A p-specific spectral multiplier theorem with sharp regularity bound for Grushin operators

Abstract

In a recent work, Chen and Ouhabaz proved a p-specific L^p-spectral multiplier theorem for the Grushin operator acting on {mathbb {R}}^{d_1}times {mathbb {R}}^{d_2} which is given by L=-∑j=1d1∂xj2-(∑j=1d1|xj|2)∑k=1d2∂yk2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} L =-\\sum _{j=1}^{d_1} \\partial _{x_j}^2 - \\Bigg ( \\sum _{j=1}^{d_1} |x_j|^2\\Bigg ) \\sum _{k=1}^{d_2}\\partial _{y_k}^2. \\end{aligned}$$\\end{document}Their approach yields an L^p-spectral multiplier theorem within the range 1< ple min { 2d_1/(d_1+2), 2(d_2+1)/(d_2+3) } under a regularity condition on the multiplier which is sharp only when d_1ge d_2. In this paper, we improve on this result by proving L^p-boundedness under the expected sharp regularity condition s>(d_1+d_2)(1/p-1/2). Our approach avoids the usage of weighted restriction type estimates which played a key role in the work of Chen and Ouhabaz, and is rather based on a careful analysis of the underlying sub-Riemannian geometry and restriction type estimates where the multiplier is truncated along the spectrum of the Laplacian on {mathbb {R}}^{d_2}.