CE40 - Mathématiques, informatique théorique, automatique et traitement du signal

# Harmonic Analysis for semigroups on commutative and non-commutative Lp spaces – HASCON

## HASCON

Harmonic Analysis of Semigroupes on commutative and noncommutative Lp spaces

### Challenges

please find these on the English website<br /><br />https://lmbp.uca.fr/~kriegler/HASCON/HASCONe.html#x1-40003

R-boundedness. This notion is a strengthening of boundedness of operators in a randomized
sense, which can be described by classical square function estimates in the setting
of classical and scalar valued, sometimes also vector valued Lp spaces.
(b) Transference principles. In some cases, semigroups or spectral multipliers f(A) can be
transferred to a much easier space by a commuting diagram usually of the form f(A)J =
JT with J an injection and T the transferred operator. This is e.g. known for positive
contractive C0 semigroups vs. bounded groups
(c) Bellman function method. This method of dominating a bilinear functional by means of a
convex functional has been surprisingly powerful very recently. It is well-known over
several decades in other elds of harmonic analysis.
(d) Littlewood-Paley decomposition. This method of building spectral blocks stems from the
classical work of Stein.
2. Question 2: Boundedness of the maximal operator.
(a) Randomization and selection process. Explicit uses of random coverings and partitions
in analytic problems stems from Lindenstrauss' work.

(b) Interplay of evolution and spatial maximal functions. This is a well-known meta principle
in harmonic analysis.
3. Question 3: Harmonic analysis objects on non-commutative Lp spaces.
(a) The notion of R-boundedness is again important here. In the non-commutative setting, a
partial description following the classical scheme has been obtained .
(b) Also the transference principle opens new ways here, as done for Schur vs. Fourier multipliers, and the Littlewood-Paley decomposition.
(c) Algebraic structures in non-commutative operator spaces. The use of tensor products with
as a consequence a wealth of different norms is intrinsic in non-commutative operator spaces
and will be useful in the second main line of the project proposition.
(d) Construction of new transference principles. To obtain such a transference, probability
theory or an ultraproduct method become important.

We have published a first result on Bochner spaces in Journal d'Anal. Math.
The second paper on complementation of radial multipliers has appeared in Archiv der Mathematik.
C. Arhancet in collaboration with Y. Raynaud have obtained results on contractively complemented noncommutative Lp-spaces.
He has also obtained new results on isometries on non-commutative Lp-spaces.

In the axis 1 of the project HASCON, the partner Luc Deleaval has obtained with his collaborator Nizar Demni a new result on generalised Bessel functions and their representation formulas.
Concerning markovian and sub-markovian semigroups, the partner Christoph Kriegler has finished a work with Komla Domelevo and Stefanie Petermichl (both exterior of the project) on H°° calculus of these semigroups on weighted L2-spaces.

In all this first half of project has been very productive and the works in collaboration between the members of project and outside of the project have been published in international mathematical journals of good renommee.

1.) Cedric Arhancet and Christoph Kriegler work on complementation of Fourier multipliers (characterization of those groups admitting a complementation).

2.) Cedric Arhancet and Christoph Kriegler work on functional calculus of the Walsh semigroups and applications of optimal H°° angle of the noncommutative Ornstein-Uhlenbeck semigroup

3.) Luc Deleaval and Christoph Kriegler work on maximal inequalities of spectral multipliers of Hörmander type and application to evolution equation of Schrödinger and wave type (continuity in time of path of the solution).

4.) Luc Deleaval and Christoph Kriegler work on weak 1,1 inequality of the Hardy-Littlewood maximal operator on L^1(R^d,Y) where Y is a UMD Banach lattice.

see the list of 12 articles in the intermediate report

The project HASCON concerns questions arising in modern harmonic analysis. The aim is to study functional calculus and maximal
operators associated with semigroups which can act on classical and non-commutative L^p spaces.
More specifically, I want to study holomorphic and Hörmander-Mihlin spectral multipliers associated with operators on UMD valued
L^p spaces, with an emphasis on Gaussian estimates, and in a different context, on non-commutative L^p operator spaces.
Such estimates play an important rôle in understanding singular integrals and partial differential equations, and more precisely, in
the convergence of Bochner-Riesz means and the solution of stochastic, parabolic and hyperbolic, PDEs.
In the latter case, the UMD space Y is a function space depending on a spatial variable.
Our functional calculus is also related to square function estimates in which case Y becomes l^2, and it is used for the study of abstract function spaces associated with a semigroup generator, where Y becomes l^q.
I also want to study the linked question when maximal operators arising in Lie and other geometric groups, or associated with
evolution semigroups, are bounded in a vector valued setting.
In another context, I study operations on non-commutative L^p spaces.
These spaces include the Schatten classes S^p and also the L^p associated with a von Neumann algebra VN(G) of a unimodular locally compact group G.
We are interested in Schur multipliers (on S^p) and non-commutative Fourier multipliers ( on L^p(VN(G)) ).

## Project coordinator

Monsieur Christoph Kriegler (LABORATOIRE DE MATHEMATIQUES BLAISE PASCAL)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

## Partner

LMBP LABORATOIRE DE MATHEMATIQUES BLAISE PASCAL

Help of the ANR 92,491 euros
Beginning and duration of the scientific project: February 2019 - 36 Months