# NCMW Harmonic Analysis (2017)

Venue: |
Indian Institute of Science, Banglore, Karnataka |

Date: |
11th, Dec 2017 to 16th, Dec 2017 |

Name |
Prof. E.K. Narayanan | Prof. P.K. Ratnakumar |

Mailing Address |
Indian Institute of Science (IISc) Bangalore |
School of Mathematics Harish-Chandra Research Institute Allahabad 211019 |

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This was an advanced workshop in four different themes in harmonic analysis. First set of lectures by Professor J Faraut focused on orbital measures and spline functions. The action of unitary group on the space of Hermitian matrices can be described in terms of the results of Olshanksii, Okunkov etc and thus involve harmonic analysis on compact Lie groups. Horns theorem describing the projection of an orbit in the space of Hermitian matrices under the action of the unitary group and the orbital measure was explained. Second set of lectures by Professor Linda Saal dealt with generalized Gelfand pairs associated to the Heisenberg group. Spherical analysis associated to these pairs was discussed in detail. Third set of lectures by Professor Shobha Madan and Professor Saurabh Srivastava focused on sparse operators and a proof of the A2 conjecture. Finally, the lectures by Professor Swagato Ray dealt with harmonic analysis on symmetric spaces. In particular, Kunze-Stein phenomenon and boundedness of convolution operators were established.

** Speakers and their affiliations with detailes of lecture**

Speakers |
Affiliations |
Details of lectures delivered |

J. Faraut, | Professor, Universite Pierre et Marie Curie, France | Introduction to orbital measures and statement of three problems related to orbital measures. Action of the unitary group on Hermitian matrices and Baryshiknov’s theorem on the joint distribution of eigenvalues. Olshanskii’s theorem on projection of orbital measures to the (1, 1)th co-ordinate of the matrices. Okunkov’s result on the distribution of X11. Action of orthogonal group on the space of real symmetric spaces. Discussion of the three problems and a formula for the projection of the orbital measure. Discussion of open problems in the case of U (p, q) action. |

Shobha Madan | Professor, IISER Mohali, India | Introduction to Euclidean Fourier analysis. Convolutions. Plancherel theorem for the Euclidean Fourier transform. Schwartz space and Paley-Wiener theorems. Hardy-Littlewood maxi- mal functions and its Lp− boundedness. Hilbert transform and its Lp boundedness properties. Introduction to singular integrals. Calderon -Zygmund singular integral operators. Calderon-Zygmund decompostions and covering lemmas. Weak type (1, 1) propertly of C–Z operators. Riesz transforms. |

Swagato Ray | Professor, ISI Kolkata, India | Iwasawa and Polar decomposition on SU (1, 1), relation with SL(2, R). Description of Haar measure in the coordinates KAN, AN K, N AK. Poincar ́e metric on the open unit disc, geodesics and horocycles. Haar measure with respect to Polar decomposition. Construction of eigenfunctions of the Laplace-Beltrami operator which are constant on horocycles. Definition of the Fourier transform. Fourier transform is well defined for Lp functions on the Poincar ́e disc. K−biinvariant functions and the spherical Fourier transform. The slice projection theorem for Radon, Fourier and the Abel transform. The Plancherel theorem for L2 functions on G/K assuming the Plancherel theorem for K−biinvariant functions.The Abel inversion and the Fourier inversion for smooth, compactly supported K−biinvariant functions on G = SU (1, 1). Harish Chandra series for the elementary spherical functions. Harish Chandras c−function and asymptotics of the elementary spherical functions. Elementary cases of Kunze–Stein phenomena. Young’s convolution inequality for unimodular and non-unimodular groups. View a function on the hyperbolic space H 2 as a function on its isometry group which is unimodular and also as a function on the associated Iwasawa N A group which is non-unimodular and use this dichotomy to obtain a new convolution inequality known as Kunze–Stein phenomenon. Mapping properties of spectral projection, restriction and extension operators on the hyperbolic space H 2 and its connection to the convolution inequality mentioned above. Hardy- Littlewood maximal function (centered and uncentered), their mapping properties. Counter example to establish the sharpness of these properties and emphasize the difference between the centered and uncentered versions. Comments on the effect of polynomial versus exponential volume growth vis-a-vis the lack of ball doubling property of the metric-measure space H 2 on some results of analysis on this space. Discussion of some open questions. |

