Illinois Journal of Mathematics

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

Yonggeun Cho and Yongsun Shim

Full-text: Open access

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


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