Variable Exponent Lebesgue Spaces Research Articles

Weighted Rellich and Hardy inequalities in L p(·) spaces

Abstract In this note we prove weighted Rellich–Sobolev and Hardy–Sobolev inequalities in variable exponent Lebesgue spaces L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} defined on stratified homogeneous groups 𝔾 {\mathbb{G}} . To derive the main results, we rely on weighted estimates for the Riesz potential operators in L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} , where 𝔾 {{\mathbb{G}}} is a general homogeneous group. The results are new even for the Abelian (Euclidean) case 𝔾 = ( ℝ d , + ) {\mathbb{G}=(\mathbb{R}^{d},+)} and the Heisenberg groups 𝔾 = ℍ n {\mathbb{G}={\mathbb{H}}^{n}} . The main statements are obtained for variable exponents satisfying the condition that the Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue spaces. We also give some quantitative estimates for the norms of integral operators involved in derived estimates.

Existence of solutions for singular elliptic equations with mixed boundary conditions

In this article, we investigate generalized solutions for elliptic equations involving double-phase ( p ( x ) , q ( x ) ) -Laplacian operators with Hardy potential in variable exponent spaces. ( p ( x ) , q ( x ) ) -Laplacian operators include p-Laplacian, q-Laplacian, p ( x ) -Laplacian and q ( x ) -Laplacian operators. Additionally, ( p ( x ) , q ( x ) ) -Laplacian elliptic equations with singular coefficients under mixed boundary conditions are seldom mentioned in previous work. Some new theorems of the existence on the generalized solutions are reestablished for such equations via variational methods when the nonlinearity satisfies suitable hypotheses in variable exponent Lebesgue spaces.

On Weighted Cauchy-Type Problem of Riemann-Liouville Fractional Differential Equations in Lebesgue Spaces with Variable Exponent

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space $L^{p(.)}$. The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the $L^{p(.)}$ is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results.

Direct Estimates of the Rate of Approximation by the Kantorovich Operator in Variable Exponent Lebesgue Spaces

Direct Estimates of the Rate of Approximation by the Kantorovich Operator in Variable Exponent Lebesgue Spaces

A characterization of the rate of approximation of Kantorovich sampling operators in variable exponent Lebesgue spaces

A characterization of the rate of approximation of Kantorovich sampling operators in variable exponent Lebesgue spaces

P(x)-biharmonic equations involving Leray–Lions type operators with Hardy potentials

In this article, we investigate p ( x ) -biharmonic equations involving Leray–Lions type operators with Hardy potentials. New criteria for the existence of at least one or at least three generalized solutions are established via variational methods when the Leray–Lions type operator and the nonlinearity f satisfying suitable hypotheses in variable-exponent Lebesgue spaces.

EXTRAPOLATION IN SOME MIXED-NORMED FUNCTION SPACES

Abstract. From the beginning of this century non-standard function spaces attracted a considerable attention of researchers. One of such spaces is variable exponent Lebesgue space (Nakano space) 𝐿𝑝(.); another one is grand Lebesgue space 𝐿𝑝0). The main reason to study these spaces was to solve a number of contemporary problems arising naturally in Applied Mathematics and PDEs (see, e.g., [1-5]). Mixed norm function spaces play an important role in Harmonic Analysis and PDEs (see, e.g. the survey by L. Huang and D. Yang (2019), papers by N. V. Krylov.

WEIGHTED VARIABLE EXPONENT LEBESGUE SPACES ON A PROBABILITY SPACE

In this paper, we introduce the weighted variable exponent Lebesgue spaces defined on a probability space and give some information about the martingale theory of these spaces. We first prove several basic inequalities for expectation operators and obtain several norm convergence conditions for martingales in weighted variable exponent Lebesgue spaces. We discuss the Hölder inequality and embedding properties in these spaces. Finally, under some conditions we investigate Doob’s maximal function.

On measure of noncompactness in variable exponent Lebesgue spaces and applications to integral equations

A novel measure of noncompactness is defined in variable exponent Lebesgue spaces Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )}$\\end{document} on an unbounded domain R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{+}$\\end{document} and its properties are examined. Using the fixed point method, we apply that measure to study the existence theorem for nonlinear integral equations. Our results can be handily applied in studying various types of (differential, integral, functional, and partial differential) equations in Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )}$\\end{document}-spaces. The Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )} $\\end{document}-spaces are natural extensions of classical constant exponent Lebesgue spaces Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{p}$\\end{document}, which allows us to bypass several restrictions that were previously discussed in the literature.

Examining Nonlinear Fredholm Equations in Lebesgue Spaces with Variable Exponents

We investigate the existence of solutions for the Fredholm integral equation Φ(ϑ)=G(ϑ,Φ(ϑ))+∫01F(ϑ,ζ,Φ(ζ))dζ, for ϑ∈[0,1], in the setting of the modular function spaces Lρ. We also derive an application of this research within the framework of variable exponent Lebesgue spaces Lp(·) subject to specific conditions imposed on the exponent function p(·) and the functions F and G.

