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

 

Maximal function estimates of
solutions to general dispersive
partial differential equations


Authors: Hans P. Heinig and Sichun Wang
Journal: Trans. Amer. Math. Soc. 351 (1999), 79-108
MSC (1991): Primary 42B25; Secondary 42A45
MathSciNet review: 1458324
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $u(x,t)=(S_\Omega f)(x,t)$ be the solution of the general dispersive initial value problem:

\begin{displaymath}\partial _tu-i\Omega(D)u=0, \quad u(x,0)=f(x), \qquad (x,t)\in \mathbb{R}^n \times \mathbb{R}\end{displaymath}

and $S^{**}_\Omega f$ the global maximal operator of $S_\Omega f$. Sharp weighted $L^p$-esimates for $S^{**}_\Omega f$ with $f\in H_s(\mathbb{R}^n)$ are given for general phase functions $\Omega$.


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Additional Information

Hans P. Heinig
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
Email: heinig@mcmail.cis.mcmaster.ca

Sichun Wang
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
Email: wangs@icarus.math.mcmaster.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02116-9
PII: S 0002-9947(99)02116-9
Keywords: Dispersive PDE, free Schr\"odinger equation, phase functions, polynomials of principal type, regular zeroes, weighted $L^p$-spaces, Sobolev spaces
Received by editor(s): May 5, 1996
Received by editor(s) in revised form: July 1, 1996
Additional Notes: The research of the first author was supported in part by NSERC grant A-4837
Article copyright: © Copyright 1999 American Mathematical Society