Sharp results for the mean summability of Fourier series on compact Lie groups

The study of polyhedral summability of Fourier series in compact Lie groups plays a particular role since it leads to results which do not have a natural counterpart in the Euclidean case. In particular some previous result states the L^p polyhedral summability of Fourier series of central function for p in a suitable neighborhood of 2. The purpose of this paper is to determine these p’s exactly. We prove that the critical index p<2 depends on the polyhedron and, for rank at least two, is always larger than the critical index for the almost everywhere convergence of polyhedral sums. This fact does not contradict the Nikishin-Stein theory since the space of central funcions is not translation invariant. However it seems to be noteworthy since for radial functions on R^n or for central functions on SU_2 the critical indices of mean and a.e. convergence coincice. We will also prove that at the critical index the polyhedral sum operators are not of weak type uniformly.