Fourier multipliers in SLn(R)

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