论文标题
概率证明Apriori $ l^p $估算的一类差异表格椭圆操作员
A probabilistic proof of apriori $l^p$ estimates for a class of divergence form elliptic operators
论文作者
论文摘要
假设$ {\ cal l} $是$ {\ cal l} f：=（1/2）e^{u} \ nabla_x \ cdot \ big [e^{e^{ - u}（i+h）\ nabla_x f \ big] $ ud $ y $ y $ hh y $ hh y y y $ hh y y $ hh y y $ hh y y $ hh y y $ hh y $ hh y是抗对称矩阵值函数。该系数不认为是有限的，而是$ c^2 $常规。我们表明，如果$ z = \ int _ {\ mathbb {r}^d} e^{ - u（x）} dx <+\ infty $以及矩阵$ - \ frac12 \ nabla^nabla^2_x u+++++++\ \ \ nabla_x x x \ x \ x \ x \ x \ x \ x的数值范围的上流h- [\ nabla_x u]^th \ right \} $满足某些指数的集成性条件，相对于尺寸$dμ= z^{ - 1} e^{ - u} dx $，然后对于任何$ 1 \ le p <q <q <q <q <+\ infty $，存在常数$ c> 0 $ c> 0 $ \ eft of eft $ \ weft \ e \ weft \ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | f \ right \ | _ {w^{2，p}（μ）} \ le c \ big（\ left \ | {\ cal l} f \ right \ | _ {l^q（μ）}+\左\ \左\ \ | f \ | f \ | c_0^\ infty（\ mathbb {r}^d）$。这里$ w^{2，p}（μ）$是$ l^p（μ）$与两个衍生物集成在一起的函数的sobolev空间。我们的证明是概率的，并且依赖于Malliavin微积分的应用。
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 $dμ=Z^{-1}e^{-U}dx$, then for any $1 \le p<q<+\infty$ there exists a constant $C>0$ such that $\left\| f\right\|_{W^{2,p}(μ)}\le C\Big(\left\|{\cal L}f\right\|_{L^q(μ)}+\left\|f\right\|_{L^q(μ)}\Big)$ for $f\in C_0^\infty(\mathbb{R}^d)$. Here $W^{2,p}(μ)$ is the Sobolev space of functions that are $L^p(μ)$ integrable with two derivatives. Our proof is probabilistic and relies on an application of the Malliavin calculus.
