Bilinear multipliers of small Lebesgue spaces

Abstract

Let G be a compact abelian metric group with Haar measure lambda and (G) over cap its dual with Haar measure mu. Assume that 1 < p(i) < infinity, p(i)' = p(i)/p(i)-1, (i = 1, 2, 3) and theta >= 0. Let L-(pi' ,L-theta (G), (i = 1, 2, 3) be small Lebesgue spaces. A bounded sequence m(xi, eta) defined on G (over cap) x G (over cap) is said to be a bilinear multiplier on G of type [(p'(1); (p'(2); (p'(3)]. if the bilinear operator B-m associated with the symbol m B-m (f, g) (x) = Sigma(delta is an element of G)Sigma(t is an element of G) (f) over cap (s) (g) over cap (t) m(s, t) (s + t, x) defines a bounded bilinear operator from L-(p'1,L- theta (G) x L-(p2',L-theta (G) into L-(p3',L-theta (G). We denote by BM theta [(p(1)' ; (p(2)' ; (p(3)'] the space of all bilinear multipliers of type [(p(1)'; (p(2)'; (p(3)'](theta). In this paper, we discuss some basic properties of the space BM. [(p(1)'; (p(2)'; (p(3)'] and give examples of bilinear multipliers.