High order Fefferman-Phong type inequalities in Carnot groups and regularity for degenerate elliptic operators plus a potential. (English) Zbl 1470.35377

Niu, Pengcheng; Zhang, Kelei

Abstr. Appl. Anal. 2014, Article ID 274859, 8 p. (2014).

Summary: Let \(\{X_1, X_2, \dots, X_m \}\) be the basis of space of horizontal vector fields in a Carnot group \(\mathbb{G} = (\mathbb{R}^n; \circ)\) \( (m < n)\). We prove high order Fefferman-Phong type inequalities in \(\mathbb{G}\). As applications, we derive a priori \(L^p(\mathbb{G})\) estimates for the nondivergence degenerate elliptic operators \(L = - \sum_{i, j = 1}^m a_{i j}(x) X_i X_j + V(x)\) with \(V M O\) coefficients and a potential \(V\) belonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

MSC:

35R03	PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
35B65	Smoothness and regularity of solutions to PDEs
35J70	Degenerate elliptic equations

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