Maximal operators associated to families of flat curves in the plane
type: Item
instance of: scholarly article
title: Maximal operators associated to families of flat curves in the plane
zbMATH Open document ID: 0827.42010
DOI: 10.1215/S0012-7094-94-07622-9
published in: Duke Mathematical Journal
publication date: 1995-02-28T00:00:00Z
review text: Let \(C\) be a smooth compact curve in the plane, with the property that the curvature vanishes at most of order \(m- 2\), \(m\geq 2\). The author shows that the maximal operator defined by \[ Mf(x)= \sup_{t> 0} |\int f(x- ty) d\sigma(y)| \] is bounded on \(L^p(\mathbb{R}^2)\) if and only if \(p> m\). This is a version of a result due to \textit{J. Bourgain} [J. Anal. Math. 47, 69-85 (1986; Zbl 0626.42012)] who proved the case \(m= 2\). In his proof the author relies on estimates of \textit{G. Mockenhaupt}, the reviewer and \textit{Ch. D. Sogge} [Ann. Math., II. Ser. 136, No. 1, 207-218 (1992; Zbl 0759.42016); J. Am. Math. Soc. 6, No. 1, 65-130 (1993; Zbl 0776.58037)] and suitable scaling arguments. He also discusses a variable coefficient analogue where the associated canonical relation is folding of order \(k\), with two-sided folds, as defined by \textit{D. H. Phong} and \textit{E. M. Stein} [Int. Math. Res. Not. 1991, No. 4, 49-60 (1991; Zbl 0761.46033)].
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 727838
zbMATH Keywords: maximal operators
zbMATH Keywords: flat curves in the plane
author: Alex Iosevich
reviewed by: Andreas Seeger
MaRDI profile type: Publication
cites work: Averages in the plane over convex curves and maximal operators
cites work: Q3714302
cites work: Q3671563
cites work: Q3671564
cites work: Q3736112
cites work: Q3752751
cites work: Estimates for translation invariant operators in \(L^p\) spaces
cites work: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle
cites work: Wave front sets, local smoothing and Bourgain's circular maximal theorem
cites work: Local Smoothing of Fourier Integral Operators and Carleson-Sjolin Estimates
cites work: Q4847895
cites work: Q3984203
cites work: Hilbert integrals, singular integrals, and Radon transforms. I
cites work: Propagation of singularities and maximal functions in the plane
cites work: Q4693724
cites work: Oscillatory integrals associated to folding canonical relations
cites work: Maximal functions: Poisson integrals on symmetric spaces
cites work: Problems in harmonic analysis related to curvature
cites work: Q3900553
full work available at URL: s0012-7094-94-07622-9
OpenAlex ID: W1515776698
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description: scientific article; zbMATH DE number 727838