A class of nonlocal hypoelliptic operators and their extensions

Abstract

In this paper we study nonlocal equations driven by the fractional powers of hypoelliptic operators in the form $$\mathcal{K}u=\mathcal{A}u-{\partial }_{t}u\stackrel{def}{=}\mathrm{tr}(Q{\nabla }^{2}u)+<BX,\nabla u>-{\partial }_{t}u,$$ introduced by Hörmander in his 1967 hypoellipticity paper. We show that the nonlocal operators $(-\mathcal{K}{)}^s$ and $(-\mathcal{A}{)}^s$ can be realized as the Dirichlet-to-Neumann map of doubly-degenerate extension problems. We solve such problems in $L^{\infty }$, and in $L^p$ for $1\le p<\infty$ when $\mathrm{tr}(B)\ge 0$. In forthcoming works we use such calculus to establish some new Sobolev and isoperimetric inequalities.

The paper is going to appear in Indiana Univ. Math. J