Let $\Omega \subset \mathbb{R}^n$ be some open set. If, for all $\psi \in L^2(\Omega)$ and some fixed integral kernel $k \in L^2(\Omega\times \Omega)$ and $\ell>0$, it is true that both $\int_{\Omega} k(\cdot,y) \psi(y)\,dy$ and $\int_{\Omega} k(y,\cdot) \psi(y)\,dy$ are $H^\ell$-functions.

Does it follow hat $k \in H^\ell(\Omega \times \Omega)?$ I doubt that this is true, because componentwise differentiability should probably not include the common differentiability. However, I am looking for the best regularity we get for $k$ with this property. (Maybe also classical properties like continuity or existence of partial derivatives could be established).