Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces

Abstract

Let ?1, ?2 be slowly increasing weight functions, and let ?3 be any weight function on Rn. Assume that m(? ,?) is a bounded, measurable function on Rn × Rn. We define Bm(f, g)(x) = Rn Rnˆ f(? )gˆ(?)m(? ,?)e2?i?+?,x d? d? for all f, g ? C? c (Rn). We say that m(? ,?) is a bilinear multiplier on Rn of type (W(p1, q1,?1; p2, q2,?2; p3, q3,?3)) if Bm is a bounded operator from W(Lp1 , L q1 ?1 ) × W(Lp2 , L q2 ?2 ) to W(Lp3 , L q3 ?3 ), where 1 ? p1 ? q1 < ?, 1 ? p2 ? q2 < ?, 1 < p3, q3 ? ?. We denote by BM(W(p1, q1,?1; p2, q2,?2; p3, q3,?3)) the vector space of bilinear multipliers of type (W(p1, q1,?1; p2, q2,?2; p3, q3,?3)). In the first section of this work, we investigate some properties of this space and we give some examples of these bilinear multipliers. In the second section, by using variable exponent Wiener amalgam spaces, we define the bilinear multipliers of type (W(p1(x), q1,?1; p2(x), q2,?2; p3(x), q3,?3)) from W(Lp1(x) , L q1 ?1 ) × W(Lp2(x) , L q2 ?2 ) to W(Lp3(x) , L q3 ?3 ), where p* 1, p* 2, p* 3 < ?, p1(x) ? q1, p2(x) ? q2, 1 ? q3 ? ? for all p1(x), p2(x), p3(x) ? P(Rn). We denote by BM(W(p1(x), q1,?1; p2(x), q2,?2; p3(x), q3,?3)) the vector space of bilinear multipliers of type (W(p1(x), q1,?1; p2(x), q2,?2; p3(x), q3,?3)). Similarly, we discuss some properties of this space.