Maximal and singular integral operators via Fourier transform estimates
type: Item
instance of: scholarly article
title: Maximal and singular integral operators via Fourier transform estimates
zbMATH Open document ID: 0568.42012
DOI: 10.1007/BF01388746
author: Javier Duoandikoetxea
author: José L. Rubio de Francia
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143351
review text: In this paper \(L^ p\) inequalities are proved for singular integral operators written as \(Tf=\sum_{k}\sigma_ k*f\) and maximal operators defined by \(Mf=\sup_{k}| \mu_ k*f|,\) where \(\sigma_ k\) and \(\mu_ k\) are Borel measures with uniformly bounded total variation, the \(\sigma_ k's\) have zero integral and the \(\mu_ k's\) are positive. The inequalities follow from the regularity at zero and the decay at infinity of the Fourier transforms of \(\sigma_ k\) and \(\mu_ k\). Several applications are given: lacunary maximal functions (which generalize lacunary spherical means), homogeneous singular integrals with size conditions on the kernel and some of their variants, including weighted inequalities for kernels bounded on the unit sphere, maximal functions and Hilbert transforms along different types of curves (homogeneous, approximately homogeneous, convex plane curves). Some of them are old resuls with easier proofs, others are new. For all the singular integral operators the almost everywhere convergence of the truncated kernels is also obtained.
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 3907088
zbMATH Keywords: singular integral operators
zbMATH Keywords: maximal operators
zbMATH Keywords: lacunary maximal functions
zbMATH Keywords: Hilbert transforms
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3907088