A Brezis-Lieb-type Lemma in Orlicz space

TL;DR

This paper extends the Brezis-Lieb Lemma to Orlicz spaces, broadening its applicability in functional analysis and variational problems.

Contribution

The paper introduces a new version of the Brezis-Lieb Lemma specifically adapted for Orlicz spaces, which were not previously covered.

Findings

Established a Brezis-Lieb-type lemma in Orlicz spaces

Provided new tools for analysis in Orlicz space settings

Enhanced understanding of convergence properties in Orlicz spaces

Abstract

In this work, we extend the well known Brezis-Lieb Lemma to the Orlicz space.

Equations

$$\mathcal{F}(u_n)-\mathcal{F}(u_n-u)\to \mathcal{F}(u)\text{mbox}in{L}^1(\Omega ),\text{mbox}asn\to +\infty .$$

$$G(2x)\le KG(x)\text{mbox}forallx\in {\mathbb{R}}_{+}.$$

$$G^{\ast }(a)=\sup \{at-G(t),t>0\}.$$

$$ab\le G(a)+G^{\ast }(b)\text{mbox}foreverya,b\ge 0.$$

$$\frac{G^{{}^{\prime }}(a)}{G(a)}\le \frac{p}{a},\forall a>0,$$

$$\underset{n\to +\infty }{\lim }{\int }_{\Omega }G(|u_n|)dx-{\int }_{\Omega }G(|u_n-u|)dx={\int }_{\Omega }G(|u|)dx.$$

$$g(a)b\le C(ϵG(a)+G(\frac{b}{ϵ}))\text{mbox}foreverya,b\ge 0,$$

$$G(a+b)$$

$$\le \frac{K}{2}(G(a)+G(b)).$$

$$G^{\ast }(g(t))\le (p-1)G(t),\forall t>0,$$

$$G^{\ast }(t)=g(t)t-G(t).$$

$$g(a)b$$

$$\le ϵG^{\ast }(g(a))+G(\frac{b}{ϵ})$$

$$\le ϵ(p-1)G(a)+G(\frac{b}{ϵ})$$

$$\le \max ((p-1),1)(ϵG(a)+G(\frac{b}{ϵ})).$$

$$G(|u_n|)=G(|u_n-u|)+g(\xi )(|u_n|-|u_n-u|),$$

$$|G(|u_n|)-G(|u_n-u|)|$$

$$\le g(|u_n|+|u_n-u|)|u|$$

$$\le g(|u|+2|u_n-u|)|u|.$$

$$g(2|u_n-u|+|u|)|u|$$

$$\le \frac{C{K}^2}{2}(ϵG(|u_n-u|)+G(|u|)+G(\frac{|u|}{ϵ}))$$

$$\le \frac{C{K}^2}{2}(ϵG(|u_n-u|)+2G(\frac{|u|}{ϵ})).$$

$$|G(|u_n|)-G(|u_n-u|)|\le \frac{C{K}^2}{2}(ϵG(|u_n-u|)+2G(\frac{|u|}{ϵ})).$$

$$W_{ϵ,n}=[|G(|u_n|)-G(|u_n-u|)-G(|u|)|-ϵ\frac{C{K}^2}{2}G(|u_n-u|){]}_{+},$$

$$|G(|u_n|)-G(|u_n-u|)-G(|u|)|$$

$$\le (C{K}^2+1)G(\frac{|u|}{ϵ})+ϵ\frac{C{K}^2}{2}G(|u_n-u|).$$

$$W_{ϵ,n}x\le (C{K}^2+1)G(\frac{|u|}{ϵ})\in L^1(\Omega ).$$

$${\int }_{\Omega }|G(|u_n|)-G(|u_n-u|)-G(|u|)|dx\le {\int }_{\Omega }(W_{ϵ,n}(x)+ϵ\frac{C{K}^2}{2}G(|u_n-u|))dx,$$

$$\underset{n\to +\infty }{\mathrm{limsup}}{\int }_{\Omega }|G(|u_n|)-G(|u_n-u|)-G(|u|)|dx\le ϵC^{{}^{\prime }}K,$$

