# Uniform Convexity in Variable Exponent Sobolev Spaces
## 4 Modular Uniform Convexity in Variable Exponent Lebesgue-Sobolev Spaces  
The class of variable exponent Lebesgue spaces was first introduced in 1931 [15]. The interested reader can consult [2; 3; 9] for an exhaustive treatment of these spaces. This section will focus on the modular uniform convexity properties of such spaces. We open the section with standard definitions. **Definition 5**. : _Let \(\Omega\subset\mathbb{R}^{n}\) be a domain. The notation \(\mathcal{M}(\Omega)\) will be used for the vector space of all real-valued, Borel-measurable functions defined on \(\Omega\). Let \(\mathcal{P}(\Omega)\) be the subset of \(\mathcal{M}\) consisting of functions \(p:\Omega\longrightarrow[1,\infty]\). For each such \(p\), define the set \(\Omega_{\infty}:=\{x\in\Omega:p(x)=\infty\}\). The function \(\varrho:\mathcal{M}(\Omega)\longrightarrow[0,\infty]\), defined by_  
\[\varrho(u)=\int\limits_{\Omega\setminus\Omega_{\infty}}|u(x)|^{p(x)}d\mu+ \sup\limits_{x\in\Omega_{\infty}}|u(x)|,\]  
_is a convex and continuous modular on \(\mathcal{M}(\Omega)\). The associated modular vector space is denoted by \(L^{p(\cdot)}(\Omega)\) or simply \(L^{p}(\Omega)\) if no confusion arises._  
**Definition 6**. : _For \(\Omega\) and \(p\) as in the preceding definition, \(W^{1,p}(\Omega)\) will stand for the vector subspace of \(L^{p}(\Omega)\) consisting of functions whose weak derivatives also belong to \(L^{p}(\Omega)\). The Sobolev space \(W^{1,p}(\Omega)\) will be equipped with the convex modular \(\rho:W^{1,p}(\Omega)\rightarrow[0,\infty]\) defined as:_\[\rho(u)=\varrho(u)+\varrho(|\
abla u|)=\varrho(u)+\varrho\left(\left(\sum\limits_{j= 1}^{n}\left(\frac{\partial u}{\partial x_{j}}\right)^{2}\right)^{\frac{1}{2}}\right)\]  
_and the corresponding Luxemburg norm introduced in Definition 2 will be denoted by \(\|\cdot\|_{\rho}\)._  
The following result follows easily from Theorem 1. **Corollary 1**. : _Set \(\Omega_{1}=\{x\in\Omega:p(x)\geq 2\}\). For \(u\in W^{1,p}(\Omega)\), \(v\in W^{1,p}(\Omega)\) it holds that_  
\[\int\limits_{\Omega_{1}}\left|\frac{\
abla(u+v)}{2}\right|^{p}dx+\int\limits _{\Omega_{1}}\left|\frac{\
abla(u-v)}{2}\right|^{p}dx\leq\frac{1}{2}\left( \int\limits_{\Omega_{1}}|\
abla u|^{p}\,dx+\int\limits_{\Omega_{1}}|\
abla v| ^{p}\,dx\right)\]  
_and_  
\[\int\limits_{\Omega\setminus\Omega_{1}}\left|\frac{\
abla(u+v)}{2 }\right|^{p}dx+\int\limits_{\Omega\setminus\Omega_{1}}\frac{p(p-1)|\
abla(u- v)|^{2}}{2^{p+1}(|\
abla u|^{2}+|\
abla v|^{2})^{\frac{p-2}{p}}}\,dx\leq\frac{1}{2} \int\limits_{\Omega\setminus\Omega_{1}}|\
abla u|^{p}\,dx\] \[+\ \frac{1}{2}\int\limits_{\Omega\setminus\Omega_{1}}|\
abla v| ^{p}\,dx.\]  
The next result will be crucial to establish the main result of this work. **Theorem 2**. : _Let \(\Omega\subseteq\mathbb{R}^{n}\) be a domain and \(p\in\mathcal{P}(\Omega)\) finite a.e. The functional \(\varrho:W^{1,p}(\Omega)\to[0,\infty]\) defined by_  
\[\varrho(\mathbf{u})=\int\limits_{\Omega}\left(\sum\limits_{1}^{n}u_{j}^{2} \right)^{\frac{p(x)}{2}}dx\]  
_(here \(u_{j}\) denotes the \(j^{th}\) partial derivative of \(u\)) is a convex pseudomodular (i.e., it has all the properties exhibited in Definition 1, except \((1)\)) and is \((UC^{*})\) provided \(p_{-}>1\). Moreover, when restricted to \(W^{1,p}_{0}(\Omega)\), \(\varrho\) is \((UC^{*})\) when \(p_{-}>1\)._  
Proof. : This proof follows along the same lines as that of Theorem 3 in [6]. We provide the details in the interest of completeness. It is obvious that \(\varrho\) is a convex modular on \(L^{p}(\Omega)\). In the course of the proof, it will be understood that \(|(x_{1},x_{2},...,x_{n})|=\left(\sum\limits_{j=1}^{n}x_{j}^{2}\right)^{\frac{ 1}{2}}\) and for a subset \(A\subseteq\Omega\), we set \(\varrho_{A}(\mathbf{u})=\int\limits_{A}|\mathbf{u}|^{p}dx\). Let \(\varepsilon\in(0,1]\).