\[Lu = \dpn{A, D^2u}{\real^{d \times d}}+ \dpn{b, Du}{\real^d}\]
be an essentially self-adjoint diffusion operator on $\real^{d}$ with smooth coefficients and symmetrising measure $\mu$.
Let $f \in \cd(\real^{d})$ and $g \in C^{\infty}(\mu) \cap L^{2}(\mu)$, then
\[\dpn{f^2g, Lg}{L^2(\mu)}\]
Proof. Since $\mu$ is a symmetrising measure, $\dpn{f^2g, Lg}{L^2(\mu)}= \dpn{L(f^2g), g}{L^2(\mu)}$.
By Lemma 2.2.3,
\[L(f^{2}g) = 2\dpn{D(f^2), ADg}{\real^d}+ f^{2}Lg + gL(f^{2})\]
$\square$