Group-free radial Kunze-Stein phenomenon on affine buildings

Abstract

We give a geometric proof of the radial Kunze--Stein phenomenon on the good vertices of a thick, locally finite, regular affine building of irreducible type. The proof uses a positive spherical function, its joint eigenvalues for sphere averages, and the growth of vector-distance spheres. Conjugation by this function and complex interpolation yield a weighted $\ell^2$ estimate and the boundedness of radial operators with $\ell^p$ kernels for $1\leq p<2$. We also present proofs by the averaging operators norms, by the Schur test and by Helgason transform. A further interpolation gives weighted mixed-exponent estimates. No transitivity assumption on the automorphism group is required; in particular the argument applies to buildings of type $\widetilde A_2$ without a Bruhat--Tits realization.