Endpoint estimates from restricted rearrangement inequalities.
type: Item
instance of: scholarly article
title: Endpoint estimates from restricted rearrangement inequalities.
zbMATH Open document ID: 1063.46061
DOI: 10.4171/RMI/383
published in: Revista Matemática Iberoamericana
publication date: 2004-09-28T00:00:00Z
full work available at URL: https://eudml.org/doc/39633
review text: There are many interesting operators in analysis for which one can prove that an inequality of the form \((Tf)^*(t)\leq h(t, | | f| | _1)\) holds, for some function \(h\) and every function \(f\) bounded by 1. This happens, for example, for operators which are of weak type \((p,p)\) with a constant \(C_p\) that blows up when \(p\) tends to 1, which is the classical case related to the theory of extrapolation. In the first part of the paper under review, the authors give very nice proofs of several previously known results by using the following interesting formula, proved previously by them [Q. J. Math. 55, No. 1, 41--45 (2004; Zbl 1056.42013)] for the decreasing rearrangement of the sum of functions: if \(f=\sum_{n}f_{n}\) and \(c_{n}>0\) are such that \(\sum_{n}c_{n}=1\), then \[ f^{* }(3t)\leq \sum_{n}\left( f_{n}^{* }(t)+{1\over t} \int_{c_{n}t}^{t}f_{n}^{* }(s)ds\right). \] In the second part, they extend the theory to more general compatible couples which include the case of Sobolev spaces. In particular, they succeed in proving an analogue of \textit{N. Y. Antonov's} result [East J. Approx. 2, No. 2, 187--196 (1996; Zbl 0919.42004)] about the convergence of the Fourier series in \(L\log L\log\log\log L(T)\) in the setting of Sobolev spaces. This extension (also useful for more general couples of Banach spaces \(\bar A=(A_0, A_1)\)) is far from being trivial, since the authors have to deal with the problem of, given an element \(a\in A_0+A_1\), finding an appropriate decomposition \(a=\sum_n 2^n a_n\), which plays the role of the classical dyadic decomposition in \(L^1+L^\infty\). The technique they use is based on the \(K\)-divisibility property of \(\bar A\).
Mathematics Subject Classification ID: 46M35
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 2104141
zbMATH Keywords: rearrangement inequality
zbMATH Keywords: real interpolation
zbMATH Keywords: Banach couples
zbMATH Keywords: extrapolation theory
zbMATH Keywords: Carleson's operator
author: María J. Carro
author: Joaquim Martin
reviewed by: Joan Cerdà
MaRDI profile type: Publication
cites work: Q4892644
cites work: POINTWISE CONVERGENCE OF FOURIER SERIES
cites work: Q4039801
cites work: Q3994467
cites work: On convergence and growth of partial sums of Fourier series
cites work: New extrapolation estimates
cites work: Extrapolation theory for the real interpolation method.
cites work: A useful estimate for the decreasing rearrangement of the sum of functions
cites work: Recent developments in the theory of Lorentz spaces and weighted inequalities
cites work: Weighted Lorentz spaces and the Hardy operator
cites work: Interpolation of linear operators on Sobolev spaces
cites work: Q5542909
cites work: Extrapolation theory with applications
OpenAlex ID: W2006280765
Recommended article: On extrapolation spaces and a.e. convergence of Fourier series
Recommended article: On properties of the space U(0,2\(\pi\) )
Recommended article: A contribution to the study of the partial sums operator \(S_{NN^ 2}\) for double Fourier series
Recommended article: Q3223328
Recommended article: Remarks on a theorem by N. Yu. Antonov
Recommended article: From weak-type weighted inequality to pointwise estimate for the decreasing rearrangement
Recommended article: Q4323084
Recommended article: Q3320848
Recommended article: Q3791637
Recommended article: Q3973244
description: scientific article; zbMATH DE number 2104141