Let $(M,g)$ be a Riemannian manifold with bounded geometry, which means that:
- the injectivity radius is positive;
- for each $l$, there exists $C_l>0$ such that $||\nabla^l R||\leq C_l$, where $R$ is the Riemann curvature tensor and the norm is taken fibrewise.
Let $T:L^2(M)\rightarrow L^2(M)$ be a bounded operator.
Question 1: Is there a general criterion for this type of setting that says when $T$ is given by a continuous Schwartz kernel $k_T:M\times M\rightarrow\mathbb{C}$?
Question 2: In particular, suppose for every $i\geq 0$ that $T$ maps from $H^i(M)$ to $C^\infty(M)$. Is this enough to guarantee the existence and continuity of $k_T$? Here $H^i(M)$ denotes the $i$-th Sobolev space on $M$.