The Amalgam spaces W(Lp(x),{pn} ) and boundedness of hardy–littlewood maximal operators

Abstract

Let Lq(x)(R)Lq(x)(R) be variable exponent Lebesgue space and l{qn}l{qn} be discrete analog of this space. In this work we define the amalgam spaces W(L p(x),L q(x)) and W(Lp(x),l{qn})W(Lp(x),l{qn}), and discuss some basic properties of these spaces. Since the global components Lq(x)(R)Lq(x)(R) and l{qn}l{qn} are not translation invariant, these spaces are not a Wiener amalgam space. But we show that there are similar properties of these spaces to the Wiener amalgam spaces. We also show that there is a variable exponent q(x) such that the sequence space l{qn}l{qn} is the discrete space of Lq(x)(R)Lq(x)(R). By using this result we prove that W(Lp(x),l{pn})=Lp(x)(R)W(Lp(x),l{pn})=Lp(x)(R). We also study the frame expansion in Lp(x)(R)Lp(x)(R). At the end of this work we prove that the Hardy–Littlewood maximal operator from W(Ls(x),l{tn})W(Ls(x),l{tn}) into W(Lu(x),l{vn})W(Lu(x),l{vn}) is bounded under some assumptions.