Orlicz Spaces Research Articles

Relevant sampling in a reproducing kernel subspace of Orlicz space

In this paper, we aim to provide a general paradigm for dealing with the sampling and random sampling problem in a reproducing kernel subspace of Orlicz space [Formula: see text]. We consider the function space [Formula: see text] as the image of an idempotent integral operator on [Formula: see text], where the integral kernel satisfies certain off-diagonal decay and regularity conditions. The model example of such reproducing kernel subspace of [Formula: see text] includes the finitely generated shift-invariant space and signal space with a finite rate of innovation. We show that a signal in [Formula: see text] can be stably reconstructed from its samples at distinct points separated by a sufficiently small gap. Next, we deduce that the random sampling inequality holds with a high probability for the class of functions in [Formula: see text] concentrated on a cube [Formula: see text], when the samples collected at i.i.d. random points are drawn on [Formula: see text] of order [Formula: see text].

Lower Local Uniform Monotonicity in F-Normed Musielak–Orlicz Spaces

Lower strict monotonicity points and lower local uniform monotonicity points are considered in the case of Musielak–Orlicz function spaces LΦ endowed with the Mazur–Orlicz F-norm. The findings outlined in this study extend the scope of geometric characteristics observed in F-normed Orlicz spaces, as well as monotonicity properties within specific F-normed lattices. They are suitable for the Orlicz spaces of ordered continuous elements, specifically in relation to the Mazur–Orlicz F-norm. In addition, in this paper presents results that can be used to derive certain monotonicity properties in F-normed Musielak–Orlicz spaces.

On a class of fractional Γ(.)-Kirchhoff-Schrödinger system type

This paper focuses on the investigation of a Kirchhoff-Schrödinger type elliptic system involving a fractional \(\gamma(.)\)-Laplacian operator. The primary objective is to establish the existence of weak solutions for this system within the framework of fractional Orlicz-Sobolev Spaces. To achieve this, we employ the critical point approach and direct variational principle, which allow us to demonstrate the existence of such solutions. The utilization of fractional Orlicz-Sobolev spaces is essential for handling the nonlinearity of the problem, making it a powerful tool for the analysis. The results presented herein contribute to a deeper understanding of the behavior of this type of elliptic system and provide a foundation for further research in related areas.

On the Boundedness of the Fractional Maximal Operator, the Riesz Potential, and Their Commutators in Orlicz Spaces

On the Boundedness of the Fractional Maximal Operator, the Riesz Potential, and Their Commutators in Orlicz Spaces

Convolutional Neural Networks Approximation in Quasi-Orlicz Spaces on Sphers

It is necessary to study the theoretical bases of an approximation deep convolutional neural networks, because of its interesting developments in vital domains. The approximation abilities of deep-convolution neural networks produced by downsampling operators in quasi- Orlicz spaces have been studied, since this space is wider and more important than other spaces. In this paper, we define quasi-Orlicz norm on spherical spaces. In addition, modulus of smoothness is also studied in terms of quasi-Orlicz norm. Finally, Function approximation theorems are studied by using convolution neural networks with

New product type operators acting between weighted Bergman–Orlicz spaces and some weighted-type spaces

New product type operators acting between weighted Bergman–Orlicz spaces and some weighted-type spaces

On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces

AbstractThis paper is concerned with a parabolic evolution equation of the form , settled in a smooth bounded domain of , , and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the ‐Laplacian for suitable , the “variable‐exponent” ‐Laplacian, or even some fractional order operators. The operator is assumed to be in the form with being measurable in and maximal monotone in . The main results are devoted to proving existence of weak solutions for a wide class of functions that extends the setting considered in previous results related to the variable exponent case where . To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so‐called ‐type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators , ) to which the result can be applied.

Brezis–Van Schaftingen–Yung formula in Orlicz spaces

In [13], Brezis–Van Schaftingen–Yung discovered new expression of the Sobolev semi-norm by replacing the Lp(R2N) norm in the Gagliardo Ws,p(RN) semi-norm with weak Lp(R2N) quasi-norm and setting s=1. In this paper we prove the same formula for functions in Orlicz–Sobolev spaces, in which the role of the power function t↦tp is performed by more general finite valued Young functions.

Weak and strong type inequalities for Orlicz spaces

We revisit the generalisation of Calderon’s Transfer Principle as espoused in [7]. This principle is used to generate weak type maximal inequalities for ergodic operators in the setting of σ-compact locally compact Hausdorff groups acting measure-preservingly on σ-finite measure spaces. In particular we develop a much more robust protocol for transferring weak and strong type inequalities from Orlicz spaces in the group setting to Orlicz spaces in the measure space setting. This is an important addition to the protocol developed in [7], which to date has only yielded weak type inequalities. The current approach also places fewer restrictions on the underlying Young functions describing the Orlicz spaces involved.

Density properties for Orlicz Sobolev spaces with fractional order

Density properties for Orlicz Sobolev spaces with fractional order

Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp-Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp(Rd) and a time weighted Lr(Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space.

