Orlicz Spaces Research Articles

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.

Solvability of product of $ n $-quadratic Hadamard-type fractional integral equations in Orlicz spaces

<abstract><p>The current study demonstrated and studied the existence of monotonic solutions, as well as the uniqueness of the solutions for a general and abstract form of a product of $ n $-quadratic fractional integral equations of Hadamard-type in Orlicz spaces $ L_\varphi $. We utilized the analysis of the measure of non-compactness associated with Darbo's fixed-point theorem and fractional calculus to obtain the results.</p></abstract>

Transference and Restriction of Bilinear Fourier Multipliers on Orlicz Spaces

Let G be a locally compact abelian group with Haar measure mG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_G$$\\end{document} and let Φi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi _i$$\\end{document}, i=1,2,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1,2,3$$\\end{document}, be Young functions. A bounded measurable function m on G×G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\ imes G$$\\end{document} is a (Φ1,Φ2;Φ3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Phi _1,\\Phi _2;\\Phi _3)$$\\end{document}-bilinear multiplier if there exists C>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C>0$$\\end{document} such that the bilinear map Bm(f,g)(γ)=∫G∫Gm(x,y)f^(x)g^(y)γ(x+y)dmG(x)dmG(y),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B_m (f,g)(\\gamma )= \\int _{G}\\int _{G} m(x,y) {{\\hat{f}}}(x) {{\\hat{g}}}(y) \\gamma (x+y) dm_G(x)dm_G(y), \\end{aligned}$$\\end{document}satisfies NΦ3(Bm(f,g))≤CNΦ1(f)NΦ2(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_{\\Phi _3}(B_m(f,g))\\le CN_{\\Phi _1}(f)N_{\\Phi _2}(g)$$\\end{document} for functions in f,g∈L1(G^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f,g\\in L^1({{\\hat{G}}})$$\\end{document} such that f^,g^∈L1(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\hat{f}}},{{\\hat{g}}}\\in L^1(G)$$\\end{document}. We denote by BM(Φ1,Φ2;Φ3)(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}{\\mathcal {M}}_{(\\Phi _1,\\Phi _2;\\Phi _3)}(G)$$\\end{document} the space of all bilinear multipliers on G×G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\ imes G$$\\end{document} and study some properties of this class. We consider (Φ1,Φ2;Φ3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Phi _1,\\Phi _2;\\Phi _3)$$\\end{document}-bilinear multipliers on various groups such as R×R,D×D,Z×Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}\ imes {\\mathbb {R}},\\, \ extbf{D}\ imes \ extbf{D},\\, {\\mathbb {Z}}\ imes {\\mathbb {Z}}$$\\end{document} and T×T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}\ imes {\\mathbb {T}}$$\\end{document}. In particular we prove, under certain assumptions involving the norm of the dilation operator on the Orlicz spaces, that regulated bilinear multipliers in BM(Φ1,Φ2;Φ3)(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}{\\mathcal {M}}_{(\\Phi _1,\\Phi _2;\\Phi _3)}({\\mathbb {R}})$$\\end{document} coincide with BM(Φ1,Φ2;Φ3)(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}{\\mathcal {M}}_{(\\Phi _1,\\Phi _2;\\Phi _3)}(\ extbf{D})$$\\end{document} with where D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{D}$$\\end{document} stands for the real line with the discrete topology. Moreover, we investigate several transference and restriction results on multipliers acting on Z×Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}\ imes {\\mathbb {Z}}$$\\end{document} and T×T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}\ imes {\\mathbb {T}}$$\\end{document}.

Sequences of independent functions and structure of rearrangement invariant spaces

The main aim of the survey is to present results of the last decade on the description of subspaces spanned by independent functions in $L_p$-spaces and Orlicz spaces on the one hand, and in general rearrangement invariant spaces on the other. A new approach is proposed, which is based on a combination of results in the theory of rearrangement invariant spaces, methods of the interpolation theory of operators, and some probabilistic ideas. The problem of the uniqueness of the distribution of a function such that a sequence of its independent copies spans a given subspace is considered. A general principle is established for the comparison of the complementability of subspaces spanned by a sequence of independent functions in a rearrangement invariant space on $[0,1]$ and by pairwise disjoint copies of these functions in a certain space on the half-line $(0,\infty)$. As a consequence of this principle we obtain, in particular, the classical Dor-Starbird theorem on the complementability of subspaces spanned by independent functions in the $L_p$-spaces. Bibliography: 103 titles.

On unconditionality of fractional Rademacher chaos in symmetric spaces

We study density estimates of an index set $\mathcal{A}$ under which the unconditionality (or even the weaker property of random unconditional divergence) of the corresponding Rademacher fractional chaos $\{r_{j_1}(t) \cdot r_{j_2}(t) \cdots r_{j_d}(t)\}_{(j_1,j_2,…,j_d) \in \mathcal{A}}$ in a symmetric space $X$ implies its equivalence in $X$ to the canonical basis in $\ell_2$. In the special case of Orlicz spaces $L_M$, unconditionality of this system is also shown to be equivalent to the fact that a certain exponential Orlicz space embeds into $L_M$.

