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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Bochner-Riesz means with respect to a rough distance function


Author: Paul Taylor
Journal: Trans. Amer. Math. Soc. 359 (2007), 1403-1432
MSC (2000): Primary 42B15; Secondary 42B25
Published electronically: November 17, 2006
MathSciNet review: 2272131
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Abstract: The generalized Bochner-Riesz operator $ S^{R,\lambda}$ may be defined as

$\displaystyle S^{R,\lambda}f(x) = \mathcal{F}^{-1}\left[\left(1-\frac{\rho}{R}\right)^{\lambda}_+ \widehat{f}\right](x) $

where $ \rho$ is an appropriate distance function and $ \mathcal F^{-1}$ is the inverse Fourier transform. The behavior of $ S^{R,\lambda}$ on $ L^p(\mathbf{R}^d\times\mathbf{R})$ is described for $ \rho(\xi',\xi_{d+1})=\max\{\vert\xi'\vert,\vert\xi_{d+1}\vert\}$, a rough distance function. We conjecture that this operator is bounded on $ \mathbf{R}^{d}\times\mathbf{R}$ when $ \lambda>\max\{d(\frac{1}{2}-\frac{1}{p})-\frac{1}{2},0\}$ and $ p<2+\frac{6}{d-3}$, and unbounded when $ p \geq 2+\frac{6}{d-3}$. This conjecture is verified for large ranges of $ p$.


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Additional Information

Paul Taylor
Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, Shippensburg University, 1871 Old Main Drive, Shippensburg, Pennsylvania 17257-2299
Email: pttaylor@ship.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03918-3
PII: S 0002-9947(06)03918-3
Keywords: Fourier analysis, multipliers, Bochner-Riesz means, cone multiplier
Received by editor(s): July 26, 2004
Received by editor(s) in revised form: November 16, 2004
Published electronically: November 17, 2006
Additional Notes: The author thanks Andreas Seeger for his guidance
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.