Consider an integral operator $$T:L^2({\mathbb R}^n)\to L^2({\mathbb R}^n),\qquad (Tf)(x)=\int_{\mathbb R^n} dy\,K(x,y)f(y),$$ with Schwartz type kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ (smooth and rapidly decaying at infinity, for all derivatives). Then $T$ is Hilbert-Schmidt (in particular, compact) with Hilbert-Schmidt norm $\|K\|_2$, i.e. the singular values $\mu_k$ of $T$ satisfy $$\sum_k\mu_k^2=\int_{\mathbb R^n} dx\int_{\mathbb R^n}dy\,|K(x,y)|^2<\infty.$$

are square summable. I wonder what the best statement on the decay of the $\mu_k$ is in this situation (Maybe $\sum_k\mu_k^p<\infty$ for all $p>0$)? I suppose this must be known (possible together with bounds on $\sum_k\mu_k^p$), but who knows a reference?

Edit: Under the stronger assumption that $K$ is Schwartz class and there exists a compact set $C\subset{\mathbb R}^n$ such that $K$ has support in $C\times{\mathbb R}^n$ or ${\mathbb R}^n\times C$, the conjectured $\sum_k\mu_k^p<\infty$ for all $p>0$ can be shown by multiplying $T$ with suitable test functions which are constant on $C$, and suitable powers of $(1-\Delta)$. But for general Schwartz kernels $K$ this argument does not apply.