Generalized Sobolev–Morrey estimates for hypoelliptic operators on homogeneous groups

Abstract

Let ${\mathbb {G}}=\big ({\mathbb {R}}^N,\circ ,\delta _{\lambda }\big )$ be a homogeneous group, Q is the homogeneous dimension of ${{\mathbb {G}}}$, $X_0, X_1, \ldots , X_m$ be left invariant real vector fields on ${\mathbb {G}}$ and satisfy Hörmander’s rank condition on ${\mathbb {R}}^N$. Assume that $X_1, \ldots , X_m$ $(m\le N-1)$ are homogeneous of degree one and $X_0$ is homogeneous of degree two with respect to the family of dilations $\big (\delta _{\lambda }\big )_{\lambda >0}$. Consider the following hypoelliptic operator with drift on ${\mathbb {G}}$

$$\begin{aligned} {\mathcal {L}}=\sum \limits _{i,j=1}^m a_{ij} X_i X_j+a_0 X_0, \end{aligned}$$

where $(a_{ij})$ is a $m \times m$ constant matrix satisfying the elliptic condition in ${\mathbb {R}}^m$ and $a_0\ne 0$. In this paper, for this class of operators, we obtain the generalized Sobolev–Morrey estimates by establishing boundedness of a large class of sublinear operators $T_{\alpha }$, $\alpha \in [0,Q)$ generated by Calderón–Zygmund operators ($\alpha =0$) and generated by fractional integral operator ($\alpha >0$) on generalized Morrey spaces and proving interpolation results on generalized Sobolev–Morrey spaces on ${\mathbb {G}}$. The sublinear operators under consideration contain integral operators of harmonic analysis such as Hardy–Littlewood and fractional maximal operators, Calderón–Zygmund operators, fractional integral operators on homogeneous groups, etc.