Maximal estimates for the Cesàro means of weighted orthogonal polynomial expansions on the unit sphere

Estimates of the maximal Cesàro means at the “critical index” are established for the orthogonal polynomial expansions (OPEs) with respect to the weight function ∏j=1d|xj|2κj on the unit sphere of RdRd. These estimates allow us to improve several known results in this area, including the almost everywhere (a.e.) convergence of the Cesàro means at the “critical index”, the sufficient conditions for the Marcinkiewitcz multiplier theorem, and a Fefferman–Stein type inequality for the Cesàro operators. In addition, several similar results for the weighted OPEs on the unit ball and on the simplex are deduced from the corresponding weighted results on the sphere.