }{e}^{\alpha {\left(\frac{p}{d}\right)}^{\beta {\delta }^{-1}|u|}}dx\le C_{\Omega }{e}^{\alpha {\left(\frac{p}{d}\right)}^{\beta {\delta }^{-1}\Vert u{\Vert }_{L^1(\Omega )}}},$$

$$\underset{{\delta }^{-p}{I}_{\delta ,p}(u,\Omega )\le M_0}{\sup }{\int }_{\Omega }{e}^{\alpha {\left(\frac{p}{d}\right)}^{{\delta }^{-1}|u|}}dx\le C_{\Omega }{e}^{\alpha {\left(\frac{p}{d}\right)}^{{\delta }^{-1}\Vert u{\Vert }_{L^1(\Omega )}}}.$$

$$\underset{{\delta }^{-d}{I}_{\delta ,p}(u,B)\le M_0}{\sup }{\text{fint}}_B{e}^{\alpha {\delta }^{-1}|u-u_B|}dx\le C,$$

$$\Vert u{\Vert }_{BMO(B)}\le C(M+\delta ),$$

$$\underset{{\delta }^{-p}{I}_{\delta ,p}(u,\Omega )\le M_0}{\sup }{\int }_{\Omega }{e}^{\alpha {\delta }^{-1}|u|}dx\le C_{\Omega }{e}^{\alpha {\delta }^{-1}\Vert u{\Vert }_{L^1(\Omega )}},$$

$$I_{\delta ,p}(u,B)\le M\text{mbox}and{\int }_B{e}^{\alpha {\gamma }^{{\delta }^{-1}|u-u_B|}}dx=+\infty .$$

$$\lim_{n\to+\infty}I_{\delta,p}(u_{n},B)=0\quad\mbox{ and }\quad\lim_{n\to+\infty}\int_{B}e^{\alpha\big{(}\delta^{-1}|u_{n}-{u_{n}}_{B}|\big{)}^{\gamma}}dx=+\infty.$$

$${\int }_{F\setminus E}\frac{dy}{{\left|x-y\right|}^{\lambda }}\ge C_{d,\lambda }{\left|E\right|}^{1-\frac{\lambda }{d}},$$

$${\int }_D{\int }_{F\setminus E}\frac{dydx}{{\left|x-y\right|}^{d+p}}\ge C_{d,p}|D|{\left|E\right|}^{-\frac{p}{d}},$$

$$\displaystyle B_{\rho}(y)=\big{(}B_{\rho}(y)\setminus E\big{)}\cup\big{(}B_{\rho}(y)\cap E\big{)}\text{ and }E=\big{(}E\setminus B_{\rho}(y)\big{)}\cup\big{(}E\cap B_{\rho}(y)\big{)}.$$

Full text

[TABLE]

for some $\delta>0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincaré inequalities, and Hardy inequalities.

1. Introduction

Let $(\rho_{n})$ be a sequence of non-negative radial functions satisfying

[TABLE]

Set

[TABLE]

for some $e\in\mathbb{S}^{d-1}$, the unit sphere in $\mathbb{R}^{d}$.

Jean Bourgain, Haim Brezis, and Petru Mironescu [10, Theorems 1 and 2] (see also [11] and [8]) proved the following BBM formula:

Proposition 1.1**.**

Let $d\geq 1$, $p>1$ and let $\Omega$ be a smooth bounded open subset in $\mathbb{R}^{d}$ or $\Omega=\mathbb{R}^{d}$. Assume that $g\in L^{p}(\Omega)$ and let $(\rho_{n})$ satisfy (1.1). Then $g\in W^{1,p}(\mathbb{R}^{d})$ if and only if

[TABLE]

Moreover, for $g\in W^{1,p}(\Omega)$,

[TABLE]

A variant of Propposition 1.1 for $p=1$ involving functions of bounded variations was obtained by Jean Bourgain, Haim Brezis, and Petru Mironescu [10] and Juan Davila [21]. Further studies related to this characterizations can be founded in [2, 6, 16, 17, 20, 25, 35, 38, 39, 42, 43, 44].

We next discuss another characterization of Sobolev spaces in the spirit of the BBM formula. To this end, for $d\geq 1$, $p\geq 1$, and $\delta>0$, for a measurable subset $O$ of $\mathbb{R}^{d}$, and for a measurable function $u$ defined in $O$, set

[TABLE]

This quantity has its root in estimates for the topological degree in [13, 12, 29, 34, 45] which has the motivation from the study of the Ginzburg Landau equation [9].

