Weighted variable exponent amalgam spaces $W(L^{p(x)},L_w^q)$

In the present paper a new family of Wiener amalgam spaces W(Lp(x),Lwq) is defined, with local component which is a variable exponent Lebesgue space Lp(x)(Rn) and the global component is a weighted Lebesgue space Lwq(Rn). We proceed to show that these Wiener amalgam spaces are Banach function spaces. We also present new Hölder-type inequalities and embeddings for these spaces. At the end of this paper we show that under some conditions the Hardy-Littlewood maximal function is not mapping the space W(Lp(x),Lwq) into itself