If I have a normed function space $\mathcal{F}(X)$ on $X$, I can always take its dual. In some cases, we can identify the dual with some known space. The most typical example is: for $p\in(1,\infty)$, $\tfrac{1}{p}+\tfrac{1}{q}=1$, and $\mathcal{F}(\mathbb{R}^n) = L^p(\mathbb{R}^n)$, then $\mathcal{F}(\mathbb{R}^n)^* = L^q(\mathbb{R}^n)$.
My question: we have two function spaces, one $\mathcal{F}_1(X)$ and one $\mathcal{F}_2(Y)$ (we can think of them as $X\subseteq\mathbb{R}^n, Y\subseteq\mathbb{R}^m$), and we know some representation of their duals $\mathcal{F}_1(X)^*$ and $\mathcal{F}_2(Y)^*$. Let's define another space $\mathcal{F}(X\times Y)$ such that $f\in\mathcal{F}(X\times Y)$ means that $f:X\times Y \to \mathbb{R}$ and $f(x,\cdot)\in\mathcal{F}_2(Y)$ for all $x$, $f(\cdot,y)\in\mathcal{F}_1(X)$ for all $y$.
For example: $\mathcal{F}_1(X)= L^{p_1}(X)$, $\mathcal{F}_2(Y)=L^{p_2}(Y)$ and $f:X\times Y \to \mathbb{R}$ such that $f(x,\cdot)\in L^{p_2}(Y), \ f(\cdot,y)\in L^{p_1}(X)$.
Can we say something about the dual space $\mathcal{F}(X\times Y)^*$?
(the example with the Lebesgue spaces is mostly there to help with reading the notation; however, I am hoping that something holds with a bit more generality)