Rearrangement Invariant Spaces Research Articles

Core decreasing functions

Given a measure space and a totally ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to define the down space of a Banach function space. This is done using a variant of the Köthe dual restricted to core decreasing functions. To study down spaces, the least core decreasing majorant construction and the level function construction, already known for functions on the real line, are extended to this general setting. These are used to give concrete descriptions of the duals of the down spaces and, in the case of universally rearrangement invariant (u.r.i.) spaces, of the down spaces themselves.The down spaces of L1 and L∞ are shown to form an exact Calderón couple with divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of level functions; and the down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property. The dual couple is also an exact Calderón couple with divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of least core decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property.

A unified approach to inequalities for K-functionals and moduli of smoothness

AbstractThe paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.

On the full range of Zippin and inclusion indices of rearrangement-invariant spaces

Let X be a rearrangement-invariant space on [0, 1]. It is known that its Zippin indices β̲X,β¯X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\underline{\\beta }{}_X,\\overline{\\beta }{}_X$$\\end{document} and its inclusion indices γX,δX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma _X,\\delta _X$$\\end{document} are related as follows: 0≤β̲X≤1/γX≤1/δX≤β¯X≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le \\underline{\\beta }{}_X\\le 1/\\gamma _X \\le 1/\\delta _X\\le \\overline{\\beta }{}_X\\le 1$$\\end{document}. We show that given β̲,β¯∈[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\underline{\\beta },\\overline{\\beta }\\in [0,1]$$\\end{document} and γ,δ∈[1,∞]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ,\\delta \\in [1,\\infty ]$$\\end{document} satisfying β̲≤1/γ≤1/δ≤β¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\underline{\\beta }\\le 1/\\gamma \\le 1/\\delta \\le \\overline{\\beta }$$\\end{document}, there exists a rearrangement-invariant space X such that β̲X=β̲\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\underline{\\beta }{}_X=\\underline{\\beta }$$\\end{document}, β¯X=β¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{\\beta }{}_X=\\overline{\\beta }$$\\end{document} and γX=γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma _X=\\gamma $$\\end{document}, δX=δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _X=\\delta $$\\end{document}.

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.

Spectrum of the Cesàro-Hardy operator in rearrangement invariant spaces

In this paper, we investigate the spectrum of the Cesàro-Hardy operator in rearrangement invariant spaces over a finite interval and a half line, thereby extending Boyd’s and Leibowitz’s results for Lp(1 < p ⩽ 8) spaces. In particular, when our rearrangement invariant space is the Lorentz space Lp,q , a full description of the spectrum and fine spectra is presented.

Boundedness of Wolff-type potentials and applications to PDEs

We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff–Havin–Maz’ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of those rearrangement invariant spaces between which the potentials are bounded via a one-dimensional inequality of Hardy-type.Since the special case of the mentioned potential is known to control precisely very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, we infer the local regularity properties of the solutions in rearrangement invariant spaces for prescribed classes of data.

Optimal Gagliardo–Nirenberg interpolation inequality for rearrangement invariant spaces

We prove the optimality of the Gagliardo–Nirenberg inequality: ‖∇u‖X≲‖∇2u‖Y1/2‖u‖Z1/2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert \ abla u\\Vert _{X}\\lesssim \\Vert \ abla ^2 u\\Vert _Y^{1/2}\\Vert u\\Vert _Z^{1/2}, \\end{aligned}$$\\end{document}where Y and Z are rearrangement invariant Banach function spaces, and X = Y^{1/2}Z^{1/2} is the Calderón–Lozanovskii space. By optimality, we mean that for a certain pair of spaces on the right-hand side, it is not possible to reduce the space on the left-hand side while remaining in the class of rearrangement invariant spaces. Our result establishes the optimality for Lorentz and Orlicz spaces, surpassing previous findings. Additionally, we discuss the significance of pointwise inequalities and present a counterexample that prohibits further improvements.

Cones generated by a generalized fractional maximal function

The paper considers the space of generalized fractional-maximal function, constructed on the basis of a rearrangement-invariant space. Two types of cones generated by a nonincreasing rearrangement of a generalized fractional-maximal function and equipped with positive homogeneous functionals are constructed. The question of embedding the space of generalized fractional-maximal function in a rearrangement invariant space is investigated. This question reduces to the embedding of the considered cone in the corresponding rearrangement-invariant spaces. In addition, conditions for covering a cone generated by a generalized fractional-maximal function by the cone generated by generalized Riesz potentials are given. Cones from non-increasing rearrangements of generalized potentials were previously considered in the works of M. Goldman, E. Bakhtigareeva, G. Karshygina and others.

Gradient continuity for p(x)-Laplacian systems under minimal conditions on the exponent

We consider solutions of p(x)-Laplacian systems with coefficients and we show that their gradient is continuous provided that the variable exponent has distributional gradient belonging to the Lorentz-Zygmund space Ln,1log⁡L and that the gradient of the coefficient belongs to the Lorentz space Ln,1. The result is new since the use of the sharp Sobolev embedding in rearrangement invariant spaces does not ensure the unique (up to now) known assumption for such result, namely the log-Dini continuity of p(⋅) and the plain Dini continuity of the coefficient. Our approach relies on perturbation arguments and allows to slightly improve results in dimension two even for the case where p(⋅) is constant.

