Noncommutative Cotlar identities for groups acting on tree-like structures

TL;DR

This paper characterizes noncommutative Cotlar identities for Fourier multipliers on groups acting on tree-like structures, establishing boundedness results and constructing new examples including Baumslag-Solitar groups and Bianchi groups.

Contribution

It provides a geometric characterization of Cotlar identities for multipliers on groups acting on rb1 trees, linking group actions to Fourier multiplier boundedness.

Findings

Characterization of Cotlar identities in terms of group elements and functions

Boundedness of multipliers on rb1 trees and rb1 actions

Construction of new Fourier multipliers for complex groups like Baumslag-Solitar and Bianchi groups

Abstract

Let $T_m$ be a noncommutative Fourier multiplier. In recent work, Mei and Ricard introduced a noncommutative analogue of Cotlar's identity in order to prove that certain multipliers are bounded on the non-commutative $L_p$-spaces of a free group. Here, we study Cotlar type identities in full generality, giving a closed characterization for them in terms of $m$: \[ \big( m(g h) - m(g) \big) \, \big( m(g^{-1}) - m(h) \big) = 0, \; \forall g \in \mathrm{G} \setminus \{e\}, h \in \mathrm{G}. \] We manage to prove, using a geometric argument, that if $X$ is a tree -- or more generally an $\mathbb{R}$-tree -- on which $\mathrm{G}$ acts and $m$ lifts to a function $\widetilde{m}: X \to \mathbb{C}$ that is constant on the connected subsets of $X \setminus \{x_0\}$, then $m$ satisfies Cotlar's identity and thus $T_m$ is bounded in $L_p$ for $1 < p < \infty$. This result establishes a new connection between groups actions on $\mathbb{R}$-trees and Fourier multipliers. We show that $m$ is trivial when the action has global fixed points. This machinery allows us to simultaneously generalize the free group transforms of Mei and Ricard and the theory of Hilbert transforms in left-orderable groups, which follows from Arveson's subdiagonal algebras. Using Bass-Serre theory, we construct new examples of Fourier multipliers in groups. Those include new families like Baumslag-Solitar groups. We also show that a natural Hilbert transform in $\mathrm{PSL}_2(\mathbb{C})$ satisfies Cotlar's identity when restricted to the Bianchi group $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{-1}])$.