Maximal function estimates of solutions to general dispersive partial differential equations

Heinig, Hans; Wang, Sichun

doi:10.1090/S0002-9947-99-02116-9

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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: 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 -esimates for $S^{**}_\Omega f$ with $f\in H_s(\mathbb{R}^n)$ are given for general phase functions $\Omega$ .

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