An invitation to harmonic analysis associated with semigroups of operators (Q2875849)

The authors of the paper under review are currently developing an ambitious generalisation of Calderón-Zygmund theory of singular integral operators in non-commutative settings. Their theory deals with operators acting on \(L^p\) spaces naturally associated with a semifinite von Neumann algebra \(\mathcal{M}\). If \(\mathcal{M} = L^{\infty}(\mathbb{T})\), then one recovers the traditional setting of Fourier multipliers acting on \(L^{p}(\mathbb{T})\). Other classical commutative settings, over locally compact abelian groups or Riemannian manifolds, can also be recovered. The scope of Junge-Mei-Parcet's theory, however, is much wider, and includes, in particular, non-commutative discrete groups. The paper under review is a survey of their theory, in the state it had reached by 2013.NEWLINENEWLINEThe central idea is to use a Markov semigroup \((e^{tL})_{t \geq 0}\) as a replacement for the geometric averaging operators used in commutative settings. This allows the authors to define spaces of bounded mean oscillations by using the semigroup to define a notion of mean, analogues of Fourier multipliers using the functional calculus of \(L\), analogues of Riesz transforms using Bakry's carré du champs (a bilinear form defined in terms of \(L\)), and analogues of Hardy spaces. Under additional assumptions on \((e^{tL})_{t \geq 0}\) that are satisfied, for instance, when \(L\) is the Laplace Beltrami operator on a complete Riemannian manifold with nonnegative Ricci curvature, they obtain a satisfying abstract theory. This theory shows that their Hardy and BMO spaces are appropriate end points for interpolation, and includes \(L^p\) boundedness results for Riesz transforms and a large class of analogues of Fourier multipliers. Junge-Mei-Parcet's theory can be seen as a generalisation of Stein's Littlewood-Paley theory associated with diffusion semigroups, Cowling's abstract spectral multiplier theory, and Bakry-Emery's carré du champs theory. It uses both an analytic and a probabilistic perspective, and develops harmonic analysis using the analogue of the heat semigroup, rather than geometric or algebraic structures.NEWLINENEWLINEWhile this abstract theory is satisfying, and already able to recover many classical results, the authors are also interested in obtaining more refined results in certain concrete situations. They study, in particular, the case of a discrete group \(G\). Given a left regular representation, a semifinite von Neumann algebra can naturally be associated with \(G\), and the corresponding \(L^{p}\) spaces generalise the \(L^{p}(\hat{G})\) spaces available when \(G\) is abelian. Cohomological concepts now enter the picture, and allow the authors to prove a striking Fourier multiplier theorem. Their theorem is a version of the Mihlin-Hörmander theorem adapted to a given affine representation of \(G\). It is not only an important result in non-commutative situations, but also proves the boundedness of Fourier multipliers that do not satisfy the conditions of the classical Mihlin-Hörmander theorem in the case \(G=\mathbb{Z}^{n}\) (which is seen as a result adapted to a natural, but specific, affine representation)!NEWLINENEWLINEThe last two sections of the paper describe directions in which their theory is now evolving: the development of an analogue of non-convolution type singular integral operator theory, and the possibility of endowing certain von Neumann algebras with a natural metric space structure. A key idea is to recognise that the standard heat semigroup is an average of geometric averaging operators. The authors define a notion of weighted spectral decomposition that, roughly speaking, encodes the properties that a family of projections needs to satisfy in order to play, for a given Markov semigroup, the role that geometric averaging operators play for the standard heat kernel. This allows the authors to prove a Hörmander type extrapolation theorem, where the standard kernel conditions are expressed in terms of such a weighted spectral decomposition. This suggests ways of uncovering some hidden metric space structure in non-commutative settings that are sketched in the last section of the paper.NEWLINENEWLINEFor the entire collection see [Zbl 1285.00036].