The well-known Schwartz kernel theorem states that a continuous operator from smooth test functions to distributions, $T \colon C^\infty_c(\mathbb{R}^n) \to C^\infty_c(\mathbb{R}^n)'$ is continuous iff it is of the form $T[f](x) = \int_{\mathbb{R}^n} T(x,y) f(y) \, dy$, where the kernel is a distribution $K(x,y) \in C^\infty_c(\mathbb{R}^n \times \mathbb{R}^n)'$.

**Q:** Is there any analog of such a theorem where smooth test functions and distributions are replaced by Sobolev spaces $H^{s}$ or $W^s_p$?

For instance, $H^{s}$ with $s\ge 0$ could play the role of test functions, while those with $s\le 0$ could play the role of distributions. The simplest example of a multiplication operator $T\colon H^0 \to H^0$, where $T[f](x) = t(x)f(x)$ with compactly supported and sufficiently regular $t(x)$, and of course $H^0 = L^2(\mathbb{R}^n)$, shows that a simple transcription of the result is false, since $T(x,y) = t(x)\delta(x-y)$, and $t(x)\delta(x-y) \not\in L^2(\mathbb{R}^n \times \mathbb{R}^n)$. But $t(x)\delta(x-y) \in H^{-1}(\mathbb{R}^n \times \mathbb{R}^n)$. So perhaps for any continuous operator $T\colon H^{s_1} \to H^{s_2}$ there necessarily exists an $s$ made out of $s_1$, $s_2$ and some appropriate shift, such that $T(x,y) \in H^{s}$.

categoricaltensor product (e.g.) of it with itself. If therewerea TVS $V\otimes V$, then we have a Cartan-Eilenberg-type adjunction ${\mathrm Hom}(V\otimes V,\mathbb C)\approx {\mathrm Hom}(V,{\mathrm Hom}(V,\mathbb C))$. That is $(V\otimes V)^*\approx {\mathrm Hom}(V,V^*)$. $\endgroup$4more comments