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Jun 2, 2020 at 8:18 comment added Mikael de la Salle @DirkWerner : thanks, I also like this argument.
Jun 2, 2020 at 0:29 comment added Yemon Choi @DirkWerner thanks! I like this argument, since in one version of the intended application, integral operators occur quite naturally, and this factorization trick might be useful for independent reasons
Jun 1, 2020 at 21:20 comment added Dirk Werner Another way to see this (for $k=1$) is by rephrasing your question in terms of integral operators; what you are asking is whether an integral operator on $L_2$ with a $C^1$-kernel is nuclear. By compactness of the support one can assume that $k\in C^1([0,1]^2)$ vanishes at the boundary; let $T_k$ be the associated integral operator. Let $k_2(s,t) = \frac{\partial}{\partial t} k(s,t)$ and $V(f)(t)=\int_0^t f(s)\,ds$. Then $T_k = T_{k_2}V$ by integration by parts; hence $T_k$ factors as a product of two Hilbert-Schmidt operators and is therefore nuclear.
Jun 1, 2020 at 15:37 comment added Yemon Choi Hi @MikaeldelaSalle - yes, your first comment was more or less the argument I had in mind, but it is good to get confirmation (I was taking $k\geq2$ just to play safe but as you observe this is unnecessary). I had overlooked your second point that the group structure is unnecessary and that one can localize to $L^2$ of a $d$-cube (hence a $d$-torus)
Jun 1, 2020 at 7:28 comment added Mikael de la Salle For your second question, the fact that you have a Lie group is not relevant: in any compact manifold (or any compactly supported function on a manifold), working in local coordinates with a partition of unity reduces the problem to $(\mathbf{R}/\mathbf{Z})^d$, in which case the above Fourier transform argument shows that $C^k$ functions work whenever $k>d/2$.
Jun 1, 2020 at 7:21 comment added Mikael de la Salle I do not know of a good reference, but the Fourier transform argument works very well for every $k>1$: by the compact support assumption, you can as well periodize the problem and work with a $C^k$ function $(\mathbf{R}/{\mathbf{Z}})^2 \to \mathbf{R}$. Then the Fourier expansion gives that $\|f\|_{L^2 \hat \otimes L^2} \leq \sum_{i,j} |\hat f(i,j)| \leq C(\varepsilon) \|f\|_{H^{1+\varepsilon}}$ for every $\varepsilon >0$ (here $H^s$ is the usual Sobolev of functions such that $((1+|i|+|j|)^s\hat f(i,j))_{i,j}$ belongs to $\ell^2$).
May 31, 2020 at 22:12 history asked Yemon Choi CC BY-SA 4.0