Endpoint estimates from restricted rearrangement inequalities.
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 \(\bar A\).
scientific article; zbMATH DE number 2104141
scientific article