Let $(M,g)$ be a complete non-compact manifold, $\Delta$ denote the Laplacian operator on smooth functions.
Q Is this true that $\int_M(\Delta f\cdot f)dvol_M\geq0$ for any function $f\in L^2(M)$ or $L^2_1(M)$.
Here $\Delta=d^*d$, I think it is not true for smooth case, e.g. $f=\exp(x)$ on $\mathbb R$.
I think it is true for $f\in L^2_2$, as $\Delta:L^2_2\to L^2$ is a continuous operator.