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November 2006 A maximal inequality associated to Schr\"{o}dinger type equation
Yonggeun CHO, Sanghyuk LEE, Yongsun SHIM
Hokkaido Math. J. 35(4): 767-778 (November 2006). DOI: 10.14492/hokmj/1285766429

Abstract

In this note, we consider a maximal operator $\sup_{t \in \mathbb{R}}|u(x,t)| = \sup_{t \in \mathbb{R}}|e^{it\Omega(D)}f(x)|$, where $u$ is the solution to the initial value problem $u_t = i\Omega(D)u$, $u(0) = f$ for a $C^2$ function $\Omega$ with some growth rate at infinity. We prove that the operator $\sup_{t \in \mathbb{R}}|u(x,t)|$ has a mapping property from a fractional Sobolev space $H^\fraca{1}{4}$ with additional angular regularity in which the data lives to $L^2((1 + |x|)^{-b}dx) (b > 1)$ . This mapping property implies the almost everywhere convergence of $u(x,t)$ to $f$ as $t \to 0$, if the data $f$ has an angular regularity as well as $H^\frac{1}{4}$ regularity.

Citation

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Yonggeun CHO. Sanghyuk LEE. Yongsun SHIM. "A maximal inequality associated to Schr\"{o}dinger type equation." Hokkaido Math. J. 35 (4) 767 - 778, November 2006. https://doi.org/10.14492/hokmj/1285766429

Information

Published: November 2006
First available in Project Euclid: 29 September 2010

zbMATH: 1122.42008
MathSciNet: MR2289360
Digital Object Identifier: 10.14492/hokmj/1285766429

Subjects:
Primary: 42A45
Secondary: 42B25

Rights: Copyright © 2006 Hokkaido University, Department of Mathematics

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Vol.35 • No. 4 • November 2006
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