A note on the spectral gap for general harmonic measures on spheres

In this paper, we consider general harmonic measures μxn,β on the unit sphere Sn−1 in Rn, where x∈Rn with 0≤|x|<1,β∈R and n≥3. Following the idea in Barthe et al. (2014) , we obtain the lower bound for the spectral gap of μxn,β. For the harmonic measure (β=0), we improve the lower bound of the spectral gap in Barthe et al. (2014) and we also improve those in Milman (2015) for general β∈R.