Dept. Math, Hokkaido Univ. EPrints Server

A maximal inequality associated to Schr\{o}dinger type equation.

Preprint Series # 685
Cho, Yonggeun and Lee, Sanghyuk and Shim, Yongsun A maximal inequality associated to Schr\{o}dinger type equation. (2005);

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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 $H^\frac14$ with additional angular regularity to $L_{loc}^2$.

Item Type:Preprint
Additional Information:10 copies
Uncontrolled Keywords:Schr\"{o}dinger type equation, maximal operator, angular regularity
Subjects:42-xx FOURIER ANALYSIS
ID Code:844