Global L 2 $L^{2}$ estimates for a class of maximal operators associated to general dispersive equations
Global L 2 $L^{2}$ estimates for a class of maximal operators associated to general dispersive equations
For a function ϕ satisfying some suitable growth conditions, consider the general dispersive equation defined by $\bigl\{ \scriptsize{ \begin{array}{l} i\partial_{t}u+\phi(\sqrt{-\Delta})u=0,\quad (x,t)\in\mathbb {R}^{n}\times\mathbb{R}, \\ u(x,0)=f(x), \quad f\in\mathcal{S}(\mathbb{R}^{n}). \end{array} }\bigr. $ (∗) In the present paper, we give some global $L^{2}$ estimate for the maximal operator $S_{\phi}^{*}$ , which is defined by …