Propagation of regularity in $\varvec{{L}^{p}}$-spaces for Kolmogorov-type hypoelliptic operators

Abstract

Consider the following Kolmogorov-type hypoelliptic operator

$$\begin{aligned} {{\mathscr {L}}}_t := {\sum _{j=2}^n}x_j\cdot \nabla _{x_{j-1}}+\mathrm {tr}(a_t \cdot \nabla ^2_{x_n}) \end{aligned}$$

on ${{\mathbb {R}}}^{nd}$, where $n\geqslant 2$, $d\geqslant 1$, $x=(x_1,\ldots ,x_n)\in ({{\mathbb {R}}}^d)^n ={{\mathbb {R}}}^{nd}$ and $a_t$ is a time-dependent constant symmetric $d\times d$-matrix that is uniformly elliptic and bounded. Let $\{{{\mathcal {T}}}_{s, t}; t\geqslant s\}$ 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 $\lambda \geqslant 0$,

$$\begin{aligned} \left\| \Delta _{x_j}^{ {1}/ {(1+2(n-j))}} \int ^{\infty }_0 {\mathrm {e}}^{-\lambda t} {{\mathcal {T}}}_{s, t+s}f(t+s, x){\mathord {\mathrm{d}}}t\right\| _p\leqslant C\Vert f\Vert _p,\quad j=1,\ldots , n, \end{aligned}$$

where $\Vert \cdot \Vert _p$ is the usual $L^p$-norm in $L^p({{\mathbb {R}}}\times {{\mathbb {R}}}^{nd}; {\mathord {\mathrm{d}}}s\times {\mathord {\mathrm{d}}}x)$. To show this type of estimates, we first study the propagation of regularity in $L^2$-space from variable $x_n$ to $x_j$, $1\leqslant j\leqslant n-1$, for the solution of the transport equation $\partial _t u+ \sum _{j=2}^nx_j\cdot \nabla _{x_{j-1}} u=f.$