Duality for vector-valued Bergman–Orlicz spaces and little Hankel operators between vector-valued Bergman–Orlicz spaces on the unit ball

Duality for vector-valued Bergman–Orlicz spaces and little Hankel operators between vector-valued Bergman–Orlicz spaces on the unit ball

Connection Between Weighted Tail, Orlicz, Grand Lorentz And Grand Lebesgue Norms

We prove that the norm of functions in a suitable Grand Lorentz space built on a measure space, equipped with sigma finite diffuse measure, coincides with the norm in a suitable exponential Grand Lebesgue Space space as well as coincides with the so-called exponential tail norm, which may be quite described as norm in a suitable Banach rearrangement invariant space. We also exhibit comparisons with exponential Orlicz norms.

Density of smooth functions in Musielak–Orlicz–Sobolev spaces Wk,Φ(Ω)$W^{k,\Phi }(\Omega)$

AbstractWe consider here Musielak–Orlicz–Sobolev (MOS) spaces , where is an open subset of , and is a Musielak–Orlicz function. The main outcomes consist of the results on density of the space of compactly supported smooth functions in . One section is devoted to compare the various conditions on appearing in the literature in the context of maximal operator and density theorems in MOS spaces. The assumptions on we apply here are substantially weaker than in the earlier papers on the topics of approximation by smooth functions. 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 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 , (A1) and that are not sufficient for the maximal operator to be bounded, the space of is dense in . In the case of variable exponent Sobolev space , we obtain the similar density result under the assumption that , , , , satisfies the log‐Hölder condition and the exponent function is essentially bounded.

Existence and regularity results for nonlinear elliptic equations in Orlicz spaces

We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on (x,u,∇u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x,u,\ abla u)$$\\end{document}, and with a convective term f. The assumptions on the members of the equation are formulated in terms of Young’s functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples.

Weighted Hessian estimates in Orlicz spaces for nondivergence elliptic operators with certain potentials

We prove interior weighted Hessian estimates in Orlicz spaces for nondivergence type elliptic equations with a lower order term which involves a nonnegative potential satisfying a reverse Hölder type condition.

On the Fenchel–Moreau conjugate of G-function and the second derivative of the modular in anisotropic Orlicz spaces

On the Fenchel–Moreau conjugate of G-function and the second derivative of the modular in anisotropic Orlicz spaces

On topological degree for pseudomonotone operators in fractional Orlicz-Sobolev spaces: study of positive solutions of non-local elliptic problems

On topological degree for pseudomonotone operators in fractional Orlicz-Sobolev spaces: study of positive solutions of non-local elliptic problems

New homogenization results for convex integral functionals and their Euler–Lagrange equations

We study stochastic homogenization for convex integral functionals u↦∫DW(ω,xε,∇u)dx,whereu:D⊂Rd→Rm,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u\\mapsto \\int _D W(\\omega ,\ frac{x}{\\varepsilon },\ abla u)\\,\ extrm{d}x,\\quad \ ext{ where }\\quad u:D\\subset {\\mathbb {R}}^d\\rightarrow {\\mathbb {R}}^m, \\end{aligned}$$\\end{document}defined on Sobolev spaces. Assuming only stochastic integrability of the map ω↦W(ω,0,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega \\mapsto W(\\omega ,0,\\xi )$$\\end{document}, we prove homogenization results under two different sets of assumptions, namely ∙1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bullet _1$$\\end{document}W satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate W∗(·,0,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^*(\\cdot ,0,\\xi )$$\\end{document} and a certain monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of u,∙2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bullet _2$$\\end{document}W is p-coercive in the sense |ξ|p≤W(ω,x,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\xi |^p\\le W(\\omega ,x,\\xi )$$\\end{document} for some p>d-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>d-1$$\\end{document}. Condition ∙2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bullet _2$$\\end{document} directly improves upon earlier results, where p-coercivity with p>d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>d$$\\end{document} is assumed and ∙1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bullet _1$$\\end{document} provides an alternative condition under very weak coercivity assumptions and additional structure conditions on the integrand. We also study the corresponding Euler–Lagrange equations in the setting of Sobolev-Orlicz spaces. In particular, if W(ω,x,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W(\\omega ,x,\\xi )$$\\end{document} is comparable to W(ω,x,-ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W(\\omega ,x,-\\xi )$$\\end{document} in a suitable sense, we show that the homogenized integrand is differentiable.

Kadec–Klee property with respect to the local convergence in measure of Orlicz–Lorentz spaces

AbstractIn this paper, we find criteria for the Kadec–Klee property with respect to the local convergence in measure in both Orlicz–Lorentz spaces as well as their subspaces of order continuous elements. In the case of Orlicz norm, the presented results are new, whereas in the case of Luxemburg norm, we rely heavily on known results, which we show for the first time as a whole. Finally, we apply the obtained results to Orlicz spaces.