Rearrangement Invariant Spaces Research Articles

Banach lattices with upper p-estimates: free and injective objects

AbstractWe study the free Banach lattice $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) together with a finite dimensional reduction, it is shown that the spaces $$\ell ^{p,\infty }(n)$$ ℓ p , ∞ ( n ) witness the universal property of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] isomorphically. As a consequence, we obtain a functional representation for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] , answering a question from Oikhberg et al. [Free Banach lattices. J Eur Math Soc (JEMS), 2024]. More generally, our proof allows us to identify the norm of any free Banach lattice over E associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] . Notably, we prove that the norm for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] cannot be isometrically witnessed by $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper p-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $$\ell ^p$$ ℓ p allows us to give an alternative proof of the subspace problem for free p-convex Banach lattices. On the other hand, we prove that $$\ell ^{p,\infty }$$ ℓ p , ∞ is not injective in the class of Banach lattices with upper p-estimates, elucidating one of many difficulties arising in the study of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] .

On the modulus of continuity of fractional Orlicz-Sobolev functions

AbstractNecessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on $${\mathbb {R}}^n$$ R n to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.

Envelopes in Banach spaces

We introduce the notion of isometric envelope of a subspace in a Banach space, establishing its connections with several key elements: (a) we explore its relation to the mean ergodic projection on fixed points within a semigroup of contractions, (b) we draw parallels with Korovkin sets from the 1970s, (c) we investigate its impact on the extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the 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}, p∉2N+4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \ otin 2{\\mathbb {N}}+4$$\\end{document} are the only separable approximately ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé Banach spaces (a strengthening of the approximately homogeneous property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope; and we show that the Gurarij space satisfies the same property. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive 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}-spaces are the only reflexive rearrangement invariant spaces on [0, 1] for which all 1-complemented subspaces are envelopes. We also identify the isometrically unique “full” quotient space of 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} by a Hilbertian subspace, for appropriate values of p, as well as the associated topological group embedding of the unitary group into the isometry group of 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}.

A note on rearrangement Poincaré inequalities and the doubling condition

We introduce Poincaré-type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure.

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.

On Isomorphic Embeddings in the Class of Disjointly Homogeneous Rearrangement Invariant Spaces

On Isomorphic Embeddings in the Class of Disjointly Homogeneous Rearrangement Invariant Spaces

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

The 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.

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.

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