Regularity of matrix coefficients of a compact symmetric pair of Lie groups

We consider Gelfand pairs ( G , K ) (G,K) where G G is a compact Lie group and K K a subgroup of fixed points of an involutive automorphism. We study the regularity of K K -bi-invariant matrix coefficients of unitary representations of G G . The results rely on the analysis of the spherical functions of the Gelfand pair ( G , K ) (G,K) . When the symmetric space G / K G/K is of rank 1 1 or isomorphic to a Lie group, we find the optimal regularity of K K -bi-invariant matrix coefficients of unitary representations. Furthermore, in rank 1 1 we also find the optimal regularity of K K -bi-invariant Herz-Schur multipliers of S p ( L 2 ( G ) ) S_p(L^2(G)) . We also give a lower bound for the optimal regularity in some families of higher rank symmetric spaces. From these results, we make a conjecture in the general case involving the root system of the symmetric space. Finally, we prove that if all K K -bi-invariant matrix coefficients of unitary representations of G G are α α -Hölder continuous for some α > 0 α >0 , then all K K -finite matrix coefficients of unitary representations are also α α -Hölder continuous.