Let $f:\mathbb{R^2}\rightarrow \mathbb{R}$ and $F:\mathbb{R}\rightarrow \mathbb{R}$ be given functions such that $\int_\mathbb{R} F(x) dx = 0$.
Find all $h:\mathbb{R^2}\rightarrow \mathbb{R}$ satisfying \begin{align} \int_\mathbb{R} h(x,y)f(x,y) dy &= F(x) \\ \int_\mathbb{R} h(x,y)f(x,y) dx &= 0 \qquad \forall y \in \mathbb{R} \end{align}
Can you please point me in the direction how tackle this kind of problem?
I don't even know if such $h$ exists. But I have felling that it exists but it is not unique, so is there an easy way how to characterize all the solutions?