Full text

aa 2010 AMS Subject Classification: Primary 46E30, Secondary 46G05, 58C20

1. Introduction and main result

The Brezis-Lieb lemma, see [1], has major applications mainly in calculus of variations, see [2, 3]. To describe a special case of this theorem, suppose that $\Omega\subset\mathbb{R}^{N}$ is a domain, $p>1$, $f(t)=|t|^{p}$ for $t\in\mathbb{R}$, $(u_{n})$ a bounded sequence in $L^{p}(\Omega)$ and $u_{n}\rightarrow u\in L^{p}(\Omega)$ a.e. on $\Omega$. If one denotes by $\mathcal{F}:L^{p}(\Omega)\rightarrow L^{1}(\Omega)$ superposition operator f generates, i.e., $\mathcal{F}(v)(x):=f((v)(x))$, then $u\in L^{p}(\Omega)$ and

[TABLE]

The main purpose of this note is to give a similar result in the Orlicz space.

To introduce our main result more precisely, we first give some basic definitions and properties concerning Orlicz space. We start by recalling the definition of the well-known Orlicz functions.

Definition 1.1**.**

$(1)$* $G:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is called an Orlicz function if it has the following properties:

$(H_{1})$ $G$ is continuous, convex, increasing and $G(0)=0$.

$(H_{2})$ $G$ satisfies the $\Delta_{2}-$condition, that is there exists $K>2$ such that*

[TABLE]

$(H_{3})$* $\displaystyle\lim_{x\rightarrow 0}\frac{G(x)}{x}=0.$

$(2)$ given $G$ an Orlicz function, we define the complementary function $G^{\ast}$ as*

[TABLE]

Remark 1.2**.**

$(i)$From the above definition it is immediate that the following Young-type inequality holds

[TABLE]

$(ii)$* It is shown in [4] that the $\Delta_{2}$ condition is equivalent to*

[TABLE]

for some $p>1$.

Now, we are ready to state our main result.

Theorem 1.3**.**

*Let $\Omega\subset\mathbb{R}^{N}$ be an open domain and $G$ be an Orlicz function. Moreover, we assume that $G$ is of class $C^{1}$ on $\mathbb{R}_{+}.$ Suppose that $(u_{n})$ is bounded in $L^{G}(\Omega)$

and $u_{n}\rightarrow u\in L^{G}(\Omega)$ a.e. on $\Omega$. Then*

[TABLE]

2. Proof of Theorem 1.3

First, we prove the following technical lemma.

Lemma 2.1**.**

*Let $G$ be an Orlicz function and $G^{\ast}$ his complementary. Then:

$(i)$ $G(a+b)\leq\frac{K}{2}(G(a)+G(b))$ for every $a,b\geq 0.$

$(ii)$ There is $C>0$ such that for every $\epsilon\in(0,1)$ we have*

[TABLE]

where $g=G^{{}^{\prime}}.$

Proof.

$(i)$ Invoking conditions $(H_{1})$ and $(H_{2})$, we deduce that

[TABLE]

$(ii)$ In what follows we show that

[TABLE]

where $p>1$ is given by (1.2). Fix $t>0$. Let $h(a):=g(t)a-G(a)$. Then, it is easy to see that $h(a)\leq h(t)$ for every $a>0$. Therefore

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Combining the above identity with (1.2) we prove (2.1). Let $a,b>0$ and $\epsilon\in(0,1)$. Then, by (1.1) and (2.1), we infer that

[TABLE]

This ends the proof. ∎

Proof of Theorem 1.3 completed:

Using the Taylor formula, for any fixed $x\in\Omega$, we have

[TABLE]

where $\xi$ is a measurable function with values between $|u_{n}(x)|$ and $|u_{n}(x)-u(x)|$. Therefore

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By Lemma 2.1, for fixed $\epsilon\in(0,1)$, we get

[TABLE]

It follows, using Lemma 2.1 and (2.2) , that

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Let

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where $[a]_{+}=\max(a,0).$ As $n\rightarrow+\infty$, $W_{\epsilon,n}(x)\rightarrow 0$ a.e. on $\Omega$. On the other hand

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Therefore

[TABLE]

Then, by Lebesgue convergence theorem, $\displaystyle\int_{\Omega}W_{\epsilon,n}(x)dx\rightarrow 0$ as $n\rightarrow+\infty.$ From the fact that

[TABLE]

we deduce that

[TABLE]

for some positive constant $C^{{}^{\prime}}$. Letting $\epsilon\rightarrow 0$ we complete the proof.