Up on Schwartz kernel theorem we know that the kernel of an integral operator belongs to distribution space S'(R^n). Moreover, we know that the kernel K is C^{\infty} off diagonal in R^n \times R^n. Now my questions are that

1-Dose Schwartz kernel of pseudo-differential operator of arbitrary order belong to Schwartz space?

2- Can we have the Schwartz kernel Theorem with kernel belong to S(R^n) Not S'(R^n)?