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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sharp maximal estimates for doubly oscillatory integrals


Author: Björn Gabriel Walther
Journal: Proc. Amer. Math. Soc. 130 (2002), 3641-3650
MSC (1991): Primary 42B25, 42B99, 35L05, 35J10, 35Q40
Published electronically: May 1, 2002
MathSciNet review: 1920044
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Abstract | References | Similar Articles | Additional Information

Abstract: We study doubly oscillatory integrals

\begin{displaymath}\int_{\mathbf R^n} \, e^{i(\xi + y\vert\xi\vert + t\vert\xi\vert^ a)} \widehat f(\xi)\,d\xi \end{displaymath}

and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on $\mathbf{R}^n$.


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Additional Information

Björn Gabriel Walther
Affiliation: Department of Mathematics, Royal Institute of Technology, SE – 100 44 Stockholm, Sweden
Address at time of publication: Department of Mathematics, Brown University, Providence, Rhode Island 02912–1917
Email: WALTHER@Math.KTH.SE, WALTHER@Math.Brown.Edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06527-9
PII: S 0002-9939(02)06527-9
Keywords: Doubly oscillatory Fourier integrals, maximal estimates, wave equation, time-dependent Schr\"odinger equation
Received by editor(s): August 10, 2000
Received by editor(s) in revised form: July 19, 2001
Published electronically: May 1, 2002
Additional Notes: This paper is a revision of [16, Chapter 9]. The author would like to thank Professor Per Sjölin, Royal Institute of Technology, Stockholm, Sweden, for patience and support. The final draft was made during visits at Brown University, Providence, RI, USA, and Univerzita Komenského, Bratislava, Slovakia.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2002 American Mathematical Society