Weighted Estimates for a Class of Global Maximal Operators Associated with Dispersive Equation

For a function ϕ satisfying some suitable growth conditions, consider the following general dispersive equation defined by {i∂tu+ϕ(−Δ)u=0,(x,t)∈ℝn×ℝ,u(x,0)=f(x),f∈S(ℝn),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{i{\partial _t}u + \phi (\sqrt { - \Delta } )u = 0,} \hfill & {(x,t) \in {\mathbb{R}^n} \times \mathbb{R},} \hfill \cr {u(x,0) = f(x),} \hfill & {f \in {\cal S}({\mathbb{R}^n}),} \hfill \cr } } \right.$$\end{document} where ϕ(−Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (\sqrt { - \Delta } )$$\end{document} is a pseudo-differential operator with symbol ϕ(∣ξ∣). In the present paper, when the initial data f belongs to Sobolev space, we give the local and global weighted Lq estimate for the global maximal operator Sϕ∗∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\phi ^{ \ast \ast }$$\end{document} defined by Sϕ∗∗f(x)=supt∈ℝ|St,ϕf(x)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\phi ^{ \ast \ast }f(x) = \mathop {\sup }\limits_{t \in \mathbb{R}} \left| {{S_{t,\phi }}f(x)} \right|$$\end{document}, where St,ϕf(x)=(2π)−n∫ℝneix⋅ξ+itϕ(|ξ|)f^(ξ)dξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_{t,\phi }}f(x) = {(2\pi )^{ - n}}\int_{{\mathbb{R}^n}} {{e^{ix \cdot \xi + it\phi (\left| \xi \right|)}}\hat f(\xi )d\xi } $$\end{document} is a formal solution of the equation (*).