While reading Grigor'yan's book on the heat kernel I have encountered the following definition of a Sobolev space on a Riemannian manifold $M$:
$W^2 (M) = \{ u \in W^1 (M) : \Delta u \in L^2 (M) \}$.
I can understand what is means, but how do I practically check this condition? Concretely, is $\int \limits _M \Bbb e ^{\Bbb - d(x,y)^2} f(y) \Bbb d y$ (with $f \in W^2 (M)$) an element of $W^2 (M)$? (Assume $M$ of finite measure).