Sobolev inequalities for harmonic measures on spheres

In this paper, we consider the harmonic measure on the unit sphere Sn−1 on Rn (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.