# NCMW - Harmonic Analysis (2020)

## Speakers and Syllabus

This will be an advanced workshop in four different themes in harmonic analysis.
The first lecture of all the topics will be accessible to any graduate student working in Analysis and related areas.

The first set of lectures will be delivered by  Professor Christoph Kreigler (University of Blaise-Pascal, France) on “Spectral multipliers for c_0-semigroups acting on Bochner spaces and non-commutative Lp-spaces”.
Since the fundamental works of Stein and Cowling, the spectral theory for semigroups T_t = exp(-tA) has become a wide mathematical field.
In this lecture series Prof. Christoh Kreigler would explore the question of finding the spectral multiplier functions which can replace the exponential function exp(-tA) to yield the boundedness on various Banach spaces.His lectures would focus on proving the boundedness of spectral multipliers  on non commutative Lp spaces  as well as the state of the art of some of these spectral multiplier theorems. This includes maximal operators, semigroups, Hormander functional calculus on Bochner spaces.

The second set of lectures will be delivered by Professor Alessio Martini  (University of Birmingham, UK)
on "Spectral multipliers for sub-Laplacians: classical results and recent progress".
In this series of lecture Prof. Alessio Martini will present  old and new results  about the functional calculus for sub-Laplacians ‘L’ on Lie groups of polynomial growth and more general sub-Riemannian manifolds. It has been known for a long time that an operator of the form ‘L’ is Lp-bounded under suitable size and smoothness assumptions on the spectral multiplier F. Nevertheless the problem of determining minimal smoothness requirements on F remains widely open for non-elliptic ‘L’ and will be the focus of the discussion.

The third set of lectures will be delivered by  Prof. Michael Ruzhansky (Ghent University, Belgium).
Title: TBA

The fourth set of lectures will be delivered by  Prof. K. Sandeep (TIFR, Bangalore) on "Inequalities, best constants and extremal functions.’’ In this series of lectures Prof. K. Sandeep  will address the question of best constant and classification of extremal functions for Sobolev type inequalities in the Euclidean space and in some general Riemannian and Sub-Riemannian geometries.

Syllabus to be covered in terms of modules of 6 lectures each :

 Name of the Speakers with affiliation. No. of Lectures Topics Prof. Christoph Kriegler,(University of Blaise-Pascal, France)(CK) six lectures = 6 hrsDiscussion= 1.5 hrs Spectral multipliers for c_0-semigroups acting on Bochner spaces and non-commutative Lp-spaces Prof. Alessio Martini(University of Birmingham, UK) (AM) six lectures = 6 hrsDiscussion = 1.5 hrs Spectral multipliers for sub-Laplacians: classical results and recent progress . Prof. Michael Ruzhansky (Ghent University, Belgium)(MR) six lectures = 6 hrsDiscussion= 1.5 hrs TBA Prof. K. Sandeep, TIFR Bangalore, India(KS) six lectures = 6 hrsDiscussion= 1.5 hrs Inequalities, best constants and extremal functions

References:

1. L. Deleaval, M. Kemppainen, C. Kriegler. Hörmander functional calculus on UMD lattice valued L^p spaces under generalised Gaussian estimates. Accepted for publication in J. Anal. Math.
2. L. Deleaval, C. Kriegler. Dimension free bounds for the vector-valued Hardy-Littlewood maximal operator. Rev. Mat. Iberoam. 35 (2019), no. 1, 101-123.
3. A. Martini and D. M¨uller, Spectral multiplier theorems of Euclidean type on new classes of two-step stratified groups. Proc. London Math. Soc. 109 (2014), 1229–1263.
4. A. Martini and D. M¨uller, Spectral multipliers on 2-step groups: topological versus homogeneous dimension. Geom. Funct. Anal. 26 (2016), 680–702.

## Time Table

Tentative time-table with names of the speakers and tutors with their affiliation:

 Day Date Lecture 1 (11:00-12:00)+12:00-12:15 discussion Lunch Break Lecture 2 (2:00-3:00)+ 3:00-3:15 discussion Lecture 3 (3:30-4:30)+4:30-4:45 discussion Lecture 4 (5:00-6:00)+6:00-6:15 discussion Name of the speaker Name of the speaker Name of the speaker Name of thespeaker Mon 7/12/2020 KS MR CK AM Tue 8/12/2020 KS MR CK AM Wed 9/12/2020 KS MR CK AM Thu 10/12/2020 KS MR CK AM Fri 11/12/2020 KS MR CK AM Sat 12/12/2020 KS MR CK AM

MR: Michael Ruzhansky (Ghent University, Belgium)

CK: Christoph Kriegler (University Blaise-Pascal, France)

KS: K. Sandeep, TIFR Bangalore, India

AM: Alessio Martini(University of Birmingham, UK)

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