Fourier multipliers in SLn(R)

Abstract

We establish precise regularity conditions for $L_p$-boundedness of Fourier multipliers in the group algebra of ${\mathrm{SL}}_n(\mathbf{R})$. Our main result is inspired by the 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. The asymptotics also match the Mikhlin formula for an exponentially growing weight with respect to the word length. Additional decay comes imposed by this growth and the Mikhlin condition for high-order terms. Lafforgue and de la Salle’s rigidity theorem fits here. The proof includes a new relation between Fourier and Schur $L_p$-multipliers for nonamenable groups. By transference, matters are reduced to a rather nontrivial $R{C}_p$-inequality for ${\mathrm{SL}}_n(\mathbf{R})$-twisted forms of Riesz transforms associated to fractional Laplacians.

Our second result gives a new and much stronger rigidity theorem for radial multipliers in ${\mathrm{SL}}_n(\mathbf{R})$. More precisely, additional regularity and Mikhlin-type conditions are proved to be necessary up to an order $\sim |\frac{1}{2}-\frac{1}{p}|(n-1)$ 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 $L_p$-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 $m\gg n$. We also prove automatic regularity and rigidity estimates for first- and higher-order derivatives of K-bi-invariant multipliers in the rank 1 groups $\mathrm{SO}(n,1)$.