Musielak-Orlicz Spaces Research Articles

Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications

The main aim of the paper is to prove a two-weight criterion for the multidimensional Hardy-type operator from weighted Lebesgue spaces into p-convex weighted Banach function spaces. The problem for the dual operator is also considered. As an application, we prove a two-weight criterion of boundedness of the multidimensional geometric mean operator from weighted Lebesgue spaces into weighted Musielak–Orlicz spaces.

Musielak-Orlicz-Sobolev spaces on metric measure spaces

Our aim in this paper is to study Musielak-Orlicz-Sobolev spaces on metric measure spaces. We consider a Hajlasz-type condition and a Newtonian condition. We prove that Lipschitz continuous functions are dense, as well as other basic properties. We study the relationship between these spaces, and discuss the Lebesgue point theorem in these spaces. We also deal with the boundedness of the Hardy-Littlewood maximal operator on Musielak-Orlicz spaces. As an application of the boundedness of the Hardy-Littlewood maximal operator, we establish a generalization of Sobolev’s inequality for Sobolev functions in Musielak-Orlicz-Hajlasz-Sobolev spaces.

Variational inequalities in Musielak-Orlicz-Sobolev spaces

In this paper we prove an existence result for some class of variational boundary value problems for quasilinear elliptic equations in the Musielak-Orlicz spaces. Some results concerning the Trace mapping have also been provided, as well as existence results for some strongly nonlinear elliptic equations in Musielak-Orlicz spaces.

Growth properties of Musielak–Orlicz integral means for Riesz potentials

In this paper we are concerned with growth properties of integral means for Riesz potentials of variable order α of functions in Musielak–Orlicz spaces LΦ(Rn), where Φ is a function of the form Φ(x,t)=tp(x)φ(x,t). Here α and p satisfy the log-Hölder condition. We also discuss the size of the exceptional sets using a capacity introduced in Maeda et al. (2011).

SOME ESTIMATES FOR SCHRÖDINGER TYPE OPERATORS ON MUSIELAK-ORLICZ-HARDY SPACES

Let $L:=-\mathrm{div}(A\nabla)+V$ be a Schrödinger type operator with the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q_0}(\mathbb{R}^n)$ for some $q_0\in[n,\infty)$ with $n\ge3$, where $A$ satisfies the uniformly elliptic condition. Assume that $\varphi:\,\mathbb{R}^n\times[0,\infty)\to[0,\infty)$ is a function such that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index $i(\varphi)\in(\frac{n}{n+\alpha_0},1]$, where $\alpha_0\in(0,1]$ measures the regularity of kernels of the semigroup generalized by $L_0:=-\mathrm{div}(A\nabla)$. In this article, we first prove that operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi,\,L}(\mathbb{R}^n)$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, we also obtain the boundedness of $VL^{-1}$ and $\nabla^2L^{-1}$ on $H_{\varphi,\,L}(\mathbb{R}^n)$. All these results are new even when $\varphi(x,t):=t^p$, with $p\in(\frac{n}{n+\alpha_0},1]$, for all $x\in\mathbb{R}^n$ and $t\in[0,\infty)$.

Equilateral Dimension of Certain Classes of Normed Spaces

The equilateral dimension of a normed space is the maximal number of pairwise equidistant points of this space. The aim of this article is to study the equilateral dimension of certain classes of finite-dimensional normed spaces. A well-known conjecture states that the equilateral dimension of any n-dimensional normed space is not less than n + 1. By using an elementary continuity argument, we establish it in the following classes of spaces: permutation-invariant spaces, Musielak-Orlicz spaces, and one codimensional subspaces of . For smooth and symmetric spaces, Musielak-Orlicz spaces satisfying an additional condition and every (n − 1)-dimensional subspace of we also provide some weaker bounds on the equilateral dimension for every space that is sufficiently close to one of these. This generalizes a result of Swanepoel and Villa concerning the spaces.

Estimates for second-order Riesz transforms associated with magnetic Schrödinger operators on Musielak-Orlicz-Hardy spaces

Let be a magnetic Schrödinger operator on , where and satisfies some reverse Hölder conditions. Assume that is a function such that is an Orlicz function, (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index . In this article, the authors prove that the operators , and are bounded from the Musielak-Orlicz-Hardy space associated with , , to the Musielak-Orlicz space , via establishing some estimates for heat kernels of . All these results are new even when , with , for all and .

Proximinality in Banach space valued Musielak-Orlicz spaces

Let be a σ-finite complete measure space and let Y be a subspace of a Banach space X. Let φ be a generalized Φ-function on . Denote by and the Musielak-Orlicz spaces whose functions take values in Y and X, respectively. Firstly, let , we characterize the distance of f from . Then, if Y is weakly -analytic and proximinal in X, we show that is proximinal in . Finally, we give the connection between the proximinality of in and the proximinality of in .

