Maximal function estimates of solutions to general dispersive partial differential equations

Heinig, Hans; Wang, Sichun

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

Maximal function estimates of solutions to general dispersive partial differential equations
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by Hans P. Heinig and Sichun Wang PDF

Trans. Amer. Math. Soc. 351 (1999), 79-108 Request permission

Abstract:

Let $u(x,t)=(S_\Omega f)(x,t)$ be the solution of the general dispersive initial value problem: \[ \partial _tu-i\Omega (D)u=0, \quad u(x,0)=f(x), \qquad (x,t)\in \mathbb {R}^n \times \mathbb {R}\] 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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