Spectral multipliers in group algebras and noncommutative Calderon-Zygmund theory

Javier Parcet教授（西班牙马德里国家科学院数学科学研究所）

We shall discuss three problems in noncommutative harmonic analysis, which are related to endpoint inequalities for singular integrals on matrix or group algebras. In first place, we find a very much expected proof of the weak type $L_1$ inequality for matrix-valued CZOs which notably avoids pseudolocalization. It uses a new CZ decomposition for martingale filtrations in von Neumann algebras and a very simple but unconventional argument. In second place, we establish the weak $L_1$ endpoint for matrix-valued CZOs over nondoubling measures of polynomial growth, in the line of Tolsa and Nazarov/Treil/Volberg. These results solve two open problems formulated in 2009. An even more interesting problem is the lack of $L_1$ endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups, formulated by Junge in 2010. Given a locally compact group G equipped with a conditionally negative length $\psi: \mathrm{G} \to \mathbb{R}_+$, we prove that Herz-Schur multipliers with symbol $m \circ \psi$ satisfying a Mikhlin condition in terms of the $\psi$-cocycle dimension are of weak type $(1,1)$. Our result extends to Fourier multipliers for amenable groups and impose sharp regularity conditions on the symbol.