BOUNDEDNESS FOR FRACTIONAL HARDY-TYPE OPERATOR ON HERZ-MORREY SPACES WITH VARIABLE EXPONENT

In this paper, the fractional Hardy-type operator of variable order <TEX>${\beta}(x)$</TEX> is shown to be bounded from the Herz-Morrey spaces <TEX>$M\dot{K}^{{\alpha},{\lambda}}_{p_1,q_1({\cdot})}(\mathbb{R}^n)$</TEX> with variable exponent <TEX>$q_1(x)$</TEX> into the weighted space <TEX>$M\dot{K}^{{\alpha},{\lambda}}_{p_2,q_2({\cdot})}(\mathbb{R}^n,{\omega})$</TEX>, where <TEX>${\omega}=(1+|x|)^{-{\gamma}(x)}$</TEX> with some <TEX>${\gamma}(x)$</TEX> > 0 and <TEX>$1/q_1(x)-1/q_2(x)={\beta}(x)/n$</TEX> when <TEX>$q_1(x)$</TEX> is not necessarily constant at infinity. It is assumed that the exponent <TEX>$q_1(x)$</TEX> satisfies the logarithmic continuity condition both locally and at infinity that 1 < <TEX>$q_1({\infty}){\leq}q_1(x){\leq}(q_1)+$</TEX> < <TEX>${\infty}(x{\in}\mathbb{R}^n)$</TEX>.

An existence theorem for a strongly nonlinear elliptic problem in Musielak–Orlicz spaces

An existence theorem for a strongly nonlinear elliptic problem in Musielak–Orlicz spaces

Complex Convexity of Musielak-Orlicz Function Spaces Equipped with thep-Amemiya Norm

The complex convexity of Musielak-Orlicz function spaces equipped with thep-Amemiya norm is mainly discussed. It is obtained that, for any Musielak-Orlicz function space equipped with thep-Amemiya norm when1≤p&lt;∞, complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in above spaces are given. Criteria for complex strict convexity and complex midpoint locally uniform convexity of above spaces are also deduced.

Asymptotically isometric copies of c_0 in Musielak-Orlicz spaces

Criteria in order that a Musielak-Orlicz function space \(L^\Phi\) as well as Musielak-Orlicz sequence space \(l^\Phi\) contains an asymptotically isometric copy of \(c_0\) are given. These results extend some results of [Y.A. Cui, H. Hudzik, G. Lewicki, Order asymptotically isometric copies of \(c_0\) in the subspaces of order continuous elements in Orlicz spaces, Journal of Convex Analysis 21 (2014)] to Musielak-Orlicz spaces.

Nonlinear parabolic problems in Musielak–Orlicz spaces

Our studies are directed to the existence of weak solutions to a parabolic problem containing a multi-valued term. The problem is formulated in the language of maximal monotone graphs. We assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of a time and space dependent N-function. This results in the formulation of the problem in generalized Musielak–Orlicz spaces. We are using density arguments, hence an important step of the proof is a uniform boundedness of appropriate convolution operators in Musielak–Orlicz spaces. For this purpose we shall need to assume a kind of logarithmic Hölder regularity with respect to t and x.

On the boundedness of sets in Musielak-Orlicz spaces

The notions of metrical and topological vector boundedness of sets are considered in Musielak-Orlicz spaces. The space is called right if both these notions are equivalent. Necessary conditions of the rightness are established, and certain sufficient conditions are found.

On Nonlinear Differential Equations in Generalized Musielak-Orlicz Spaces

We consider ordinary differential equations \(u'(t)+(I-T)u(t)=0\), where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and \(T\) is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.

Approximate identities and Young type inequalities in Musielak-Orlicz spaces

We discuss the convergence of approximate identities in Musielak-Orlicz spaces extending the results given by Cruz-Uribe and Fiorenza (2007) and the authors F.-Y. Maeda, Y. Mizuta and T. Ohno (2010). As in these papers, we treat the case where the approximate identity is of potential type and the case where the approximate identity is defined by a function of compact support. We also give a Young type inequality for convolution with respect to the norm in Musielak-Orlicz spaces.

Smoothness of the Orlicz norm in Musielak–Orlicz function spaces

In this paper, we present a characterization of support functionals and smooth points in , the Musielak–Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of is also obtained. Some expressions involving the norms of functionals in , the topological dual of , are proved for arbitrary Musielak–Orlicz functions.

Criteria of Two-weighted Inequalities for Multidimensional Hardy Type Operator in Weighted Musielak-Orlicz Spaces and Some Application

In this paper a two-weight criterion for multidimensional Hardy type operator and its dual operator acting from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces is proved. As application we prove the boundedness of multidimensional geometric mean operator in the weighted Musielak-Orlicz spaces. In particular, from obtained results implies the boundedness of multidimensional Hardy operator and its dual operator acting from usual weighted Lebesgue spaces into weighted variable Lebesgue spaces. In this paper we establish integral-type neces- sary and sufficient condition on weights, which provides the boundedness of the multidimensional Hardy type operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.

One-Complemented Subspaces in Musielak-Orlicz Sequence Spaces with a General Smoothness Condition

The aim of this article is to obtain a characterization (in the spirit of Jamison-Kamińska-Lewicki) of one-complemented subspaces in Musielak-Orlicz sequence spaces defined by Musielak-Orlicz functions satisfying a general smoothness condition. The result obtained is an important extension of the Jamison-Kamińska-Lewicki Theorem. Some applications are also discussed.

Trudinger's inequality for Riesz potentials of functions in Musielak–Orlicz spaces

In this paper we are concerned with Trudinger's inequality for Riesz potentials of functions in Musielak–Orlicz spaces.