# Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{th}century these spaces were brought into the center stage of mathematical research as they were realized to be the natural solution space for partial differential equations exhibiting non-standard growth. The first systematic treatment of variable exponent spaces was given in [2]. In 1997, while studying differential equations in electromagnetism, V. Zhikov’s work [3] led to the minimization of integrals of the form

## 2. Modular Spaces

**Definition**

**1.**

- $\rho \left(x\right)=0\iff x=0,$
- $\rho \left(tx\right)=\left|t\right|\rho \left(x\right)$ for any $x\in V$, $\left|t\right|=1,$
- $\rho (tx+(1-t)y)\le {t}^{s}\rho \left(x\right)+{(1-t)}^{s}\rho \left(y\right)$ for all $x,y\in V$ and $t\in (0,1]$.

**Definition**

**2.**

#### Modular Uniform Convexity

**Definition**

**3.**

**Definition**

**4.**

## 3. Lebesgue Spaces with Variable Exponent

**Theorem**

**1.**

## 4. Uniform Convexity

**Theorem**

**2.**

- (i)
- ${L}^{p(\xb7)}\left(\mathsf{\Omega}\right)$ is uniformly convex;
- (ii)
- $1<{p}_{-}\le {p}_{+}<\infty ;$
- (iii)
- The modular ${\rho}_{p}$ satisfies the ${\Delta}_{2}$-condition. More precisely, there exists a positive constant K such that for any $v\in {L}^{p(\xb7)}\left(\mathsf{\Omega}\right)$ it holds that ${\rho}_{p}\left(2v\right)\le K{\rho}_{p}\left(v\right).$

## 5. Modular Uniform Convexity

**Lemma**

**1.**

- (i)
- If $p\ge 2$ [17], it holds that$${\left(\right)}^{\frac{a+b}{2}}p\le \frac{1}{2}\left(\right)open="("\; close=")">{\left|a\right|}^{p}+{\left|b\right|}^{p}$$
- (ii)
- If $1<p\le 2$ and $\left|a\right|+\left|b\right|\ne 0$ [20], then$${\left(\right)}^{\frac{a+b}{2}}p{\left(\right)}^{\frac{a-b}{2}}p$$

**Theorem**

**3.**

**Remark**

**1.**

**Proof**

**of**

**Theorem**

**3.**

## 6. Applications

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**5.**

**Theorem**

**6.**

**Proof**

**of**

**Theorem**

**6.**

**Definition**

**5.**

**Theorem**

**7.**

**Proof**

**of**

**Theorem**

**7.**

**Theorem**

**8.**

**Proof**

**of**

**Theorem**

**8.**

**Theorem**

**9.**

**Proof**

**of**

**Theorem**

**9.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Bachar, M.; Mendez, O.; Bounkhel, M.
Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability. *Symmetry* **2018**, *10*, 708.
https://doi.org/10.3390/sym10120708

**AMA Style**

Bachar M, Mendez O, Bounkhel M.
Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability. *Symmetry*. 2018; 10(12):708.
https://doi.org/10.3390/sym10120708

**Chicago/Turabian Style**

Bachar, Mostafa, Osvaldo Mendez, and Messaoud Bounkhel.
2018. "Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability" *Symmetry* 10, no. 12: 708.
https://doi.org/10.3390/sym10120708