Project ANR JCJC HASCON (2019 -
Harmonic Analysis for Semigroups on Commutative
and Non-commutative Lp spaces
- Cédric Arhancet
- Luc Deleaval
- Adrián González-Pérez (postdoctorant year 2019-2020)
- Christoph Kriegler (project coordinator)
2 Key words of research
- Harmonic Analysis
- Semigroups of operators
- Non-commutative geometry
- Functional calculus
- Maximal operators
- Non-commutative Lp spaces
- Operator spaces
- Riesz transforms
- Singular integrals
- Geometry of Banach spaces
- Fourier multipliers on groups and Schur multipliers
3 Post Doc Position
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,
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
4 Presentation of the project
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
- Under which circumstances (e.g. which underlying classical or
non-commutative Lp 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 Lp(Ω)), or
also be a non-commutative Lp 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 = Lp(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 Tt; for square function estimates,
where Y = ℓ2 (then the interesting functional calculus question involves
a sequence of spectral multipliers (fk)k); and lastly for descriptions of
abstract function spaces associated with the generator, such as Sobolev
and Triebel-Lizorkin spaces, where Y = ℓq.
- For which cases is an evolution maximal operator, in the most classical
form MTf = supt>0|Ttf| or spatial maximal operator MHLf =
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 Lp(Ω) above
to a sectorial maximal operator. Then under the presence of integral
kernel estimates of Tt, 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 MHL,MT. From a general
point of view, standard maximal operators are important in several
branches of harmonic and real analysis (e.g. singular integrals, multipliers,
- What kind of operations on non-commutative Lp 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.
6 Publications and Preprints
- L. Deleaval and C. Kriegler: Dunkl spectral multipliers with values in
UMD lattices, Journal of Functional Analysis, 272(5):2132–2175, 2017.
- 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 Lp 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 ℝn, to appear in Archiv
der Mathematik. Preprint on HAL, arXiv.org.
- C. Arhancet and C. Kriegler: Projections, multipliers and decomposable
maps on non-commutative Lp-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.