Leray–Lions type [formula omitted]-biharmonic equations involving Hardy potentials

In this article, we investigate p(x)-biharmonic equations involving Leray–Lions type operators and Hardy potentials. Leray–Lions type operators include p-Laplacian, p-Laplacian-like, p(x)-Laplacian, (p,q)-Laplacian, (p(x),q(x))-Laplacian operators and so forth. Moreover, fourth-order Leray–Lions type elliptic equations with variable exponents are seldom mentioned in previous papers. Some new theorems of the existence on the generalized solutions are reestablished for such equations when the Leray–Lions type operator and the nonlinearity satisfy suitable hypotheses in variable exponent Lebesgue spaces.

Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces

AbstractThe main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.

Local grand variable exponent Lebesgue spaces

We introduce local grand variable exponent Lebesgue spaces, where the variable exponent Lebesgue space is “aggrandized” only at a given closed set $F$ of measure zero. It is shown that, for different aggrandizers with positive Matuszewska–Orlicz indices, the corresponding local grand variable exponent Lebesgue spaces coincide. We show that the maximal operator, singular operators, and maximal singular operators are bounded in such spaces. Lastly, an application to a Dirichlet problem for the Poisson equation, where $F$ may be chosen as the boundary of the domain, is provided within the framework of such local grand spaces.

Stability of p(·)-Integrable Solutions for Fractional Boundary Value Problem via Piecewise Constant Functions

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (Lp(·)). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers are established. Our results are obtained using two fixed-point theorems, then illustrating the results with a comprehensive example.

Terminal value problem for Riemann‐Liouville fractional differential equation in the variable exponent Lebesgue space Lp(.)$$ {L}^{p(.)} $$

In this manuscript, the existence, uniqueness, and stability of solutions to the terminal value problem of Riemann‐Liouville fractional equations are established in the variable exponent Lebesgue spaces . We convert the variable exponent Lebesgue spaces to the Lebesgue spaces using the generalized intervals and piece‐wise constant function. Further, the Banach contraction principle is used, the Ulam‐Hyers‐stability is examined, and finally, we construct an example.

Dual descent regularization algorithms in variable exponent Lebesgue spaces for imaging

Dual descent regularization algorithms in variable exponent Lebesgue spaces for imaging

On approximation properties of functions by means of Fourier and Faber series in weighted Lebesgue spaces with variable exponent

In this paper the approximation of functions by linear means of Fourier series in weighted variable exponent Lebesgue spaces was studied. This result was applied to the approximation of the functions by linear means of Faber series in Smirnov classes with variable exponent defined on simply connected domain of the complex plane.

The infimum eigenvalue for degenerate p(x)-biharmonic operator with the Hardy potentiel

The aim of this article is to study the existence of at least one unbounded nondecreasing sequence of nonnegative eigenvalues (λk)k≥1 for a class of elliptic Navier boundary value problems involving the degenerate p(·)-biharmonic operator with q(x)-Hardy inequality by using the variational technique based on the Ljusternik-Schnirelmann theory on C1-manifolds and the theory of the variable exponent Lebesgue spaces. Also, we obtain the positivity of the infimum eigenvalue for the problem.

Modular-Proximal Gradient Algorithms in Variable Exponent Lebesgue Spaces

We consider structured optimisation problems defined in terms of the sum of a smooth and convex function, and a proper, l.s.c., convex (typically non-smooth) one in reflexive variable exponent Lebesgue spaces $L_{p(\cdot)}(\Omega)$. Due to their intrinsic space-variant properties, such spaces can be naturally used as solution space and combined with space-variant functionals for the solution of ill-posed inverse problems. For this purpose, we propose and analyse two instances (primal and dual) of proximal gradient algorithms in $L_{p(\cdot)}(\Omega)$, where the proximal step, rather than depending on the natural (non-separable) $L_{p(\cdot)}(\Omega)$ norm, is defined in terms of its modular function, which, thanks to its separability, allows for the efficient computation of algorithmic iterates. Convergence in function values is proved for both algorithms, with convergence rates depending on problem/space smoothness. To show the effectiveness of the proposed modelling, some numerical tests highlighting the flexibility of the space $L_{p(\cdot)}(\Omega)$ are shown for exemplar deconvolution and mixed noise removal problems. Finally, a numerical verification on the convergence speed and computational costs of both algorithms in comparison with analogous ones defined in standard $L_{p}(\Omega)$ spaces is presented.

Density of smooth functions in Musielak–Orlicz spaces

We provide necessary and sufficient conditions for the space of smooth functions with compact supports $$C^\infty _c(\Omega )$$ to be dense in Musielak–Orlicz spaces $$L^\Phi (\Omega )$$ where $$\Omega $$ is an open subset of $${\mathbb {R}}^d$$ . In particular, we prove that if $$\Phi $$ satisfies condition $$\Delta _2$$ , the closure of $$C^\infty _c(\Omega )\cap L^\Phi (\Omega )$$ is equal to $$L^\Phi (\Omega )$$ if and only if the measure of singular points of $$\Phi $$ is equal to zero. This extends the earlier density theorems proved under the assumption of local integrability of $$\Phi $$ , which implies that the measure of the singular points of $$\Phi $$ 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.