The Generalized Grand Wiener Amalgam Spaces and the boundedness of Hardy-Littlewood maximal operators

In \cite{g5}, we defined and investigated the grand Wiener amalgam space $W(L^{p),{\theta }_1}(\mathrm{\Omega }),L^{q),{\theta }_2}(\mathrm{\Omega }))$ , where $1<p,q<\mathrm{\infty },{\theta }_1>0,{\theta }_2>0$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and the Lebesgue measure of $\mathrm{\Omega }$ is finite. In the present paper we generalize this space and define the generalized grand Wiener amalgam space $W(L_a^{p)}({\mathbb{R}}^n),L_b^{q)}({\mathbb{R}}^n)),$ where $L_a^{p)}({\mathbb{R}}^n)$ and $L_b^{q)}({\mathbb{R}}^n),$ are the generalized grand Lebesgue spaces, (see \cite{u}, \cite{su3}). Later we investigate some basic properties. Next we study embeddings for these spaces and we discuss boundedness and unboundedness of the Hardy-Littlewood maximal operator between some generalized grand Wiener amalgam spaces.