# Uniform Convexity in Variable Exponent Sobolev Spaces
## 3 Variable Exponent Spaces  
It is by today's standards abundantly clear that the normed space structure is much too stringent to completely capture certain mathematical subtleties that are only visible under a more flexible lens. To name an example (in fact, it may be the most important to understand the aim of this work), the variable exponent \(p\)-Laplacian is modular in nature. With this in mind, we set out to a present brief summary of definitions and known results. The reader is referred to [2; 3; 9; 10] for a more detailed discussion of the topics briefly outlined in this section. Definition 1: [10; 11; 12; 13] A convex modular on a real vector space \(X\) is a function \(\varrho:X\to[0,\infty]\) satisfying the following conditions:  
(1) \(\varrho(x)=0\) if and only if \(x=0\);  
(2) \(\varrho(\alpha x)=\varrho(x)\), if \(|\alpha|=1\);  
(3) \(\varrho(\alpha x+(1-\alpha)y)\leq\alpha\varrho(x)+(1-\alpha)\varrho(y)\), for any \(\alpha\in[0,1]\) and any \(x,y\in X\). Furthermore, it is considered that \(\varrho\) exhibits left-continuity when, for all \(x\in X\),  
\[\lim_{r\to 1^{-}}\ \varrho(rx)=\varrho(x).\]  
A modular function defined on a vector space \(X\) naturally gives rise to a modular space. Definition 2: When a convex modular function \(\varrho\) is defined on the vector space \(X\), the resulting modular space consists of the following set:  
\[X_{\varrho}=\{x\in X;\ \lim_{\alpha\to 0}\ \varrho(\alpha x)=0\}.\]  
The Luxemburg norm, denoted as \(\|\cdot\|\varrho\) and defined on the vector space \(X_{\varrho}\), is given by the following expression:  
\[\left\|x\right\|_{\varrho}:=\inf\Bigl{\{}\alpha>0;\ \varrho\left(\frac{x}{ \alpha}\right)\leq 1\Bigr{\}}.\]  
In preparation for the next section, a concept related to the geometry of modular spaces is introduced [14]. Specifically, for \(x\in X\) and \(r>0\), it is a natural and, as shall be seen in Section 5, relevant question, whether the modular ball \(\{y\in X:\rho(x-y)<r\}\) is uniformly convex in the modular sense. Though an exhaustive discussion of this subject is beyond the scope of this work [2; 10], the following type of uniform convexity introduced in [14] will have far reaching consequences in the applications to be discussed in Section 5. Notice that one can routinely verify that Definition 3 generalizes the idea of the norm-uniform convexity of a ball in a normed space. **Definition 3** ([14]). : _Given a modular \(\varrho\) on a vector space \(X\), we introduce the following uniform convexity-type properties of \(\varrho\):_  
_(a) Let_ \(r>0\) _and_ \(\varepsilon>0\) _be given. Define_  
\[D_{2}(r,\varepsilon):=\bigg{\{}(x,y)\text{; }x,y\in X_{\varrho},\varrho(x) \leq r,\varrho(y)\leq r,\varrho\bigg{(}\frac{x-y}{2}\bigg{)}\geq\varepsilon r \bigg{\}}.\]  
_Set_  
\[\delta_{2}(r,\varepsilon)=\inf\biggl{\{}1-\frac{1}{r}\;\varrho\bigg{(}\frac{x +y}{2}\bigg{)}\text{; }(x,y)\in D_{2}(r,\varepsilon)\bigg{\}}\]  
_if_ \(D_{2}(r,\varepsilon)\
eq\emptyset\) _and write_ \(\delta_{2}(r,\varepsilon)=1\) _otherwise._ \(\varrho\) _is said to satisfy_ \((\text{UC2})\) _if for every_ \(r>0\) _and_ \(\varepsilon>0\)_, one has_ \(\delta_{2}(r,\varepsilon)>0.\) _Observe that given_ \(r>0\)_,_ \(\varepsilon>0\) _can be chosen small enough so that_ \(D_{2}(r,\varepsilon)\
eq\emptyset\)_._ _(b)_ \(\varrho\) _is said to satisfy_ \((\text{UUC2})\) _if for every_ \(s\geq 0\) _and_ \(\varepsilon>0\)_, there exists_ \(\eta_{2}(s,\varepsilon)>0\) _depending on_ \(s\) _and_ \(\varepsilon\) _such that_  
\[\delta_{2}(r,\varepsilon)>\eta_{2}(s,\varepsilon)>0\;for\;r>s.\]  
**Definition 4** ([2]).