A Hörmander–Mikhlin theorem for higher rank simple Lie groups

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Abstract We establish regularity conditions for ‐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 . 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.