The principle of general localization on unit sphere (Q1023026)

The author establishes the following results, which he claims to be the fundamental one in this paper: Let \(f\in L^p(S^N)\), \(1<p<2\), where \(S^N\) denotes the unit sphere in \(\mathbb{R}^{N+1}\). Assume \(f(x)=0\) on \(\Omega\in S^N\). Then the Fourier-Laplace series of the function \(f\) converges to 0 a.e. on \(\Omega\) by Riesz means of order \(\alpha=(N-1) (1/p-1/2)\). These means are defined by the formula \[ E^\alpha_nf(x)= \sum^n_{k=0}\left(1-\frac{\lambda_k}{\lambda_n} \right)^\alpha Y_k (f,x), \] with \(\lambda_k\) and \(Y_k(f;x)\) defined in terms of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator \(\Delta_S\) on \(S^N\). Further results are concerned with maximal operators in the sense of Hardy-Litt1ewood, obtaining several \(L_1\) and \(L_2\) estimates in regard to the Riesz means. The last section of the paper deals with an analytic family of operators, pursuing the same topics.