Propagation of regularity in $L^p$-spaces for Kolmogorov type hypoelliptic operators

Abstract

Consider the following Kolmogorov type hypoelliptic operator $$ \mathscr L_t:=\mbox{$\sum_{j=2}^n$}x_j\cdot\nabla_{x_{j-1}}+{\rm Tr} (a_t \cdot\nabla^2_{x_n}), $$ where $n\ge 2$, $x=(x_1,\cdots ,x_n)\in ({\mathbb{R}}^d{)}^n={\mathbb{R}}^{nd}$ and $a_t$ is a time-dependent constant symmetric $d\times d$-matrix that is uniformly elliptic and bounded.. Let $\{{\mathcal{T}}_{s,t};t\ge s\}$ be the time-dependent semigroup associated with ${\mathcal{L}}_t$; that is, ${\partial }_s{\mathcal{T}}_{s,t}f=-{\mathcal{L}}_s{\mathcal{T}}_{s,t}f$. For any $p\in (1,\infty )$, we show that there is a constant $C=C(p,n,d)>0$ such that for any $f(t,x)\in L^p(\mathbb{R}\times {\mathbb{R}}^{nd})=L^p({\mathbb{R}}^{1+nd})$ and every $\lambda \ge 0$, $${\Vert {\Delta }_{x_j}^{1/(1+2(n-j))}{\int }_0^{\infty }{e}^{-\lambda t}{\mathcal{T}}_{s,s+t}f(t+s,x)dt\Vert }_p\le C\Vert f{\Vert }_p,j=1,\cdots ,n,$$ where $\Vert \cdot {\Vert }_p$ is the usual $L^p$-norm in $L^p({\mathbb{R}}^{1+nd};ds\times dx)$. To show this type of estimates, we first study the propagation of regularity in $L^2$-space from variable $x_n$ to $x_1$ for the solution of the transport equation ${\partial }_{t}u+{\sum }_{j=2}^n{x}_j\cdot {\nabla }_{x_{j-1}}u=f$.

arXiv comment25 pages

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