A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth

Abstract

AbstractIn this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most $d$ in a complete noncompact pseudohermitian $(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.