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.



Nuclearity properties of Integral operators with Schwartz type kernels
To answer my own question: If the kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ is of Schwartz type, then the associated operator $T$ is $p$-nuclear for all $p>0$, i.e. its singular values satisfy $\sum\_k\mu\_k^p<\infty$ for all $p>0$, i.e. they decay faster than any inverse power of $k$.


One way of seeing this is as follows: Consider the decomposition of $K$ into the orthonormal basis $\phi\_{\underline{\alpha}}\otimes\phi\_{\underline{\beta}}:=\phi\_{\alpha\_1}\otimes...\otimes\phi\_{\alpha\_{n}}\otimes \phi\_{\beta\_1}\otimes...\otimes\phi\_{\beta\_{n}}$ of $L^2({\mathbb R}^{2n})$, where the $\phi\_i\in L^2({\mathbb R})$ denote the Hermite functions. Because $K$ is Schwartz class, one has $K=\sum\_{\underline{\alpha},\underline{\beta}}c\_{\underline{\alpha},\underline{\beta}}\cdot\phi\_{\underline{\alpha}}\otimes\phi\_{\underline{\beta}}$ with rapidly decreasing coefficients $c\_{\underline{\alpha},\underline{\beta}}$, i.e. for each $p>0$ one has $\sum\_{\underline{\alpha},\underline{\beta}}\left|c\_{\underline{\alpha},\underline{\beta}}\right|^p<\infty$. This statement can for example be found in Reed/Simon I Thm. V.13.


Hence the operator $T$ can be written as $T\xi=\sum\_{\underline{\alpha},\underline{\beta}}c\_{\underline{\alpha},\underline{\beta}}\langle\phi\_{\underline{\alpha}},\xi\rangle\,\phi\_{\underline{\beta}}$, so that $\langle\phi\_{\underline{\alpha}},T\phi\_{\underline{\beta}}\rangle=c\_{\underline{\beta},\underline{\alpha}}$. But the singular values satisfy $\sum\_k\mu\_k^p\leq\sum\_{\underline{\alpha},\underline{\beta}}\left|\langle\phi\_{\underline{\alpha}},T\phi\_{\underline{\beta}}\rangle\right|^p$, which is finite by the previous argument.


