Transference and restriction of Fourier multipliers on Orlicz spaces

Abstract

Let G be a locally compact abelian group with Haar measure (Formula presented.) and (Formula presented.) be Young functions. A bounded measurable function m on G is called a Fourier (Formula presented.) -multiplier if (Formula presented.) defined for functions in (Formula presented.) such that (Formula presented.), extends to a bounded operator from (Formula presented.) to (Formula presented.). We write (Formula presented.) for the space of (Formula presented.) -multipliers on G and study some properties of this class. We give necessary and sufficient conditions for m to be a (Formula presented.) -multiplier on various groups such as (Formula presented.), and (Formula presented.). In particular, we prove that regulated (Formula presented.) -multipliers defined on (Formula presented.) coincide with (Formula presented.) -multipliers defined on the real line with the discrete topology D, under certain assumptions involving the norm of the dilation operator acting on Orlicz spaces. Also, several transference and restriction results on multipliers acting on (Formula presented.) and (Formula presented.) are achieved.