Subelliptic operators on weighted Folland-Stein spaces

Abstract

The CR positive mass problem plays an essential role in CR geometry. As well known, we need a CR positive mass theorem to solve the CR Yamabe problem for the cases which either the CR dimension n = 1 or the CR manifoldM is spherical with higher CR dimension. When n = 1, this was shown by Cheng, Malchiodi and Yang [3]. On the other hand, when n ≥ 2 and M is spherical, this was finished by Cheng, Yang and the author [2] through showing that the developing map is injective. However in the case n = 2, we need an extra condition on the growth rate of the Green’s function on the universal cover of M . So in the case n = 2, the CR positive mass theorem is not really completed. In the paper [1], Cheng and the author showed that for n = 2, M being spherical, if moreover M has a spin structure, then we have the CR positive mass theorem built up through a spinorial approach. Recall that in 1982, E. Witten described a proof of the positive mass theorem using spinors (see [8, 7]). Applying a Weitzenbock-type formula to ψ satisfying Dψ = 0 and approaching a constant spinor at infinity and integrating after taking the inner product with ψ, we then pick up the p-mass from the boundary integral and obtain a Witten-type formula for the p-mass. So the non-negativity of p-mass follows. Therefore, for the CR positive mass theorem, it suffices to show that the square of the Dirac operator D on some suitable weighted Folland-Stein spaces is an isomorphism. In this paper we prove that both sublaplacian ∆b (see Theorem 3.2) and D (see Theorem 3.3) on suitable spaces are isomorphisms.