## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 48, Number 4 (2004), 1081-1092.

### Weighted {$L\sp 2$} estimates for maximal operators associated to dispersive equations

#### Abstract

Let $Tf(x,t) = e^{2\pi it\phi(D)}f(x)$ be the solution of the general dispersive equation with phase $\phi$ and initial data $f$ in the Sobolev space $H^s$. We prove a weighted $L^2$ estimate for the global maximal operator $T^{**}$ defined by taking the supremum over the time variable $t \in \mathbb{R}$ so that $ \|T^{**}f\|_{L^2(w\,dx)} \le C\|f\|_{H^s}$. The exponent $s$ depends on the phase function $\phi$, whose gradient may vanish or have singularities.

#### Article information

**Source**

Illinois J. Math., Volume 48, Number 4 (2004), 1081-1092.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138500

**Digital Object Identifier**

doi:10.1215/ijm/1258138500

**Mathematical Reviews number (MathSciNet)**

MR2113666

**Zentralblatt MATH identifier**

1070.42011

**Subjects**

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

Secondary: 35J10: Schrödinger operator [See also 35Pxx] 35Q40: PDEs in connection with quantum mechanics 42A45: Multipliers

#### Citation

Cho, Yonggeun; Shim, Yongsun. Weighted {$L\sp 2$} estimates for maximal operators associated to dispersive equations. Illinois J. Math. 48 (2004), no. 4, 1081--1092. doi:10.1215/ijm/1258138500. https://projecteuclid.org/euclid.ijm/1258138500