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

假设 $\mathcal{大号}$ 是形式的散度形式微分算子 $\mathcal{大号}F：=（\mathrm{1个}/2）{Ë}^{ü}{\mathrm{\nabla }}_X\cdot [{Ë}^{-ü}（\mathrm{一世}+H）{\mathrm{\nabla }}_{X}F]$，其中U是标量值，I是单位矩阵，H是反对称矩阵值函数。系数不被认为是有界的，而是$C^2$定期。我们证明如果$ž={\int }_{{\mathbb{[R}}^d}{Ë}^{-ü（X）}dX<+\infty$ 和矩阵的数值范围的最大值 $-\frac{\mathrm{1个}}{2}{\mathrm{\nabla }}_X^2ü+\frac{\mathrm{1个}}{2}{\mathrm{\nabla }}_X\{{\mathrm{\nabla }}_X\cdot H-{[{\mathrm{\nabla }}_Xü]}^{Ť}H\}$ 满足关于度量的某些指数可积性条件 $d\mu ={ž}^{-\mathrm{1个}}{Ë}^{-ü}dX$，那么对于任何 $\mathrm{1个}\le p<q<+\infty$ 存在一个常数 $C>0$ 这样 ${\Vert F\Vert }_{{\mathrm{w\; ^}}^{2，p}（\mu ）}\le C（{\Vert \mathcal{大号}F\Vert }_{{\mathrm{大号}}^q（\mu ）}+{\Vert F\Vert }_{{\mathrm{大号}}^q（\mu ）}）$ 对于 $F\in C_0^{\infty }（{\mathbb{[R}}^d）$。这里${\mathrm{w\; ^}}^{2，p}（\mu ）$ 是函数的Sobolev空间 ${\mathrm{大号}}^p（\mu ）$与两个导数可积分。我们的证明是概率性的，并且依赖于Malliavin演算的应用。