# Nonlocal Hormander's hypoellipticity theorem

Research paper by **Xicheng Zhang**

Indexed on: **07 Apr '14**Published on: **07 Apr '14**Published in: **Mathematics - Probability**

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#### Abstract

Consider the following nonlocal integro-differential operator: for
$\alpha\in(0,2)$, $$ \cal L^{(\alpha)}_{\sigma,b} f(x):=\mbox{p.v.}
\int_{\mathbb{R}^d-\{0\}}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}d
z+b(x)\cdot\nabla f(x), $$ where
$\sigma:\mathbb{R}^d\to\mathbb{R}^d\times\mathbb{R}^d$ and
$b:\mathbb{R}^d\to\mathbb{R}^d$ are two $C^\infty_b$-functions, and p.v. stands
for the Cauchy principal value. Let $B_1(x):=\sigma(x)$ and
$B_{j+1}(x):=b(x)\cdot\nabla B_j(x)-\nabla b(x)\cdot B_j(x)$ for
$j\in\mathbb{N}$. Under the following H\"ormander's type condition: for any
$x\in\mathbb{R}^d$ and some $n=n(x)\in\mathbb{N}$, $$ \mathrm{Rank}[B_1(x),
B_2(x),\cdots, B_n(x)]=d, $$ by using the Malliavin calculus, we prove the
existence of the heat kernel $\rho_t(x,y)$ to the operator $\cal
L^{(\alpha)}_{\sigma,b}$ as well as the continuity of $x\mapsto
\rho_t(x,\cdot)$ in $L^1(\mathbb{R}^d)$ for each $t>0$. Moreover, when
$\sigma(x)=\sigma$ is constant, under the following uniform H\"ormander's type
condition: for some $j_0\in\mathbb{N}$, $$
\inf_{x\in\mathbb{R}^d}\inf_{|u|=1}\sum_{j=1}^{j_0}|u B_j(x)|^2>0, $$ we also
prove the smoothness of $(t,x,y)\mapsto\rho_t(x,y)$ with
$\rho_t(\cdot,\cdot)\in C^\infty_b(\mathbb{R}^d\times\mathbb{R}^d)$ for each
$t>0$.