Oscillating singular integral operators on compact Lie groups revisited

Abstract

Fefferman (Acta Math 24:9–36, 1970, Theorem 2') has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian Delta , namely, operators of the form 0.2Tθ(-Δ):=(1-Δ)-nθ4ei(1-Δ)θ2,0≤θ<1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} T_{\ heta }(-\\Delta ):= (1-\\Delta )^{-\\frac{n\ heta }{4}}e^{i (1-\\Delta )^{\\frac{\ heta }{2}}},\\,0\\le \ heta <1. \\end{aligned}$$\\end{document}The aim of this work is to extend Fefferman’s result to oscillating singular integrals on any arbitrary compact Lie group. We also consider applications to oscillating spectral multipliers of the Laplace–Beltrami operator. The proof of our main theorem illustrates the delicate relationship between the condition on the kernel of the operator, its Fourier transform (defined in terms of the representation theory of the group) and the microlocal/geometric properties of the group.