On functions with Fourier transforms in Generalized Grand Lebesgue space

Abstract: Let $1<p,q<\infty ,\ \theta_1 \geq 0,\ \theta_2 \geq 0$ and let $a(x), b(x)$ be a weight functions. In the present paper we intend to study the function space $A_{q),\theta {2}}^{p),\theta _{1}}\left( \mathbb R^n\right)$ consisting of all functions $f\in L_a^{p),\theta_1 }\left( \mathbb R^n\right) $ whose generalized Fourier transforms $\widehat{f}$ belong to grand $L_b^{q),\theta_2 }\left( \mathbb R^n\right), $ where $ L_a^{p),\theta_1 }\left( \mathbb R^n\right)$ and $L_b^{q),\theta_2 }\left( \mathbb R^n\right) $ are generalized grand Lebesgue spaces. In the second section some definitions and notations used in this work are given. In the third and fourth sections we discuss some basic properties and inclusion properties of $A{q),\theta {2}}^{p),\theta _{1}}\left( \mathbb R^n\right)$. In the fifth section we characterize the multipliers from ${L^{1 }(\mathbb R^{n}, a^{\frac{\varepsilon}{p}})}$ to $( L_a^{p),\theta}\left(\mathbb R^{n}\right))^{\ast}$ and from ${L^{1 }(\mathbb R^{n}, a^{\frac{\varepsilon}{p}})}$ into $(A{q),\theta_2}^{p),\theta_1}\left(\mathbb R^{n}\right))^{\ast}$ for ${0<\varepsilon \leq p-1}.$ The importance of this section is that, it gives us some insight into the structure of the dual space $( L_a^{p),\theta}\left(\mathbb R^{n}\right))^{\ast}$ of the generalized grand Lebesgue space, the properties of which are not yet known. Later we discuss duality and reflexivitiy properties of the space $A_{q),\theta _{2}}^{p),\theta _{1}}\left( \mathbb R^n\right)$.