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Journal of Evolution Equations

, Volume 19, Issue 4, pp 1041–1069 | Cite as

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

  • Zhen-Qing ChenEmail author
  • Xicheng Zhang
Article
  • 71 Downloads

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 \( \partial _t u+ \sum _{j=2}^nx_j\cdot \nabla _{x_{j-1}} u=f.\)

Keywords

Kolmogorov’s hypoelliptic operators Fefferman–Stein’s theorem Propagation of \(L^p\)-regularity 

Mathematics Subject Classification

Primary 42B20 60H30 Secondary 35H10 35Q20 

Notes

Acknowledgements

We thank the referees for helpful comments.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China

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