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 $$\mathscr K u = \mathscr A u - \partial_t u \overset{def}{=} \operatorname{tr}(Q \nabla^2 u) + <BX,\nabla u> - \partial_t u,$$ introduced by Hörmander in his 1967 hypoellipticity paper. We show that the nonlocal operators $(-\mathscr K)^s$ and $(-\mathscr A)^s$ 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 $1\leq p<\infty$ when $\operatorname{tr}(B)\geq 0$. In forthcoming works we use such calculus to establish some new Sobolev and isoperimetric inequalities.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1811.02968
- arXiv:
- arXiv:1811.02968
- Bibcode:
- 2018arXiv181102968G
- Keywords:
-
- Mathematics - Analysis of PDEs
- E-Print:
- The paper is going to appear in Indiana Univ. Math. J