Sobolev inequalities for harmonic measures on spheres

• DOI: 10.1016/j.spl.2015.02.008
• Publication Date: 2015-02-16T03:17:58Z
• Authors: Jing Du; Liangzhen Lei; Yutao Ma
• Keywords: Unit sphere; Sobolev inequality; Constant (computer programming); Exponent; Poincaré inequality; Harmonic

Abstract In this paper, we consider the harmonic measure on the unit sphere S n − 1 on R n ( n ≥ 2 ) and offer a two-sided estimate of precise order on the Sobolev constant with exponent p ∈ ( 1 , 2 ) . As special cases for p = 1 and p tending to 2 , our estimates recover those in Barthe et al. (2104) for n ≥ 3 and in Ma and Zhang (2014) for n = 2 .