$$L_p$$–$$L_q$$ Fourier Multipliers on Locally Compact Quantum Groups

Abstract

Let $$\mathbb {G}$$ G be a locally compact quantum group with dual $$\widehat{\mathbb {G}}$$ G ^ . Suppose that the left Haar weight $$\varphi $$ φ and the dual left Haar weight $$\widehat{\varphi }$$ φ ^ are tracial, e.g. $$\mathbb {G}$$ G is a unimodular Kac algebra. We prove that for $$1<p\le 2 \le q<\infty $$ 1 < p ≤ 2 ≤ q < ∞ , the Fourier multiplier $$m_{x}$$ m x is bounded from $$L_p(\widehat{\mathbb {G}},\widehat{\varphi })$$ L p ( G ^ , φ ^ ) to $$L_q(\widehat{\mathbb {G}},\widehat{\varphi })$$ L q ( G ^ , φ ^ ) whenever the symbol x lies in $$L_{r,\infty }(\mathbb {G},\varphi )$$ L r , ∞ ( G , φ ) , where $$1/r=1/p-1/q$$ 1 / r = 1 / p - 1 / q . Moreover, we have $$\begin{aligned} \Vert m_{x}:L_p(\widehat{\mathbb {G}},\widehat{\varphi })\rightarrow L_q(\widehat{\mathbb {G}},\widehat{\varphi })\Vert \le c_{p,q} \Vert x\Vert _{L_{r,\infty }(\mathbb {G},\varphi )}, \end{aligned}$$ ‖ m x : L p ( G ^ , φ ^ ) → L q ( G ^ , φ ^ ) ‖ ≤ c p , q ‖ x ‖ L r , ∞ ( G , φ ) , where $$c_{p,q}$$ 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 $$\mathbb {R}^n$$ 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 $$L_p$$ L p -Fourier multipliers over discrete group von Neumann algebras. A similar result for $$\mathcal {S}_p$$ S p -$$\mathcal {S}_q$$ S q Schur multipliers is also proved.