A method for proving \(L^p\)-boundedness of singular Radon transforms in codimension 1
type: Item
instance of: scholarly article
title: A method for proving \(L^p\)-boundedness of singular Radon transforms in codimension 1
zbMATH Open document ID: 1015.42008
DOI: 10.1215/S0012-7094-01-10826-0
author: Michael Greenblatt
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The author considers the following operator: \[ Tf(x,t)=\int_{\mathbb R^n} f(t+S(t,x,y),y)K(t,x,y) dy \] , where \(x,y\in \mathbb R^n\), \(t\in \mathbb R^n\), \(K(t,x,y)\) is a distribution supported on a compact set \(U\) and satisfies \[ |\partial_x^\alpha K(t,x,y)|\leq M|x-y|^{-n-|\alpha|}, \] \[ |\partial_y^\alpha K(t,x,y)|\leq M|x-y|^{-n-|\alpha|}, \] \[ |\partial_t^\alpha K(t,x,y)|\leq M|x-y|^{-n} \] for \(0\leq |\alpha|\leq 1\), \[ \left|\int_{|y-x|=a}K(t,x,y) dx\right|, \left|\int_{|y-x|=a}K(t,x,y) dy\right|\leq Ma \] for \(a>0\), and \(S(t,x,y)\) is a \(C^\infty\)-function satisfying \(S(t,x,x)=0\), \(|\partial_tS|<1/2\), \[ \sup\{|\partial_x^\alpha\partial_y^\beta\partial_t^\gamma S(t,x,y)|;\;(t,x,y)\in U, |\alpha|+|\beta|+|\gamma|\leq d+1\}\leq M \] for some positive integer \(d\), and some curvature condition equivalent to a finite-type condition given by \textit{M. Christ, A. Nagel, E. M. Stein}, and \textit{S. Wainger} [Ann. Math. (2) 150, No.~2, 489-577 (1999; Zbl 0960.44001)]. For the above operator, he gives another proof of the \(L^p\)-boundedness by geometric and analytic methods, without using the method of lifting to nilpotent Lie groups. He discusses a few directions in which generalizations of his arguments might be attempted.
reviewed by: Kôzô Yabuta
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 1820824
zbMATH Keywords: singular integral
zbMATH Keywords: Calderón-Zygmund operators
zbMATH Keywords: Radon transform
zbMATH Keywords: \(L^p\)-estimate
zbMATH Keywords: \(L^p\)-boundedness
MaRDI profile type: Publication
cites work: Q3034300
cites work: Hilbert transforms along curves. I: Nilpotent groups
cites work: Singular and maximal Radon transforms: Analysis and geometry
cites work: Sobolev estimates for fractional and singular Radon transforms
cites work: Maximal and singular integral operators via Fourier transform estimates
cites work: Singular integrals and partial differential equations of parabolic type
cites work: On X-ray transforms for rigid line complexes and integrals over curves in $\mathbb {R}^4$
cites work: Estimates for singular Radon transforms and pseudodifferential operators with singular symbols
cites work: Balls and metrics defined by vector fields. I: Basic properties
cites work: Uniform estimates for oscillatory integral operators
cites work: Q4847895
cites work: Singular integrals related to the Radon transform and boundary value problems
cites work: Hilbert integrals, singular integrals, and Radon transforms. I
cites work: The Newton polyhedron and oscillatory integral operators
cites work: Harmonic analysis on nilpotent groups and singular integrals. I: Oscillatory integrals
cites work: Degenerate Fourier integral operators in the plane
cites work: Radon transforms and finite type conditions
cites work: Classes of singular integrals along curves and surfaces
cites work: Maximal functions associated to smooth curves
cites work: Singular integrals associated to hypersurfaces: \(L^2\) theory
full work available at URL: s0012-7094-01-10826-0
OpenAlex ID: W2070715919
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description: scientific article; zbMATH DE number 1820824