It was shown [28, Theorems 2 and 5] and [7, Theorem 1] that

Proposition 1.2**.**

Let $d\geq 1$ and $\Omega$ be a smooth bounded open subset in $\mathbb{R}^{d}$ or $\Omega=\mathbb{R}^{d}$ and let $p>1$ and $g\in L^{p}(\Omega)$. Then $u\in W^{1,p}(\Omega)$ if and only if

[TABLE]

Moreover, for $g\in W^{1,p}(\Omega)$,

[TABLE]

where $K_{d,p}$ is defined by (1.2). We also have, for all $\delta>0$,

[TABLE]

for some positive constant $C_{d,p}$ depending only on $d$ and $p$.

The case $p=1$ is more delicate. One has [28, Theorem 8] (see also [18, Proposition 2]), for $u\in W^{1,1}(\Omega)$,

[TABLE]

and (see [28, Theorem 8] and [7, Theorem 1]) that $u\in BV(\Omega)$ provided that $u\in L^{1}(\Omega)$ and $\liminf_{\delta\to 0}I_{\delta,1}(u,\Omega)<+\infty$. Let $B_{r}$ denote the ball centered at 0 and of radius $r$. An example due to Augusto Ponce presented in [28] showed that there exists $u\in W^{1,1}(B_{1})$ such that $\lim_{\delta\to 0}I_{\delta,1}(u,B_{1})=+\infty$. When $d=1$, there exists $u\in W^{1,1}(0,1)$ [18, Pathology 2] such that

[TABLE]

It turns out that the concept of $\Gamma$-convergence fits very well this setting. It was shown [30, 32] that the $\Gamma$-limit exists for $p\geq 1$. Surprisingly, the $\Gamma$-limit, which is positive, is strictly less than the pointwise limit [32, 30]. The quantity $I_{\delta,1}$ has a similar form with non-local filters using in denoising process [19], in particular with Yaroslavsky’s ones [47, 48]. A discussion on a connection between nonlocal filters using $I_{\delta,1}$ and local ones involving the total variations via the $\Gamma$-convergence theory is given in [18, Section 5.2]. Further interesting investigations related to the $\Gamma$-limit of $I_{\delta,p}$ are given in [3, 4, 5, 18].

One can obtain new and improved variants of Poincaré’s inequality, Sobolev’s inequality and Rellich-Kondarachov’s compactness criterion using the information of $I_{\delta,p}$ instead of the one of the gradient [33, Theorems 1, 2, and 3]. Concerning the Sobolev inequality, one has

Proposition 1.3**.**

Let $1<p<d$ and set $q=dp/(d-p)$ and fix $\delta>0$ arbitrary. We have, for $u\in L^{p}(\mathbb{R}^{d})$,

[TABLE]

for some positive constants $\lambda$ and $C$ independent of $u$.

Concerning the Poincaré inequality, one obtains

Proposition 1.4**.**

Let $d\geq 1$, $p\geq 1$, $\delta>0$, let $B$ be an open ball of $\mathbb{R}^{d}$, and let $u\in L^{p}(B)$. There exists a positive constant $C_{d,p}$ depending only on $d$ and $p$ such that

[TABLE]

The proof of Sobolev’s inequality (1.6) is based on the one of Poincaré’s inequality (1.7) and uses the theory of sharp functions due to Charles Fefferman and Elias Stein [23] and the method of truncation due to Vladimir Mazya [26]. The proof of Poincaré’s inequality (1.7) has its roots in [7] and uses John-Nirenberg’s inequality [24].

Remark 1.1**.**

For a measurable function defined in $B$, by applying (1.7) for $u_{k}$ with

u_{k}=\min\big{\{}k,\max\{u,-k\}\big{\}} and letting $k\to+\infty$, one also obtains (1.7) for measurable functions.

With Marco Squassina, the second author also established new and improved variants of Hardy and Caffarelli, Kohn, Nirenberg’s inequality [35] using the quantity $I_{\delta,p}$. The approach used in [35] does not involve the integration-by-parts arguments and can be extended for the fractional Sobolev spaces [36]. Other investigations related to $I_{\delta,p}$ can be found in [14, 18, 31, 33, 36, 37, 38].

Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^{d}$ and $p\geq d$. It follows from (1.7) that $u\in BMO(\Omega)$ provided that $u\in L^{1}(\Omega)$ and $I_{\delta,p}(u,\Omega)<+\infty$. More precisely, one has, for $p\geq d$,

[TABLE]

where

[TABLE]

Here, for a given a measurable set $O$ of $\mathbb{R}^{d}$ and a function $u\in L^{1}(O)$, one sets

[TABLE]

One can then derive the exponential integrability of $u$ from John-Nirenberg’s inequality:

[TABLE]

for some positive constant $c$ and $C$ depending only on $d$ and for any open ball $B$.

