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

Cho, Yonggeun ; Shim, Yongsun

Illinois J. Math., Tome 48 (2004) no. 3, p. 1081-1092 / Harvested from Project Euclid

Résumé

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.

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Publié le : 2004-10-15
Classification:  42B25,  35J10,  35Q40,  42A45