A class of nonlocal hypoelliptic operators and their extensions

In this paper we study nonlocal equations driven by the fractional powers of hypoelliptic operators in the form $$\mathcal{K}u=\mathcal{A}u-{\mathrm{\partial }}_{t}u\mathrm{tr}(Q{\mathrm{\nabla }}^{2}u)+<BX,\mathrm{\nabla }u>-{\mathrm{\partial }}_{t}u,$$ introduced by H"ormander 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 $L^{\mathrm{\infty }}$, and in $L^p$ for $1\le p<\mathrm{\infty }$ when $\mathrm{tr}(B)\ge 0$. In forthcoming works we use such calculus to establish some new Sobolev and isoperimetric inequalities.