Using the Poincaré inequality, one can prove that $u\in W^{1,p}(\Omega)$ then $u\in BMO(\Omega)$, this yields the exponential integrability of $u$ in (1.10). In fact, for $u\in W^{1,p}(\Omega)$ with $p\geq d$, one can improve (1.10). First, Morrey’s inequality (see, e.g., [15]) states that $u\in C^{\alpha}(\Omega)$ with $\alpha=1-d/p$ if $u\in W_{0}^{1,p}(\Omega)$ for $p>d$. Second, Moser-Trudinger’s inequality [27, 46, 40, 41] confirms that

[TABLE]

for some positive constants $\alpha$ and $C$ depending only on $\Omega$.

The goal of this paper is to understand whether or not a better integrability property of $u$ than (1.10) inequality holds when $u\in L^{p}(\Omega)$ and $I_{\delta,p}(u,\Omega)<+\infty$. It is worth noting that, for all $\delta>0$, there exists $u\in L^{\infty}(\Omega)\setminus C(\bar{\Omega})$ such that $I_{\delta,p}(u,\Omega)=0$ for all $p\geq 1$. A simple example is the function $u=\delta\mathds{1}_{B}\mbox{ in }\Omega,$ for some ball $B\Subset\Omega$, where $\mathds{1}_{O}$ denotes the characteristic function of a subset $O$ of $\mathbb{R}^{d}$. One can also show that there exists a function $u$ such that $I_{\delta}(u,\Omega)<+\infty$ and $u\not\in L^{\infty}(\Omega)$. An example for this is the function $u(x)=\left(\ln\lambda\right)^{-1}\ln\ln|x|^{-1}$ for $x\in B_{1/e}$ and $\lambda>p/d$ (the verification is given in Section 3).

In this work, we address the gap between the exponential integrability (1.10) and the boundedness for functions $u$ with $I_{\delta,p}(u,\Omega)<+\infty$ for some $\delta>0$ and $p\geq d$. Our first result is

Theorem 1.1**.**

Let $p>d\geq 1$, $\delta>0$, and let $B$ be a an open ball of $\mathbb{R}^{d}$. We have,

$i)$* for $M>0$ and $\alpha>0$, there exists a constant $0\leq\beta=\beta(\alpha,M)\leq 1$ depending only on $M$ and $\alpha$ such that*

[TABLE]

$ii)$* given $\alpha>0$, there exists a positive constant $M_{0}$ (small) depending only on $\alpha$, $d$, and $p$ such that*

[TABLE]

Here $C$ denotes a positive constant depending only on $d$, $p$, and $\alpha$.

As a consequence of Theorem 1.1, we obtain

Proposition 1.5**.**

Let $p>d\geq 1$, $\delta>0$, and let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^{d}$. We have,

$i)$* for $M>0$ and $\alpha>0$, there exists a constant $0\leq\beta=\beta(\alpha,M)\leq 1$ depending only on $\alpha$ and $M$ such that*

[TABLE]

$ii)$* given $\alpha>0$, there exists a positive constant $M_{0}$ (small) depending only on $\alpha$, $d$, $p$, and $\Omega$ such that*

[TABLE]

Here $C_{\Omega}$ denotes a positive constant depending only on $d$, $p$, $\alpha$, and $\Omega$.

Here is a variant of $ii)$ of Theorem 1.1.

Theorem 1.2**.**

Let $p=d\geq 1$, $\delta>0$, and let $B$ be a an open ball of $\mathbb{R}^{d}$. Given $\alpha>0$, there exists a positive constant $M_{0}$ (small) depending only on $\alpha$ and $d$ such that

[TABLE]

*for some positive constant $C$ depending only on $d$ and $\alpha$. *

Remark 1.2**.**

Inequality (1.13) shares some similarities with John-Nirenberg’s inequality but is different. In fact, fixing $\delta>0$, as a consequence of (1.7), we have

[TABLE]

if $I_{\delta,p}(u)\leq M$. Hence $\|u\|_{BMO(B)}$ does not generally converge to [math] and (1.13) cannot be derived from (1.10).

As a consequence of Theorem 1.2, we have

Proposition 1.6**.**

Let $p=d\geq 1$, $\delta>0$, and let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^{d}$. Given $\alpha>0$, there exists a positive constant $M_{0}$ (small) depending only on $\alpha$, $d$, and $\Omega$ such that

[TABLE]

for some positive constant $C_{\Omega}$ depending only on $d$, $\alpha$, and $\Omega$.

