Schwartz kernel

Question

By the Schwartz Kernel Theorem, we know that the kernel of an integral operator belongs to distribution space $\mathscr{S}'(\mathbb{R}^n)$. Moreover, we know that the kernel $K$ is $C^{\infty}$ off the diagonal in $\mathbb{R}^n \times \mathbb{R}^n$.

Now my questions are:

1. Does the Schwartz kernel of a pseudo-differential operator of arbitrary order belong to the Schwartz space?

2. Can we have the Schwartz Kernel Theorem be true with the kernel belonging to $\mathscr{S}(\mathbb{R}^n)$ as opposed to $\mathscr{S}'(\mathbb{R}^n)$?

Answer 1

Your preliminary statement needs clarification: a linear continuous operator $T$ from $\mathscr S(\mathbb R^n)$ into $\mathscr S'(\mathbb R^n)$ has a kernel $k$ in $\mathscr S'(\mathbb R^n\times\mathbb R^n )$ so that for $\phi, \psi\in \mathscr S(\mathbb R^n)$, $$ \langle T\psi,\phi\rangle_{\mathscr S'(\mathbb R^n),\mathscr S(\mathbb R^n)} =\langle k(x,y),\phi(x) \psi (y)\rangle_{\mathscr S'(\mathbb R^{2n}),\mathscr S'(\mathbb R^{2n})}. $$ So some operators may have a very singular kernel everywhere and not only on the diagonal. A simple example of an operator with a singular kernel off diagonal is the translation of vector $h$: $$ (T\phi)(x)=\phi(x-h), $$ whose kernel is $k(x,y)=\delta_0(x-h-y)$, singular at $x=h+y$.

Your statement (1) is incorrect: the identity has the kernel $\delta_0(x-y)$ and this answers negatively (2).

What is true about pseudodifferential operators is that their kernels are indeed smooth off diagonal but more and more singular at the diagonal when you increase the order of the operator. The simplest example is to consider, say in one dimension, the operator $(d/dx)^N$, a differential operator of order $N$: its kernel is $$ \delta_0^{(N)}(x-y). $$ Operators with kernels in $\mathscr S(\mathbb R^{2n})$ are some particular type of regularizing operators.