Topologically multiply recurrent sequence of cosine operators on Orlicz spaces

Topologically multiply recurrent sequence of cosine operators on Orlicz spaces

Best Spline Approximation in Besov-Orlicz Space

Orlicz spaces have got the attention of many researchers over the decades. Many spaces have been defined in terms of Orlicz spaces, in particular Besov-Orlicz spaces. In this paper, we discuss this class of spaces with modulus of continuity. We study the best approximation of Besov-Orlicz spaces with splines and polynomials. The degree of best approximation depends on modulus of continuity.

Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory

<p>In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a $ L^p-L^q $ estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.</p>

On the Compactness and Continuity of Uryson's Operator in Orlicz Spaces

Uryson's operators are very famous in the theory of fuzzy functional analysis. This paper is dedicated to studying and generalizing many results about the compactness and the continuity of Uryson's operator in two-variables defined with integral equation on G with the norm ‖u‖f= sup(ρ(v,f)≤) |∫u(x)v(x)dxG | ;v(x)∈L*g ,u(x)∈Lf*. Also, we study the convergence of Urysons' sequences Kn defined with the family of functions Kn (x, y; u) by using the convergence with respect to the defined measure and Caratheodory Condition.

On the basicity of one trigonometric system in Orlicz spaces

On the basicity of one trigonometric system in Orlicz spaces

Weighted subsequential ergodic theorems on Orlicz spaces

Weighted subsequential ergodic theorems on Orlicz spaces

Solvability of Riemann Boundary Value Problems and Applications to Approximative Properties of Perturbed Exponential System in Orlicz Spaces

Solvability of Riemann Boundary Value Problems and Applications to Approximative Properties of Perturbed Exponential System in Orlicz Spaces

Existence and multiplicity results for parameter Kirchhoff double phase problem with Hardy–Sobolev exponents

The solvability of a class of parameter Kirchhoff double phase Dirichlet problems with Hardy–Sobolev terms is considered. We focus on the existence of at least one solution, two solutions, three solutions, and infinitely many solutions to the problem, as the nonlinear terms satisfy different growth conditions, respectively. Our tools are mainly based on variational methods and critical point theory. In particular, in order to establish the relationship between singular terms and the norm of the Musielak–Orlicz–Sobolev space, we extend the Sobolev–Hardy inequality from W01,p to W01,H.

Multiple Solutions for a Nonlocal Kirchhoff Problem in Fractional Orlicz-Sobolev Spaces

In this paper, using the three critical points theorem we obtain the existence of three weak solutions for a Kirchhoff type problem driven by a nonlocal operator of the elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.

Hölder's inequalities and multilinear singular integrals on generalized Orlicz spaces

Hölder's inequalities and multilinear singular integrals on generalized Orlicz spaces

APPROXIMATION OF FUNCTIONS BY THE MATRIX MEANS IN WEIGHTED ORLICZ SPACES

Abstract. In this work, the approximation properties of the matrix submethods in weighted Orlicz spaces L T M , are investigated. We obtain some results related to trigonometric approximation using matrix submethods of partial sums of Fourier series of functions in weighted Orlicz spaces.The degree of trigonometric approximations by the matrix methods to the functions have been investigated in weighted Orlicz spaces. The error of estimations in this work is obtained in more general terms.

On subspaces of Orlicz spaces spanned by independent copies of a mean zero function

We study the subspaces of the Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,…$, of a function $f\in L_M$, $\int_0^1 f(t) dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions guaranteeing that the unit ball of such a subspace consists of functions with equicontinuous norms in $L_M$ are found. In particular, we prove that there is a wide class of Orlicz spaces $L_M$ (containing the $L^p$-spaces, $1\le p< 2$), for which each of the above properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $\psi$ satisfy $\alpha_\psi^0>\beta_M^\infty$.

On the obstacle problem in fractional generalised Orlicz spaces

On the obstacle problem in fractional generalised Orlicz spaces

Weighted Orlicz regularity for fully nonlinear elliptic equations with oblique derivative at the boundary via asymptotic operators

We prove weighted Orlicz-Sobolev regularity for fully nonlinear elliptic equations with oblique boundary condition under asymptotic conditions of the following problem{F(D2u,Du,u,x)=f(x)inΩβ⋅Du+γu=gon∂Ω, where Ω is a bounded domain in Rn (n≥2), under suitable assumptions on the source term f, data β,γ and g. Our approach guarantees such estimates under conditions where the governing operator F does not require a convex (or concave) structure. We also deal with weighted Orlicz-type estimates for the obstacle problem with oblique derivative condition on the boundary. As a final application, the developed methods provide weighted Orlicz-BMO regularity for the Hessian, provided that the source lies in that space and in weighted Orlicz space associated.

Local integrability of $$G(\cdot )$$-superharmonic functions in Lebesgue and Musielak–Orlicz spaces

Local integrability of $$G(\cdot )$$-superharmonic functions in Lebesgue and Musielak–Orlicz spaces