The exponential growths in (1.12) and (1.13) are optimal. In fact, we have

Proposition 1.7**.**

Let $p\geq d\geq 1$, $\gamma>p/d$, and $\alpha>0$, and let $B$ be a an open ball of $\mathbb{R}^{d}$.

i) If $p>d$ then for any $M>0$ there exists $u\in L^{1}(B)$ such that

[TABLE]

i) If $p=d$ then there exists a bounded sequence $(u_{n})\subset L^{1}(B)$ such that

[TABLE]

2. Proofs of Theorems 1.1 and 1.2

This section contains the proof of the Theorems 1.1 and 1.2. We first establish two lemmas used in the proof of (1.11), (1.12), and (1.13) and then establish Theorems 1.1 and 1.2.

2.1. Two useful lemmas

For $x\in\mathbb{R}^{d}$ and $\rho>0$, let $B_{\rho}(x)$ denote the ball in $\mathbb{R}^{d}$ centered at $x$ and of radius $\rho$. We have

Lemma 2.1**.**

Let $d\geq 1$, $\lambda>0$, and let $E\subset F\subset\mathbb{R}^{d}$ be measurable subsets of $\mathbb{R}^{d}$ with $0<\left|E\right|<\left|F\right|<\infty$. Let $\rho>0$ be such that $|E|=|B_{\rho}|$ and let $x\in\mathbb{R}^{d}$ be such that $B_{2\rho}(x)\subset F$. Then

[TABLE]

for some positive constant $C_{d,\lambda}$ depending only on $d$ and $\lambda$. As a consequence, if $p\geq 1$, $|E|=|B_{\rho}|$ for some $\rho>0$ and, for $p\geq 1$, $D$ is measurable subset of $F$ such that for almost every $x\in D$, the ball $B_{2\rho}(x)\subset F$, then

[TABLE]

for some positive constant $C_{d,p}$ depending only on $d$ and $p$.

Proof.

For $y\in\mathbb{R}^{d}$, we have

[TABLE]

Since $|E|=\left|B_{\rho}(y)\right|$, it follows that

[TABLE]

Fix $x$ such that $B_{2\rho}(x)\subset F$. We have

[TABLE]

Then

[TABLE]

Combining (2.1) and (2.5) yields

[TABLE]

This yields

[TABLE]

which is (2.1).

Integrating (2.1) w.r.t. $x$ in $D$, we obtain (2.2). ∎

Remark 2.1**.**

A similar version of inequality (2.1) has played crucial roles in deriving fractional versions of Sobolev [22] and Hardy [1] inequalities.

The following simple lemma is also used in the proof of Theorem 1.1.

Lemma 2.2**.**

Let $d\geq 1$, $p>1$, $\delta>0$, and let $O$ be a ball in $\mathbb{R}^{d}$. Let $g\in L^{1}_{loc}(O)$. We have, $k\in\mathbb{N}_{+}$,

[TABLE]

Proof.

By considering the function $u/\delta$ and by the recurrence, it suffices to consider the case $\delta=1$ and $k=1$. We have

[TABLE]

By a change of variables $z=x/2+y/2$, we obtain

[TABLE]

which yields the conclusion for $\delta=1$ and $k=1$. ∎

2.2. Proof of part i) of Theorem 1.1.

In this proof, for notational ease, we denote $I_{\delta,p}$ by $I_{\delta}$ for $\delta>0$. Without loss of generality we can assume $B=B_{1}$, $u_{B}=0$, and $\delta=1$. Define $\tilde{u}$ in $B_{3/2}$ by

[TABLE]

We have, for all $\tau>0$,

[TABLE]

and, see e.g., [18, Lemma 17],

[TABLE]

Using John-Nirenberg’s inequality, we have

[TABLE]

if $\ell\geq c_{1}M$ for some $c_{1}>0$.

We claim that, for $\ell\geq c_{1}M$ and $\lambda>2$,

[TABLE]

In fact, fix an arbitrary $x\in B_{5/4}$ and let $\rho$ be such that |B_{\rho}(x)|=\big{|}\big{\{}x\in B_{3/2};|u|\geq(\lambda-1)\ell\big{\}}\big{|}. Since $\lambda>2$, it follows from (2.9) that $\rho<1/8$, which yields $B_{2\rho}(x)\subset B_{3/2}$. Applying Lemma 2.1 with $D=\{x\in B_{5/4};|\tilde{u}|\geq\lambda\ell\}\cap O,$ $E=\{x\in B_{3/2};|\tilde{u}|\geq(\lambda-1)\ell\}$ and $F=B_{3/2}$, and using (2.7) and (2.8), we obtain (2.10).

