Approximation of hardy space on the unit sphere

The authors discuss the boundedness and approximation properties of translation and mean operator onH1(Σ) by the estimates of high degree difference on ultraspherical polynomials, atom decomposition and construct properties on sphere. Also the boundedness and approximation of linear means at all kinds of indexes onHp(0<p<1) and the almost everywhere convergence of Cesàro means are established.