Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation $$\partial_t u -\Delta_gu=0 \quad \text{on $(0,T)\times M$},$$ subject to initial data $f$ at times $t=0$. Let the map $$ G: L^2(M) \to L^2(M)$$ be defined via $Gf=u(T,\cdot)$ on $M$.
Is it true that $G$ is compact?
Parabolic regularity show that $u$ is regular for all positive times; in particular $u(t,\cdot) \in W^{1,2}(M)$ for all $t > 0$. Interior parabolic estimates additionally show that there is a constant $C = C(g,T) > 0$ so that \begin{equation} \lvert u(t,\cdot) \rvert_{W^{1,2}} \leq C \lvert f \rvert_{L^2} \quad \text{ for all $t \in [T/2,2T]$.} \end{equation} Rellich–Kondrachov implies that the map $G: f \in L^2(M) \mapsto u(T,\cdot) \in L^2(M)$ is compact.
