On functions with Fourier transforms in Generalized Grand Lebesgue space
Let $1<p,q<\infty ,\ θ_1 \geq 0,\ θ_2 \geq 0$ and let $a(x), b(x)$ be a weight functions. In the present paper we intend to study the function space $A_{q),θ_{2}}^{p),θ_{1}}\left( \mathbb R^n\right)$ consisting of all functions $f\in L_a^{p),θ_1 }\left( \mathbb R^n\right) $ whose generalized Fourier transforms $\widehat{f}$ belong to grand $L_b^{q),θ_2 }\left( \mathbb R^n\right), $ where $ L_a^{p),θ_1 }\left( \mathbb R^n\right)$ and $L_b^{q),θ_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 $A_{q),θ_{2}}^{p),θ_{1}}\left( \mathbb R^n\right)$. In the fifth section we characterize the multipliers from ${L^{1 }(\mathbb R^{n}, a^{\frac{\varepsilon}{p}})}$ to $( L_a^{p),θ}\left(\mathbb R^{n}\right))^{\ast}$ and from ${L^{1 }(\mathbb R^{n}, a^{\frac{\varepsilon}{p}})}$ into $(A_{q),θ_2}^{p),θ_1}\left(\mathbb R^{n}\right))^{\ast}$ 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 $( L_a^{p),θ}\left(\mathbb R^{n}\right))^{\ast}$ of the generalized grand Lebesgue space, the properties of which are not yet known. Later we discuss duality and reflexivitiy properties of the space $A_{q),θ_{2}}^{p),θ_{1}}\left( \mathbb R^n\right)$.