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

Abstract

Suppose that $\mathcal{L}$ is a divergence form differential operator of the form $\mathcal{L}f:=(1/2)e^U{\nabla }_x\cdot [e^{-U}(I+H){\nabla }_{x}f]$, 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 $Z={\int }_{{\mathbb{R}}^d}{e}^{-U(x)}dx<+\infty$ and the supremum of the numerical range of matrix $-\frac{1}{2}{\nabla }_x^{2}U+\frac{1}{2}{\nabla }_x\left\{{\nabla }_x\cdot H-[{\nabla }_{x}U{]}^{T}H\right\}$ 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 ${\Vert f\Vert }_{W^{2,p}(\mu )}\le C({\Vert \mathcal{L}f\Vert }_{L^q(\mu )}+{\Vert f\Vert }_{L^q(\mu )})$ 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.

15 pages