Fourier multipliers in SLn(R)

Abstract

We establish precise regularity conditions for Lp-boundedness of Fourier multipliers in the group algebra of SLn(R). Our main result is inspired by Hörmander-Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity order. The asymptotics also match Mikhlin formula for a exponentially growing metric with respect to the word length. Additional decay comes imposed by this growth and Mikhlin condition for high order terms. Lafforgue/de la Salle’s rigidity theorem fits here. The proof includes a new relation between Fourier and Schur Lp-multipliers for nonamenable groups. In SLn(R), this holds in terms of Harish-Chandra’s almost L2 matrix coefficients. By transference, matters are reduced to a rather nontrivial RCp-inequality for SLn(R)-twisted forms of Riesz transforms associated to fractional laplacians. Our second result gives a new and much stronger rigidity theorem for radial multipliers in SLn(R). More precisely, additional regularity and Mikhlin type conditions are proved to be necessary up to an order ∼ | 1 2 − 1 p |(n − 1) for large enough n in terms of p. Locally, necessary and sufficient growth rates match up to that order. Asymptotically, extra decay for the symbol and its derivatives imposes more accurate and additional rigidity in a wider range of Lp-spaces. This rigidity increases with the rank, so we can construct radial generating functions satisfying our Hörmander-Mikhlin sufficient conditions in a given rank n and failing the rigidity conditions for ranks m >> n. We also prove automatic regularity and rigidity estimates for first and higher order derivatives of K-biinvariant multipliers in the rank 1 groups SO(n, 1).