Boundedness of projection operators and Cesàro means in weighted 𝐿^{𝑝} space on the unit sphere

Abstract

For the weight function ∏ i = 1 d + 1 | x i | 2 κ i \prod _{i=1}^{d+1}|x_i|^{2\kappa _i} on the unit sphere, sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the Cesàro ( C , δ ) (C,\delta ) means in the weighted L p L^p space for δ \delta above the critical index. Similar results are also proved for corresponding weight functions on the unit ball and on the simplex.