Linda Saal | Professor, University of Cordoba, Argentina | Introduction to Gelfand pairs, class one representations and spherical functions. Examples, (M (n), SO(n)), (HM (n), U (n)) etc. General examples of the type (G K, K) where G is a two step nilpotent Lie group. Spherical functions in explicit form. Bessel functions and Laguerre functions. Generalized Gelfand pairs. U (p, q) action on the Heisenberg group. Associated generalized spherical distribution and spherical analysis associated to this pair. Open problems. |

Saruabh Srivastava | Associate Professor, IISER Bhopal, India | Introduction to sparse operators. Calderon- Zygmund operators. Sparse bounds for C–Z operators with smooth kernels. A2 conjecture. Proof of the A2 conjecture due to A. K. Lerner. Discussion of some open problems related to rough singluar integrals. |

Time Table

Date |
Lecture 19:30-10:30 |
Tea10:30 -11 |
Lecture 2 11-12 |
Lecture 3 12-1 |
Lunch1- 2:30 |
Lecture 42:30-3:30 |
Tea3:30 - 4 |
Lecture 5 4-5 |

11-12-2017 | JF | LS | SM | SR | SS | |||

12-12-2017 | JF | LS | SM | SR | SS | |||

13-12-2017 | JF | LS | SM | SR | SS | |||

14-12-2017 | JF | LS | SM | SR | SS | |||

15-12-2017 | JF | LS | SM | SR | SS | |||

16-12-2017 | JF | LS | SM | SR | SS |

Actual Participants |

Sr.n |
Name |
Affiliation |
Position held |

1 | Anoop V. P. | NISER, Bhubaneswar | Research scholar |

2 | Sayan Bagchi | ISI, Kolkata | Inspire faculty |

3 | Rijju Basak | IISER Bhopal | BS-MS student |

4 | Mithun Bhowmick | ISI, Kolkata | Research scholar |

5 | Santanu Debnath | Kolkata University | Research scholar |

6 | Venku Naidu Dogga | IIT Hyderabad | Assistant Professor |

7 | Abhishek Ghosh | IIT Kanpur | Research scholar |

8 | Qaiser Jahan | IIT Mandi | Assistant Professor |

9 | Jotsaroop Kaur | IIT Bombay | Assistant Professor |

10 | Ashisha Kumar | IIT Indore | Assistant Professor |

11 | Shravan Kumar | IIT Delhi | Assistant Professor |

12 | Lakshmi Lavanya | IISER Tirupati | Assistant Professor |

13 | Arup Kumar Maity | Harish-Chandra Rese- arch Institute, Allahabad |
Research scholar |

14 | S. Pitchai Murugan | RIASM, Chennai | Research scholar |

15 | Muna Naik | ISI, Kolkata | Research scholar |

16 | Sanjay Parui | NISER, Bhubaneswar | Assistant Professor |

17 | Partha Sarathi Patra | IIT Hyderabad | Research scholar |

18 | Savan Patel | St Xavier’s college, Ahemdabad | Assistant Professor |

19 | Sanjay Pusti | IIT, Bombay | Associate Professor |

20 | Senthil Raani | ISI, Bangalore | Postdoctoral Fellow |

21 | Sivaramakrishna C | IIT Hyderabad | Research scholar |

22 | Ratnakumar P. K. | HCRI, Allahabad | Professor |

23 | Kalachand Shuin | IISER Bhopal | Research scholar |

24 | Sumit Kumar Rano | IIT Guwahati | Research scholar |

25 | Samya Kumar Ray | IIT Kanpur | Research scholar |

26 | Sanjay P. K | NIT, Kozhikode, Kerala | Assistant Professor |

27 | Jayanta Sarkar | ISI, Kolkata | Research scholar |

28 | Rajesh Kumar Singh | IIT Kanpur | Research scholar |

29 | Devendra Tiwari | Delhi University | Research scholar |

**How to reach**

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