Applying Lemma 2.2, we have, for $k\in\mathbb{N}$,

[TABLE]

Fix $k_{0}$ be such that for $k\geq k_{0}$, one has $c_{2}2^{-k(p-1)}M\leq e^{-2\alpha}$, which yields

[TABLE]

Set

[TABLE]

Then, for some $c_{3}$ larger than $c_{2}$,

[TABLE]

Using (2.10), (2.11), and a standard iterative process, we have, for $\lambda\in\mathbb{N}$ and

[TABLE]

This implies

[TABLE]

This implies the conclusion of part $i)$ with $\beta(\alpha,M)=\ell_{0}^{-1}$ where $\ell_{0}$ is given by (2.12). ∎

2.3. Proof of part $ii)$ of Theorem 1.1.

The proof of part $ii)$ is in the spirit of part $i)$. In fact, noting that if $M_{0}$ is small enough then (2.13) holds with $\ell_{0}=1$. The conclusion then follows. ∎

2.4. Proof of (1.13) of Theorem 1.2.

The proof is similar to the one of part $ii)$ of Theorem 1.1 and is omitted. ∎

2.5. Proof of Propositions 1.5 and 1.6

Propositions 1.5 and 1.6 can be derived from Theorems 1.1 and 1.2 respectively after using local charts and appropriately extending $u$ in a neighborhood of $\Omega$ (see, e.g., [18, Lemma 17]. The details are omitted. ∎

3. Proof of Proposition 1.7

Without loss of generality, one might assume that $B=B_{1/e}$ and $\delta=1$.

Proof of assertion (1.14). Fix $\gamma>\lambda>p/d>1$, set, for $x\in B_{1/e}$,

[TABLE]

It is clear that $g\in L^{1}(I)$. Using polar coordinates, we have

[TABLE]

We have, for $0<r_{1}<r_{2}<e^{-1}$,

[TABLE]

this yields

[TABLE]

for some positive constant $C$ depending only on $d$, $p$, and $\lambda$. It follows that, for $0<r_{1}<r_{2}<e^{-1}$ and $|g(r_{1})-g(r_{2})|>1$,

[TABLE]

We derive from (3) and (3.2) that

[TABLE]

We have

[TABLE]

since $r_{1}^{1/\lambda}-r_{1}\geq Cr_{1}^{1/\lambda}$ and $e^{-1}-r_{1}\geq C$ for $r_{1}\in(0,e^{-\lambda})$.

On the other hand, for any $\tau\in I$, we have, with $\rho=\frac{\ln\gamma}{\ln\lambda}-1$,

[TABLE]

since \lim_{r\to 0_{+}}\big{(}\log r^{-1}\big{)}^{1+\rho}/\log r^{-1}=+\infty.

Set, for $0<\tau<e^{-1}$,

[TABLE]

Then

[TABLE]

Combining (3.5) and (3.6) yields the conclusion since for any $M>0$ we can choose $\tau>0$ small enough so that $I_{1,p}(u_{\tau},B_{e^{-1}})\leq M$. ∎

Proof of assertion (1.15). Let $n\in\mathbb{N}$ large and fix $1<q<\gamma$ and denote $q^{\prime}=q/(q-1)$. Define

[TABLE]

As in (3.3), we have

[TABLE]

where $I=(0,1/e)$.

We now estimate $I_{1,1}(g_{n},I)$. Denote $J_{n}=(0,1/n)$, and $K_{n}=I\setminus J_{n}$. We have

[TABLE]

where

[TABLE]

We next estimate $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$. We begin with $\mathcal{I}_{1}$. For $(r_{1},r_{2})\in J_{n}\times K_{n}$, we have $|g_{n}(r_{1})-g_{n}(r_{2})|>1$ if and only if

[TABLE]

It follows that

[TABLE]

We next deal with $\mathcal{I}_{2}$. For $(r_{1},r_{2})\in K_{n}\times K_{n}$ with $r_{2}\geq r_{1}$, we have $|g_{n}(r_{1})-g_{n}(r_{2})|>1$ if and only if

[TABLE]

We then have

[TABLE]

Combining (3.8), (3.9), and (3.10) yields

[TABLE]

which yields, by (3.7),

[TABLE]

We have

[TABLE]

This implies

[TABLE]

On the other hand, since $\gamma>q$, we have

[TABLE]

The conclusion now follows from (3.11), (3.12), and (3.13). ∎