arXiv subjects

Mariusz Żyluk

Publications and source records attributed to Mariusz Żyluk.

5 recordsLinked to original sources

arXiv2024

Density of compactly supported smooth functions $C_C^\infty(\mathbb{R}^d)$ in Musielak-Orlicz-Sobolev spaces $W^{1,Φ}(Ω)$

We investigate here the density of the set of the restrictions from $C_C^\infty(\mathbb{R}^d)$ to $C_C^\infty(Ω)$ in the Musielak-Orlicz-Sobolev space $W^{1,Φ}(Ω)$. It is a continuation of article \cite{KamZyl3}, where we have studied density of $C_C^\infty(\mathbb{R}^d)$ in $W^{k, Φ}(\mathbb{R}^d)$ for $k\in\mathbb{N}$. The main theorem states that for an open subset $Ω\subset \mathbb{R}^d$ with its boundary of class $C^1$, and Musielak-Orlicz function $Φ$ satisfying {\rm condition (A1)} which is a sort of log-Hölder continuity and the growth condition $Δ_2$, the set of restrictions of functions from $C_C^\infty(\mathbb{R}^d) $ to $Ω$ is dense in $W^{1,Φ}(Ω)$. We obtain a corresponding result in variable exponent Sobolev space $W^{1,p(\cdot)}(Ω)$ under the assumption that the exponent $p(x)$ is essentially bounded on $Ω$ and $Φ(x,t) = t^{p(x)}$, $t\ge 0$, $x\inΩ$, satisfies the log-Hölder condition.

math.FA↗

arXiv2023

Density of smooth functions in Musielak-Orlicz-Sobolev spaces $W^{k,Φ}(Ω)$

We consider here Musielak-Orlicz Sobolev (MOS) spaces $W^{k,Φ}(Ω)$, where $Ω$ is an open subset of $\mathbb{R}^d$, $k\in\mathbb{N}$ and $Φ$ is a Musielak-Orlicz function. The main outcomes consist of the results on density of the space of compactly supported smooth functions $C_C^\infty(Ω)$ in $W^{k,Φ}(Ω)$. One section is devoted to compare the various conditions on $Φ$ appearing in the literature in the context of maximal operator and density theorems in MOS spaces. The assumptions on $Φ$ we apply here are substantially weaker than in the earlier papers on the topics of approximation by smooth functions \cite{Ahmida, Has1, Hasbook}. One of the reasons is that in the process of proving density theorems we do not use the fact that the Hardy-Littlewood maximal operator on Musielak-Orlicz space $L^Φ(Ω)$ is bounded, a standard tool employed in density results for different types of Sobolev spaces. We show in particular that under some regularity assumptions on $Φ$, (A1) and $Δ_2$ that are not sufficient for the maximal operator to be bounded, the space of $C_C^\infty(\mathbb{R}^d)$ is dense in $W^{k,Φ}(Ω)$. In the case of variable exponent Sobolev space $W^{k,p(\cdot)}(\mathbb{R}^d)$, we obtain the similar density result under assumption that $Φ(x,t) = t^{p(x)}$, $p(x)\ge 1$, $t\ge 0$, $x\in \mathbb{R}^d$, satisfies the log-Hölder condition and the exponent function $p$ is essentially bounded.

math.FA↗

arXiv2022

Density of smooth functions in Musielak-Orlicz spaces

We provide necessary and sufficient conditions for the space of smooth functions with compact supports $C^\infty_C(Ω)$ to be dense in Musielak-Orlicz spaces $L^Φ(Ω)$ where $Ω$ is an open subset of $\mathbb{R}^d$. In particular we prove that if $Φ$ satisfies condition $Δ_2$, the closure of $C^\infty_C(Ω)\cap L^Φ(Ω)$ is equal to $L^Φ(Ω)$ if and only if the measure of singular points of $Φ$ is equal to zero. This extends the earlier density theorems proved under the assumption of local integrability of $Φ$, which implies that the measure of the singular points of $Φ$ is zero. As a corollary we obtain analogous results for Musielak-Orlicz spaces generated by double phase functional and we recover the well known result for variable exponent Lebesgue spaces.

math.FA↗

arXiv2021

Uniform convexity, reflexivity, supereflexivity and $B$ convexity of generalized Sobolev spaces $W^{1,Φ}$

We investigate Sobolev spaces $W^{1,Φ}$ associated to Musielak-Orlicz spaces $L^Φ$. We first present conditions for the boundedness of the Voltera operator in $L^Φ$. Employing this, we provide necessary and sufficient conditions for $W^{1,Φ}$ to contain isomorphic subspaces to $\ell^\infty$ or $\ell^1$. Further we give necessary and sufficient conditions in terms of the function $Φ$ or its complementary function $Φ^*$ for reflexivity, uniform convexity, $B$-convexity and superreflexivity of $W^{1,Φ}$. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces $W^{1,φ}$ where $φ$ is an Orlicz function, the variable exponent Sobolev spaces $W^{1,p(\cdot)}$ and the Sobolev spaces associated to double phase functionals.

math.FA↗

arXiv2019

Local geometric properties in quasi-normed Orlicz spaces

Several local geometric properties of Orlicz space $L_ϕ$ are presented for an increasing Orlicz function $ϕ$ which is not necessarily convex, and thus $L_ϕ$ does not need to be a Banach space. In addition to monotonicity of $ϕ$ it is supposed that $ϕ(u^{1/p})$ is convex for some $p>0$ which is equivalent to that its lower Matuszewska-Orlicz index $α_ϕ>0$. Such spaces are locally bounded and are equipped with natural quasi-norms. Therefore many local geometric properties typical for Banach spaces can also be studied in those spaces. The techniques however have to be different, since duality theory cannot be applied in this case. In this article we present complete criteria, in terms of growth conditions of $ϕ$, for $L_ϕ$ to have type $0<p\le2$, cotype $q\ge 2$, to be (order) $p$-convex or $q$-concave, to have an upper $p$-estimate or a lower $q$-estimate, for $0<p,q<\infty$. We provide detailed proofs of most results, avoiding appealing to general not necessary theorems.

math.FA↗