## Electronic Theses and Dissertations

#### Date

2021

#### Date of Award

2021

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Committee Chair

Anna Kamininska

#### Abstract

\ind \.Zyluk, Mariusz Ph.D. The University of Memphis. May 2021. On density of smooth functions in Musielak Orlicz Sobolev spaces and uniform convexity. Major Professor: Anna Kami\'nnska, Ph.D.\ind We study density of smooth functions in Musialka Orlicz Sobolev $W^{k,\Phi}$ spaces as well as uniform convexity in those spaces.\ind We start with studying Musielak Orlicz spaces $L^\Phi$ and their basic properties. We improve some basic results by dropping the assumption of local integrability. Next, we characterize the density of compact supported smooth functions ($C_C^\infty$) in a Musielak Orlicz spaces $L^\Phi$ generated by a functions $\Phi$ that are not necessarily locally integrable.\ind We proceed to study Musielak Orlicz Sobolev spaces $W^{k,\Phi}$. We start with characterizing those Musielak Orlicz spaces that embed into the space of locally integrable functions. We introduce the Musielak Orlicz Sobolev Space $W^{k,\Phi}$ and state some basic properties of those spaces. We connect the problem of density of $C_C^\infty$ in $W^{k,\Phi}$ with the problem of boundedness of the Hardy-Litllewood maximal operator $\mathcal{M} $ on $L^{\Phi}$. We analyze the (A) conditions on the Musielak Orlicz function and compare them with other known conditions on Musielak Orlicz functions that are connected to the problem of density. We use the (A) condition to establish a fundamental convolution inequality in $L^\Phi$. We use this result to prove results about approximation of compactly supported elements of $W^{k,\Phi}$ by elements of $C_C^\infty$. We prove density of $C_C^\infty(\R^d)$ in $W^{k,\Phi}(\R^d)$. We then proceed to prove density in $W^{1,\Phi}(\Omega)$ of restrictions of compactly supported smooth functions on $\R^d$.\ind Finally we study uniform convexity of $W^{1,\Phi}$. We establish sufficient conditions for boundedness of integral operators, in particular Voltera operator on $L^\Phi$. We discuss the $\Delta_2$ condition for double phase Musielak Orlicz functions and investigate the boundedness of the Voltera operator on $L^\Phi$ spaces generated by those functions. We study existence of isomorphic copies of $\ell^\infty$ and $\ell^1$ in $W^{1,\Phi}$. Next, we provide the necessarily and sufficient conditions for $W^{1,\Phi}$ to be reflexive. We characterize uniform convexity of $W^{1,\Phi}$ for those spaces where the Voltera operator is bounded.

#### Library Comment

Dissertation or thesis originally submitted to ProQuest

#### Recommended Citation

Żyluk, Mariusz, "ON DENSITY OF SMOOTH FUNCTIONS IN MUSIELAK ORLICZ SOBOLEVSPACES AND UNIFORM CONVEXITY OF THOSE SPACES" (2021). *Electronic Theses and Dissertations*. 2847.

https://digitalcommons.memphis.edu/etd/2847

## Comments

Data is provided by the student.