The principle of general localization on unit sphere

In this paper we study the general localization principle for Fourier–Laplace series on unit sphere S N ⊂ R N + 1 . Weak type ( 1 , 1 ) property of maximal functions is used to establish the estimates of the maximal operators of Riesz means at critical index N − 1 2 . The properties Jacobi polynomials are used in estimating the maximal operators of spectral expansions in L 2 ( S N ) . For extending positive results on critical line α = ( N − 1 ) ( 1 p − 1 2 ) , 1 ⩽ p ⩽ 2 , we apply interpolation theorem for the family of the linear operators of weak types. The generalized localization principle is established by the analysis of spectral expansions in L 2 . We have proved the sufficient conditions for the almost everywhere convergence of Fourier–Laplace series by Riesz means on the critical line.