A probabilistic proof of apriori $l^p$ estimates for a class of divergence form elliptic operators

Suppose that ${\cal L}$ is a divergence form differential operator of the form ${\cal L}f:=(1/2) e^{U}\nabla_x\cdot\big[e^{-U}(I+H)\nabla_x f\big]$, where $U$ is scalar valued, $I$ identity matrix and $H$ an anti-symmetric matrix valued function. The coefficients are not assumed to be bounded, but are $C^2$ regular. We show that if $Z=\int_{\mathbb{R}^d}e^{-U(x) }dx<+\infty$ and the supremum of the numerical range of matrix $-\frac12\nabla^2_x U+\frac12\nabla_x\left\{\nabla_x\cdot H-[\nabla_x U]^TH\right\}$ satisfies some exponential integrability condition with respect to measure $d\mu=Z^{-1}e^{-U}dx$, then for any $1 \le p<q<+\infty$ there exists a constant $C>0$ such that $\left\| f\right\|_{W^{2,p}(\mu)}\le C\Big(\left\|{\cal L}f\right\|_{L^q(\mu)}+\left\|f\right\|_{L^q(\mu)}\Big)$ for $f\in C_0^\infty(\mathbb{R}^d)$. Here $W^{2,p}(\mu)$ is the Sobolev space of functions that are $L^p(\mu)$ integrable with two derivatives. Our proof is probabilistic and relies on an application of the Malliavin calculus.