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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Classes of singular integrals along
curves and surfaces


Authors: Andreas Seeger, Stephen Wainger, James Wright and Sarah Ziesler
Journal: Trans. Amer. Math. Soc. 351 (1999), 3757-3769
MSC (1991): Primary 42B20, 42B15
DOI: https://doi.org/10.1090/S0002-9947-99-02496-4
Published electronically: May 20, 1999
MathSciNet review: 1665337
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Abstract: This paper is concerned with singular convolution operators in $\mathbb{R}^{d}$, $d\ge 2$, with convolution kernels supported on radial surfaces $y_{d}=\Gamma (|y'|)$. We show that if $\Gamma (s)=\log s$, then $L^{p}$ boundedness holds if and only if $p=2$. This statement can be reduced to a similar statement about the multiplier $m(\tau ,\eta )=|\tau |^{-i\eta }$ in $\mathbb{R}^{2}$. We also construct smooth $\Gamma $ for which the corresponding operators are bounded for $p_{0}<p\le 2$ but unbounded for $p\le p_{0}$, for given $p_{0}\in [1,2)$. Finally we discuss some examples of singular integrals along convex curves in the plane, with odd extensions.


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Additional Information

Andreas Seeger
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: seeger@math.wisc.edu

Stephen Wainger
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: wainger@math.wisc.edu

James Wright
Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, Australia
Email: jimw@maths.unsw.edu.au

Sarah Ziesler
Affiliation: Department of Mathematics, University College Dublin, Dublin 4, Ireland
Address at time of publication: Department of Mathematics, Dominican University, River Forest, Illinois 60305
Email: ziessara@email.dom.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02496-4
Received by editor(s): May 27, 1997
Published electronically: May 20, 1999
Additional Notes: Research supported in part by grants from the National Science Foundation (A. S. & S. W.), the Australian Research Council (J. W.), and the Faculty of Arts, University College Dublin (S. Z.)
Article copyright: © Copyright 1999 American Mathematical Society