Multipliers of Lipschitz spaces on a certain class of groups

Abstract

Zygmund [lo], Mizuhara [7], and Anderson [l] have studied various multiplier problems of Lipschitz spaces defined on the circle group T. In [8,9], the authors have characterized the multipliers of Lipschitz spaces defined on a Vilenkin group G (i.e., G is an infinite compact O-dimensional metric Abelian group). In this paper we define a certain class S of metric locally compact Abelian groups, and study the multipliers from one Lipschitz space Lip(a,p; G) to another Lipschitz space Lip@, q; G) for G in .Y. The class .Y contains the classical groups T” and R”, as well as many other interesting groups. Our main result is a characterization of the multipliers from Lip(a,p; G) to Lip@‘, q; G), as restrictions of multipliers from Lp(G) to L9(G), for GE ,Y, 0 < a </3 < 1, 1 <p < co, and l<q<co.