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

 

A sharp bound for the Stein-Wainger oscillatory integral


Author: Ioannis R. Parissis
Journal: Proc. Amer. Math. Soc. 136 (2008), 963-972
MSC (2000): Primary 42A50; Secondary 42A45
Published electronically: November 16, 2007
MathSciNet review: 2361870
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Abstract: Let $ \mathcal{P}_d$ denote the space of all real polynomials of degree at most $ d$. It is an old result of Stein and Wainger that

$\displaystyle \sup_ {P\in\mathcal{P}_d} \bigg\vert p.v.\int_{\mathbb{R}} {e^{iP(t)}\frac{dt}{t}} \bigg\vert\leq C_d$

for some constant $ C_d$ depending only on $ d$. On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is $ \log d$. We prove that

$\displaystyle \sup_ {P\in\mathcal{P}_d}\bigg\vert p.v. \int_{\mathbb{R}}{e^{iP(t)}\frac{dt}{t}}\bigg\vert\sim \log{d}.$


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Additional Information

Ioannis R. Parissis
Affiliation: Department of Mathematics, University of Crete, Knossos Avenue, 71409 Iraklio, Crete, Greece
Email: ypar@math.uoc.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09013-2
PII: S 0002-9939(07)09013-2
Received by editor(s): November 20, 2006
Published electronically: November 16, 2007
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.