Spectrum of the Cesaro-Hardy operator in Lorentz Lp,q(0,1) spaces

The aim of this paper is to investigate the spectrum of the Cesaro-Hardy operator in Lorentz Lp,q spaces over (0,1). In this paper we extended Leibowitz's results for Lp space to Lorentz spaces. Note that Lp space is a special case of Lorentz spaces when indexes p and q coincide. Interestingly, we obtained the same results as for Lp space. The point spectrum is obtained by solving an Euler differential equation of first order. We used the operator Pξ to find the resolvent set of the Cesaro-Hardy operator. This operator was defned in Boyd's work in [1]. Boundedness of the operator Pξ on Lp was proved in the same paper. But its boundedness on Lp,q was proved in this paper by using Lp,q norm of the dilation operator. Here, we also used the Boyd's theorem, which describes boundedness of operators on rearrangement invariant spaces. We verified conditions of Boyd's theorem. It allows us to obtain a bounded inverse of the operator λI −C for some complex numbers λ.

The order-type Banach-Saks properties

The study of the Banach-Saks property in Banach spaces has a long and illustrious history. Of late, motivated by applications in financial mathematics, interest has arisen in the Banach-Saks type properties with respect to order convergence. This paper presents a study of order Banach-Saks properties in Banach function spaces, and in particular in rearrangement invariant spaces. Among the results obtained, we provide some sufficient conditions for the (weak) order Banach-Saks property. We also characterize the (weak) order Banach-Saks property in Orlicz spaces. It is also shown that the (weak) order Banach-Saks property is equivalent to its hereditary version.

ON THE DENSITY OF LAGUERRE FUNCTIONS IN SOME BANACH FUNCTION SPACES

Let λ &gt; 0 and Φλ := {ϕ1,λ, ϕ2,λ, . . . } be the system of dilated Laguerre functions. We show that if L1 (R+) ∩ L∞(R+) is embedded into a separable Banach function space X(R+), then the linear span of Φλ is dense in X(R+). This implies that the linear span of Φλ is dense in every separable rearrangement-invariant space X(R+) and in every separable variable Lebesgue space Lp(·) (R+)

Sobolev Embeddings for Fractional Hajłasz-Sobolev Spaces in the Setting of Rearrangement Invariant Spaces

We obtain symmetrization inequalities in the context of Fractional Hajłasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure.

On Local Solvability of Higher Order Elliptic Equations in Rearrangement Invariant Spaces

On Local Solvability of Higher Order Elliptic Equations in Rearrangement Invariant Spaces

Weakly compact sets and weakly compact pointwise multipliers in Banach function lattices

AbstractWe prove that the class of Banach function lattices in which all relatively weakly compact sets are equi‐integrable sets (i.e. spaces satisfying the Dunford–Pettis criterion) coincides with the class of 1‐disjointly homogeneous Banach lattices. New examples of such spaces are provided. Furthermore, it is shown that Dunford–Pettis criterion is equivalent to de la Valleé Poussin criterion in all rearrangement invariant spaces on the interval. Finally, the results are applied to characterize weakly compact pointwise multipliers between Banach function lattices.

Compactness of Sobolev embeddings and decay of norms

We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with $d$-Ahlfors measures under certain restriction on the speed of their decay o

Rosenthal's space revisited

Let $E$ be a rearrangement invariant (r.i.) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\infty )$ such that $f^*\chi _{[0,1]}\in E$ and $f^*\chi _{[1,\infty )}\in L^2$. We reveal close connections between proper

The space is primary for 1 &lt; p &lt; ∞

Abstract The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which $$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx &lt; \infty.\end{align*} $$ We show that $L_1(L_p) (1 &lt; p &lt; \infty )$ is primary, meaning that whenever $L_1(L_p) = E\oplus F$ , where E and F are closed subspaces of $L_1(L_p)$ , then either E or F is isomorphic to $L_1(L_p)$ . More generally, we show that $L_1(X)$ is primary for a large class of rearrangement-invariant Banach function spaces.

Reiteration theorem for [formula omitted] and [formula omitted]-spaces with the same parameter

Let E,F,E0,E1 be rearrangement invariant spaces; let a,b,b0,b1 be slowly varying functions and 0<θ0,θ1<1. We characterize the interpolation spaces(X‾θ0,b0,E0,a,FR,X‾θ1,b1,E1,a,FL)η,b,E,0≤η≤1, when the parameters θ0 and θ1 are equal (under appropriate conditions on bi(t), i=0,1). This completes the study started in [11,12,22], which only considered the case θ0<θ1. As an application we recover and generalize interpolation identities for grand and small Lebesgue spaces given by [26].

Fine spectra and compactness of generalized Cesàro operators in Banach lattices in [formula omitted

The generalized Cesàro operators Ct, for t∈[0,1), introduced in the 1980's by Rhaly, are natural analogues of the classical Cesàro averaging operator C1 and act in various Banach sequence spaces X⊆CN0. In this paper we concentrate on a certain class of Banach lattices for the coordinate-wise order, which includes all separable, rearrangement invariant sequence spaces, various weighted c0 and ℓp spaces and many others. In such Banach lattices X the operators Ct, for t∈[0,1), are always compact (unlike C1) and a full description of their point, continuous and residual spectrum is given. Estimates for the operator norm of Ct are also presented.