Journal of Evolution Equations

, Volume 19, Issue 4, pp 1041–1069

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

• Zhen-Qing Chen
• Xicheng Zhang
Article

## 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

## References

1. 1.
R. Alexander: Fractional order kinetic equations and hypoellipcity. Anal. Appl. (Singap.) 10, no.3 (2012), 237-247.
2. 2.
F. Bouchut: Hypoelliptic regularity in kinetic equations. J. Math. Pures Appl. 81 (2002), 1135-1159.
3. 3.
M. Bramanti, G. Cupini, E. Lanconelli and E. Priola: Global $$L^p$$-estimate for degenerate Ornstein-Uhlenbeck operators. Math Z.266 (2010), 789-816.
4. 4.
M. Bramanti, G. Cupini, E. Lanconelli and E. Priola: Global $$L^p$$-estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients. Math. Nachr.286 (2013), no. 11-12, 1087-1101.
5. 5.
Z.-Q. Chen and X. Zhang: $$L^p$$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators. J. Math. Pures et Appliquées 116 (2018), 52-87.
6. 6.
E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle: Regularity of stochastic kinetic equations. Electron. J. Probab. Volume 22 (2017), paper no. 48, 42 pp.Google Scholar
7. 7.
L. Huang, S. Menozzi and E. Priola: $$L^p$$-estimates for degenerate non-local Kolmogorov operators. J. Math. Pures Appl.121 (2019), 162-215.
8. 8.
A. N. Kolmogorov: Zufállige Bewegungen. Ann. Math. 35 (1934), 116-117.
9. 9.
A. Lunardi: Schauder’s estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $${\mathbb{R}}^n$$. Annali della Scuola Normale Superiore di Pisa , Vol. 24 (1997), 133-164.
10. 10.
E. M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton 1993.
11. 11.
X. Zhang: $$L^p$$-maximal regularity of nonlocal parabolic equations and applications. Ann. I. H. Poincare-AN 30 (2013), 573-614.
12. 12.
X. Zhang: Stochastic Hamiltonian flows with singular coefficients. Sci. China. Math. 61 (2018), 1353-1384.