A Hörmander–Mikhlin theorem for higher rank simple Lie groups

Abstract

We establish regularity conditions for $L_p$-boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural Hörmander–Mikhlin (HM) criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard, and de la Salle for $S{L}_n\left(R\right)$. Our approach is partly based on a sharp local HM theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular non-Toeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle-based approach to Fourier multipliers in group algebras by Junge, Mei, and Parcet. A few related open problems are also discussed.