A homomorphism theorem for multipliers

Nakhlé Habib Asmar
1989 Proceedings of the Edinburgh Mathematical Society
Notation and introduction Throughout the paper, the symbols G % and G 2 will denote two locally compact abelian groups with character groups X^ and X 2 , respectively. Haar measures on Gj are denoted by fiy, the ones on Xj are denoted by 6j (/ = 1,2). The measures fij and dj are normalized so that the Plancherel Theorem holds (see [7, p. 226, Theorem 31.18]). If G is a locally compact abelian group with character group X, and if / is a complex-valued function on G, then / is said to be
more » ... said to be measurable means that / is measurable with respect to Haar measure on G. The class of measurable functions on G, with integrable pth power, is denoted by if p (G) l^p<oo; the class of essentially bounded measurable functions by &JJ3); the class of continuous functions with compact support by ^Jfi). If A is a subset of G, the complement of A in G is denoted by A' or G\A. The symbol l A will denote the indicator function of the set A. All other notation used in this paper without explanation is as in  and  . A bounded measurable function m on X is called an i? p (G)-multiplier, lgp<oo, if for every / in y^G) n £f 2 (G) there is a g in y p {G) such that g = mf, and ||g|| p^A T p (»i)||/|| p , where N p (m) is the norm of the unique extension of the bounded linear operator f-*g to all of i? p (G). We shall denote this extension by T m . The set of all multipliers on SC P (G) will be denoted by M P (G). Suppose that i is a continuous nonzero homomorphism from X 2 into X t . A well-known theorem for multipliers asserts that if m is continuous on X u then m n is in M p (G 2 ) and JV p (m o T) ^ N p (m). (See [5, Theorem B.2.1, p. 187]). We will refer to this fact as the homomorphism theorem for continuous multipliers. Many interesting multipliers are not continuous; e.g. the sgn function on U which is an £f p (R)-multiplier for l<p<oo. Our goal, in this essay, is to give a new proof of the homomorphism theorem for continuous multipliers based on the so-called transference methods, then derive a more general version that applies to multipliers like the sgn function. The homomorphism theorem We continue with the notation of Section 1: m is a bounded continuous function on X lt and T is a continuous nonzero homomorphism from X 2 into X l .