Project ANR JCJC HASCON (2019 -
2021)

Harmonic Analysis for Semigroups on Commutative
and Non-commutative L^{p} spaces

Contact:

1 Members

2 Key words of research

3 Post Doc Position

4 Presentation of the project

5 Events

6 Publications and Preprints

2 Key words of research

3 Post Doc Position

4 Presentation of the project

5 Events

6 Publications and Preprints

- Cédric Arhancet
- Luc Deleaval
- Adrián González-Pérez (postdoctorant year 2019-2020)
- Christoph Kriegler (project coordinator)

- Harmonic Analysis
- Semigroups of operators
- Non-commutative geometry
- Functional calculus
- Maximal operators
- Non-commutative L
^{p}spaces - Operator spaces
- Riesz transforms
- Singular integrals
- Geometry of Banach spaces
- Fourier multipliers on groups and Schur multipliers
- R-boundedness

There is a one year postdoctoral position to be filled within the ANR project.

Start: September or October 2019, Duration: 12 months.

Place: Laboratoire de Mathématiques Blaise Pascal in Clermont-Ferrand,
France.

Deadline for application: March 15, 2019.

We are now accepting applications for a postdoctoral research fellowship (without teaching duty) for one year. We are looking for applicants who received their Ph.D. recently (after Sept. 1, 2014) or will receive it until August 2019. The fellow is expected to carry out a research project on the topics of the project HASCON (see key words of research above). The fellowship provides an overall salary of 50 000 Euros, which is about 2 250 Euros per month after social costs. It also includes an additional travelling allowance.

For more details please contact the project members Cédric Arhancet, Luc Deleaval and Christoph Kriegler under

Applications (in English or French) should contain a curriculum vitae, a research statement (max. 5 pages), a list of publications if applicable, a recommendation letter sent by a senior mathematician (or a recommendation e-mail contact), and the desired starting date. Please send it to

Since the fundamental works of Stein and Cowling, the spectral theory for semigroups has become a wide mathematical field and a lot of mathematicians work in that field today. Much progress has been achieved over the last four decades, many beautiful connections have been proven to be fruitful in solving problems inside and outside harmonic analysis. The aim of our project HASCON is to answer the following questions, which arise in the context of spectral theory, functional calculus, harmonic analysis or abstract partial differential equations:

- Under which circumstances (e.g. which underlying classical or
non-commutative L
^{p}or Banach space) does the generator of a semigroup admit an H^{∞}or Hormander(-Mihlin) functional calculus? This is a property well-known to be of great importance in theoretical aspects and for many applications. The answer depends on the underlying space X, which can consist of functions over some measure space Ω (often L^{p}(Ω)), or also be a non-commutative L^{p}space, i.e. a space of (un)bounded operators affiliated with a von Neumann algebra. Hereby, geometrical properties of the Banach space X usually play an important rôle, and we give a particular emphasis to Bochner spaces X = L^{p}(Ω,Y ). Then the property of Y being a UMD space becomes important, and also its Rademacher-type and -cotype as well as related notions such as p-convexity and q-concavity if Y is moreover a lattice. Our motivation for Bochner spaces comes from their importance in applications to abstract Cauchy problems, where Y takes over the rôle of a spatial variable, whereas the time variable is the parameter t of the semigroup T_{t}; for square function estimates, where Y = ℓ^{2}(then the interesting functional calculus question involves a sequence of spectral multipliers (f_{k})_{k}); and lastly for descriptions of abstract function spaces associated with the generator, such as Sobolev and Triebel-Lizorkin spaces, where Y = ℓ^{q}. - For which cases is an evolution maximal operator, in the most classical
form M
_{T}f = sup_{t>0}|T_{t}f| or spatial maximal operator M_{HL}f = sup_{r>0}∫_{B(x,r)}|f(y)|dy bounded? This and the above functional calculus question are closely linked and reinforce each other. Namely, on the one hand, an H^{∞}calculus with a good angle allows to extend the boundedness of the evolution maximal operator on L^{p}(Ω) above to a sectorial maximal operator. Then under the presence of integral kernel estimates of T_{t}, evolution and spatial maximal operators are simultaneously bounded. On the other hand, boundedness of a maximal operator plays sometimes a crucial tool in establishing H^{∞}calculus and Hormander-Mihlin calculus. Again we pay a particular attention to Bochner-space valued boundedness of M_{HL},M_{T}. From a general point of view, standard maximal operators are important in several branches of harmonic and real analysis (e.g. singular integrals, multipliers, Littlewood-Paley theory). - What kind of operations on non-commutative L
^{p}spaces yield bounded and completely bounded maps? The most prominent examples of such mappings important in harmonic analysis are Schur multipliers, non-commutative Fourier multipliers or operations stemming from second quantization, such as q-Ornstein-Uhlenbeck semigroups. Schur multipliers provide a surprisingly rich class of mappings and they have a longstanding usage in various fields of analysis such as complex function theory, Banach spaces, operator theory, multivariate analysis, theory of absolutely summing operators and functional calculus. Fourier multipliers on non-commutative groups and second quantizations are a rather new field in harmonic analysis. Non-commutative harmonic analysis involves more algebraic and also combinatoric structure.

There will be a two days meeting of the ANR project during the year 2020. Further information to follow.

- L. Deleaval and C. Kriegler: Dunkl spectral multipliers with values in UMD lattices, Journal of Functional Analysis, 272(5):2132–2175, 2017. Preprint here.
- L. Deléaval and C. Kriegler: Dimension free bounds for the vector-valued Hardy-Littlewood maximal operator, Rev. Mat. Iberoam., 35(1):101–123, 2019. Preprint on HAL, arxiv.org.
- L. Deleaval, M. Kemppainen and C. Kriegler: Hormander functional
calculus on UMD lattice valued L
^{p}spaces under generalised Gaussian estimates, submitted, Preprint here. - C. Arhancet and C. Kriegler: Complementation of the subspace of radial
multipliers in the space of Fourier multipliers on ℝ
^{n}, to appear in Archiv der Mathematik. Preprint on HAL, arXiv.org. - C. Arhancet and C. Kriegler: Projections, multipliers and decomposable
maps on non-commutative L
^{p}-spaces, submitted, Preprint on arXiv.org. - C. Arhancet and C. Kriegler: Riesz transforms, Hodge-Dirac operators and functional calculus for multipliers I, Preprint on arXiv.org.
- C. Arhancet: Dilations of markovian semigroups of Fourier multipliers on locally compact groups, Preprint on arXiv.org.

Website updated on 30/08/2019.