L_p–L_q Fourier Multipliers on Locally Compact Quantum Groups

Abstract

Let mathbb {G} be a locally compact quantum group with dual widehat{mathbb {G}}. Suppose that the left Haar weight varphi and the dual left Haar weight widehat{varphi } are tracial, e.g. mathbb {G} is a unimodular Kac algebra. We prove that for 1<ple 2 le q<infty , the Fourier multiplier m_{x} is bounded from L_p(widehat{mathbb {G}},widehat{varphi }) to L_q(widehat{mathbb {G}},widehat{varphi }) whenever the symbol x lies in L_{r,infty }(mathbb {G},varphi ), where 1/r=1/p-1/q. Moreover, we have ‖mx:Lp(G^,φ^)→Lq(G^,φ^)‖≤cp,q‖x‖Lr,∞(G,φ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert m_{x}:L_p(\\widehat{\\mathbb {G}},\\widehat{\\varphi })\\rightarrow L_q(\\widehat{\\mathbb {G}},\\widehat{\\varphi })\\Vert \\le c_{p,q} \\Vert x\\Vert _{L_{r,\\infty }(\\mathbb {G},\\varphi )}, \\end{aligned}$$\\end{document}where c_{p,q} is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93–140, 1960) for mathbb {R}^n, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of L_p-Fourier multipliers over discrete group von Neumann algebras. A similar result for mathcal {S}_p-mathcal {S}_q Schur multipliers is also proved.