Regularity of integral kernel

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).

The assumptions imply that $k\in H^\ell(\Omega)\hat\otimes L^2(\Omega) \cap L^2(\Omega) \hat\otimes H^\ell(\Omega)$, where $\hat\otimes$ is the (completed) Hilbert space tensor product. This space equals $H^\ell(\Omega_x;L^2(\Omega_y))\cap L^2(\Omega_x;H^\ell(\Omega_y))$ which, by interpolation, is contained in $H^{\ell-t}(\Omega_x;H^t(\Omega_y))$, for any $0\leq t\leq \ell$. So, indeed, your kernel belongs to $H^\ell(\Omega\times \Omega)$.