# 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$${\left|\frac{a+b}{2}\right|}^{p}+{\left|\frac{a-b}{2}\right|}^{p}\le \frac{1}{2}\left({\left|a\right|}^{p}+{\left|b\right|}^{p}\right).$$
- (ii)
- If $1<p\le 2$ and $\left|a\right|+\left|b\right|\ne 0$ [20], then$${\left|\frac{a+b}{2}\right|}^{p}+\frac{p(p-1)}{2}\phantom{\rule{4pt}{0ex}}{\left|\frac{a-b}{\left|a\right|+\left|b\right|}\right|}^{2-p}{\left|\frac{a-b}{2}\right|}^{p}\le \frac{1}{2}\left({\left|a\right|}^{p}+{\left|b\right|}^{p}\right).$$

**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

- Orlicz, W. Über konjugierte Exponentenfolgen. Stud. Math.
**1931**, 3, 200–211. [Google Scholar] [CrossRef] - Kováčik, O.; Rákosník, J. On spaces L
^{p(x)},W^{k,p(x)}. Czechoslov. Math. J.**1991**, 41, 592–618. [Google Scholar] - Zhikov, V.V. On some variational problems. Rus. J. Math. Phys.
**1997**, 5, 105–116. [Google Scholar] - Rajagopal, K.; Ruzicka, M. On the modeling of electrorheological materials. Mech. Res. Commun.
**1996**, 23, s401–s407. [Google Scholar] [CrossRef] - Ruzicka, M. Electrorheological Fluids: Modeling and Mathematical Theory; Lecture Notes in Mathematics 1748; Springer: Berlin, Germany, 2000. [Google Scholar]
- Bansevicius, R.; Virbalis, J.A. Two-dimensional Braille readers based on electrorheological fluid valves controlled by electric field. Mechatronics
**2007**, 17, 570–577. [Google Scholar] [CrossRef] - Chen, J.Z.; Liao, W.H. Design, testing and control of a magnetorheological actuator for assistive knee braces. Smart Mater. Struct.
**2010**, 19, 035029. [Google Scholar] [CrossRef] - Choi, S.H.; Kim, S.; Kim, P.; Park, J.; Choi, S.B. A new visual feedback-based magnetorheological haptic master for robot-assisted minimally invasive surgery. Smart Mater. Struct.
**2015**, 24, 065015. [Google Scholar] [CrossRef] - Spencer, B.; Yang, G.; Carlson, J.; Sain, M. “Smart” Dampers for Seismic Protection of Structures: A Full-Scale Study. In Proceedings of the Second World Conference on Structural Control, Kyoto, Japan, 28 June–1 July 1998. [Google Scholar]
- Khamsi, M.A.; Kozlowski, W. Fixed Point Theory in Modular Function Spaces; Birkäuser: Basel, Switzerland, 2015. [Google Scholar]
- Musielak, J. Orlicz Spaces and Modular Spaces; Lecture Notes in Mathematics, No. 1034; Springer: Berlin/Heidelberg, Germany, 1983. [Google Scholar]
- Nakano, H. Topology of Linear Topological Spaces; Maruzen: Tokyo, Japan, 1951. [Google Scholar]
- Diening, L.; Harjulehto, P.; Hästö, P.; Růžička, M. Lebesgue and Sobolev Spaces with Variable Exponents; Lecture Notes in Mathematics; Springer: Heidelberg, Germany, 2011; Volume 2017. [Google Scholar]
- Edmunds, D.E.; Lang, J.; Méndez, O. Differential Operators on Spaces of Variable Integrability; World Scientific: Singapore, 2014. [Google Scholar]
- Méndez, O.; Lang, J. Analysis on Function Spaces of Musielak–Orlicz Type; Taylor and Francis: Didcot, UK, 2018; in press. [Google Scholar]
- Nakano, H. Modulared Semi-Ordered Linear Spaces; Maruzen: Tokyo, Japan, 1950. [Google Scholar]
- Clarkson, J. Uniformly Convex Spaces. Trans. Am. Math. Soc.
**1936**, 40, 396–414. [Google Scholar] [CrossRef] - James, R. Uniformly non-square Banach spaces. Ann. Math.
**1964**, 80, 542–550. [Google Scholar] [CrossRef] - Lukeš, J.; Pick, L.; Pokorný, D. On geometric properties of the spaces L
^{p(x)}(Ω). Rev. Mat. Complut.**2011**, 24, 115–130. [Google Scholar] [CrossRef] - Sundaresan, K. Uniform convexity of Banach spaces ℓ({p
_{i}}). Stud. Math.**1971**, 39, 227–231. [Google Scholar] [CrossRef]

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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