Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers

Abstract

We introduce a notion of Ricci curvature lower bound for symmetric sub-Markovian semigroups. We use this notion to investigate functional calculus of the Hodge-Dirac operator associated to the semigroup in link with the boundedness of suitable Riesz transforms. Our paper offers a unified framework that not only encapsulates existing results in some contexts but also yields new findings in others. This is demonstrated through applications in the frameworks of Riemannian manifolds, compact (quantum) groups, noncommutative tori, Ornstein-Uhlenbeck semigroup, q-Ornstein-Uhlenbeck semigroups and semigroups of Schur multipliers. We also provide an L p -Poincaré inequality that is applicable to all previously discussed contexts under assumptions of boundedness of Riesz transforms and uniform exponential stability. Finally, we prove the boundedness of some Riesz transforms in some contexts as compact Lie groups.

Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

“…We will use the notation of [ArK22]. Let G be a discrete group and (T t ) t 0 be a markovian semigroup of Fourier multipliers on the group von Neumann algebra VN(G).…”

Dilations of Markovian semigroups of measurable Schur multipliers

“…Finally, note that Schur multipliers are crucial operators in noncommutative analysis and are connected to a considerable number of topics as harmonic analysis, double operator integrals, perturbation theory, and Grothendieck's theorem. More recently, the semigroups of Schur multipliers considered in this paper were connected to noncommutative geometry in the memoir [ArK22a].…”

“…This point is important for applications in vector-valued noncommutative -spaces since we need injective von Neumann algebras in this context (see [Pis98]). We refer to the papers [AFM17, ALM14, Arh13b, HaM11] for related things. Finally, note that Schur multipliers are crucial operators in noncommutative analysis and are connected to a considerable number of topics as harmonic analysis, double operator integrals, perturbation theory, and Grothendieck’s theorem.…”