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

Estimates of the maximal Cesaro means at the "critical index" are established for the orthogonal polynomial expansions (OPEs) with respect to the weight function Pi(d)(j=1) vertical bar x(j)vertical bar(2 kappa j) on the unit sphere of R-d. These estimates allow us to improve several known results in this area, including the almost everywhere (a.e.) convergence of the Cesaro means at the "critical index", the sufficient conditions for the Marcinkiewitcz multiplier theorem, and a Fefferman-Stein type inequality for the Cesaro 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. (C) 2013 Elsevier Inc. All rights reserved.