This is in relation with the question in Compact "Hilbert-Schmidt'' integral operators on Sobolev spaces where I asked about compactness of integral operators. Since I did not get an answer, here, I am asking if boundedness is preserved when considering the spaces $H^s(\mathbb{R}), s>1$. More precisely:
If one has an integral operator with kernel $K\in L^2(\mathbb{R}^2)$, then it is known as a Hilbert-Schmidt integral operator and it is compact on $L^2(\mathbb{R})$ and thus bounded.
My operator has more regularity. In the case where $K\in H^1(\mathbb{R}^2)$, it is to show that it is bounded on $H^1(\mathbb{R})$. What about if $K\in H^s(\mathbb{R}^2)$, $s>1$. Is the operator then bounded on $H^s(\mathbb{R})$?