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 $n\geq 2$, $x=(x_1,\cdots,x_n)\in(\mathbb R^d)^n =\mathbb R^{nd}$ and $a_t$ is a time-dependent constant symmetric $d\times d$-matrix that is uniformly elliptic and bounded.. Let $\{\mathcal T_{s,t}; t\geq s\}$ be the time-dependent semigroup associated with $\mathscr L_t$; that is, $\partial_s {\mathcal T}_{s, t} f = - {\mathscr L}_s {\mathcal T}_{s, t}f$. 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 $\lambda \geq 0$, $$ \left\|\Delta_{x_j}^{{1}/{(1+2(n-j)})}\int^{\infty}_0 e^{-\lambda t} {\mathcal T}_{s, s+t }f(t+s, x)dt\right\|_p\leq C\|f\|_p,\quad j=1,\cdots, n, $$ where $\|\cdot\|_p$ is the usual $L^p$-norm in $L^p(\mathbb R^{1+nd}; d s\times d x)$. 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 $\partial_t u+\sum_{j=2}^nx_j\cdot\nabla_{x_{j-1}} u=f$.