Regularities of General Maximal Wave Operators

Abstract

In this note, we consider a general maximal wave operator defined by

\begin{array}{c}\hfill W_{a,t}f(x)={\int }_{{\mathbb{R}}^n}{e}^{i(x·\xi +t(x)|\xi |)}a(x,\xi )\widehat{f}(\xi )d\xi ,\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} W_{a,t}f(x)=\int _{\mathbb {R}^{n}}e^{i(x\cdot \xi +t(x)|\xi |)}a(x,\xi )\widehat{f}(\xi )d\xi , \end{aligned}$$\end{document}

where the amplitude

a\in L^{\infty }{S}_{\rho }^m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a\in L^{\infty }S^{m}_{\rho }$$\end{document}

and

t\in L^{\infty }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t\in L^{\infty }$$\end{document}

. We prove that this operator is bounded on

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{2}$$\end{document}

provided

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} m<\frac{(n-1)\rho -n}{2}. \end{aligned}$$\end{document}

As a direct application, we obtain the well-known result that the maximal wave operator

W^{\ast }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$W^*$$\end{document}

is bounded from the Sobolev space

H^s=W^{s,2}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^s=W^{s,2}$$\end{document}

to

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}

if

s>\frac{1}{2}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s>\frac{1}{2}$$\end{document}

. This result is known to be sharp for

s>\frac{1}{2}.

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s>\frac{1}{2}.$$\end{document}

.

Keywords

Maximal wave operator / Fourier integral operator / Rough Hörmander class / Rough k-corank condition / 42B20 / 42B37

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