Convergence and summability of Fourier transforms and Hardy spaces
This book deals with the convergence and summability of Fourier transforms in any dimension, for both Fourier series of periodic functions and the Fourier transform in \(\mathbb R^n\). The author studies these kind of questions in the whole range of exponents \(p>0\), for this reason the spaces under consideration includes Hardy spaces. Summability methods for Fourier series and integrals are established, in one and higher dimensions, in this almost self-contained book. The book is structured as follows: Chapter 1 is devoted to the introduction of Hardy spaces, \(H^p\) on the real line, establishing different maximal characterizations of the spaces and their atomic decomposition. Also, interpolation spaces of Hardy spaces are given, and also sufficient conditions for the boundedness of operators defined from \(H^p\) into \(L^p\). In the second chapter, the one-dimensional Fourier transform is considered and its extensions on tempered distributions. The general summability method, which includes Fejér, Riesz, Weierstrass, Abel, Picard, Bessel, Rogosinski and de La Vallée-Poussin summation is presented. Also, it includes the boundedness of maximal operators of the summability means from \(H^p\) into \(L^p\) whenever \(p>p_0\), for some critical index \(p_0<1\), which depends on the summability method. The methods applied are not valid in higher dimensions, and this reason justifies the inclusion of all results concerning \(H^p\) in several variables in a separate chapter. In Chapter 3, different types of Hardy-Littlewood maximal operators and multidimensional Hardy spaces are introduced, also atomic decompositions of each Hardy space and the interpolation between them are verified. In Chapter 4, the multi-dimensional Fourier transform and problems concerning the convergence of multidimensional Fourier series and the inverse Fourier transform are considered. The study of several multidimensional summability kernels are especially relevant in this chapter. Finally, in Chapters 5 and 6, the problem of multidimensional summability of Fourier series and integrals is investigated. Two types of convergence and maximal operators are considered, namely, the convergence over the diagonal or more generally over a cone, called the restricted convergence, and the unrestricted one which is the convergence over all \(\mathbb{R}^n\). It is shown that the boundedness from \(H^p\) into \(L^p\) of the maximal operators associated to rectangular summability means, for \(p\) above a critical exponent \(p_0<1\), implies the almost everywhere convergence of the summability means. The results about convergence are formulated for Lebesgue points.
scientific article; zbMATH DE number 6758719