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

Suppose that $\mathcal{L}$ is a divergence form differential operator of the form $\mathcal{L}f:=(1\mathrm{/}2)e^U{\mathrm{\nabla }}_x\cdot [e^{-U}(I+H){\mathrm{\nabla }}_{x}f]$, 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 $Z={\int }_{{\mathbb{R}}^d}{e}^{-U(x)}dx<+\mathrm{\infty }$ 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 $1\le p<q<+\mathrm{\infty }$ 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}$^d)$\mathrm{.}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.