## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 35, Number 4 (2006), 767-778.

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

Yonggeun CHO, Sanghyuk LEE, and Yongsun SHIM

#### 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.

#### Article information

**Source**

Hokkaido Math. J., Volume 35, Number 4 (2006), 767-778.

**Dates**

First available in Project Euclid: 29 September 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1285766429

**Digital Object Identifier**

doi:10.14492/hokmj/1285766429

**Mathematical Reviews number (MathSciNet)**

MR2289360

**Zentralblatt MATH identifier**

1122.42008

**Subjects**

Primary: 42A45: Multipliers

Secondary: 42B25: Maximal functions, Littlewood-Paley theory

**Keywords**

Schr\"{o}dinger type equation maximal operator angular regularity

#### Citation

CHO, Yonggeun; LEE, Sanghyuk; SHIM, Yongsun. A maximal inequality associated to Schr\"{o}dinger type equation. Hokkaido Math. J. 35 (2006), no. 4, 767--778. doi:10.14492/hokmj/1285766429. https://projecteuclid.org/euclid.hokmj/1285766429