Sharp Estimates for Schrödinger Groups on Hardy Spaces for 0

Abstract

Let X be a space of homogeneous type with the doubling order n. Let L be a nonnegative self-adjoint operator on L^2(X) and suppose that the kernel of e^{-tL} satisfies a Gaussian upper bound. This paper shows that for 0<ple 1 and s=n(1/p-1/2), ‖(I+L)-seitLf‖HLp(X)≲(1+|t|)s‖f‖HLp(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}\\Vert (I+L)^{-s}e^{itL}f\\Vert _{H^p_L(X)} \\lesssim (1+|t|)^{s}\\Vert f\\Vert _{H^p_L(X)} \\end{aligned}$$\\end{document}for all tin {mathbb {R}}, where H^p_L(X) is the Hardy space associated to L. This recovers the classical results in the particular case when L=-Delta and extends a number of known results.