Weighted estimates for singular integrals via Fourier transform estimates
type: Item
instance of: scholarly article
title: Weighted estimates for singular integrals via Fourier transform estimates
zbMATH Open document ID: 0711.42025
DOI: 10.1215/S0012-7094-90-06015-6
author: David K. Watson
published in: Duke Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: We consider weighted norm inequalities for singular integral operators \(Tf(x)=p.\nu.f*k(x)\), for kernels k which satisfy no smoothness estimates. Among the results we obtain is the following: If \(1<r\leq \infty\) and \(k(x)=h(| x|)\Omega (x/| x|)/| x|^ n,\) where \(\Omega \in L^ r(S^{n-1})\) has mean value 0, and \(\int^{2R}_{R}| h(t)|^ rdt\leq CR\) uniformly in \(R>0\), then T is bounded on \(L^ p(w)\) provided that either (A) \(p>r'\) and \(w\in A_{p/r'}\), or (B) \(1<p<r\) and \(w^{-1/(p-1)}\in A_{p/r'}\), or (C) \(1<p<\infty\) and \(w^{r'}\in A_ p\). In particular, letting \(h=1\) gives the standard weighted norm inequalities for homogeneous singular operators without the usual smoothness requirements on \(\Omega\). We prove our results by deriving Fourier transform estimates for the kernel and showing that these estimates enable us to approximate T by operators whose kernels satisfy appropriate smoothness conditions, by a procedure similar to that used in the unweighted setting to derive norm inequalities for Hilbert transforms along curves. In a subsequent paper (to appear), the author derives the corresponding estimates for the corresponding maximal singular integral operator and related maximal operators.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 4169360
zbMATH Keywords: weighted norm inequalities
zbMATH Keywords: singular integral operators
zbMATH Keywords: Fourier transform estimates
MaRDI profile type: Publication
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full work available at URL: s0012-7094-90-06015-6
OpenAlex ID: W1526067066
description: scientific article; zbMATH DE number 4169360