Clemson University Analysis Seminar Schedule

Spring 2022

Necessary Conditions for Two Weight Weak Type Norm Inequalities for Multilinear Singular Integral Operators

Speaker:                John-Oliver MacLellan - University of Alabama
Time:                     Friday, January 28, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
https://clemson.zoom.us/rec/share/slM0FvcoHVTT--d87dOnab7AtHrON0xsehyJBeXOJCUsl3-0uVRAB4DgVOBBfqCh.W2Uu-WB4JW-MgFDR?startTime=1643386654000
Abstract:
A central problem in harmonic analysis is to characterize the pairs of weights (u,v) so that a Calderón Zygmund operator maps L^p(v)-> L^p(u). In this talk we will discuss necessary conditions for a multilinear Calderón Zygmund operator T to satisfy two weight weak type norm inequalities provided the kernel of T satisfies a weak non degeneracy condition. We generalize results from our recent paper, and earlier results from Lacey, Sawyer, and Uriarte-Tuero, and by Stein in the linear case.  As an application of our techniques, we will show that in general a multilinear Calderón Zygmund does not satisfy a two-weight strong endpoint estimate.

The C*-algebra generated by Toeplitz operators with quasi-radial symbol

Speaker:               Vishwa Dewage - Louisiana State University
Time:                     Friday, February 4, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
https://clemson.zoom.us/rec/share/Nq04iBzZKhcLpQG5PBkArCUNJWRBURNNSX8yGjc561VYADRZAkt6B_UPasg24hvR.tU6tbaWNhb9eCmd5?startTime=1643991390000
Abstract:
 In this talk we discuss Toeplitz operators with k-quasi-radial symbols acting on the Fock space $\mathcal{F}(\mathbb{C}^n)$. Toeplitz operators with k-quasi-radial symbols generate a commutative C*-algebra that is isometrically isomorphic to $C_{b,u}(\mathbb{N}_0^k)$ of bounded functions on $\mathbb{N}_0^k$ that are uniformly continuous with respect to the square root metric. In fact, the spectral functions (multi-sequences of eigenvalues) of these Toeplitz operators are dense in the space $C_{b,u}(\mathbb{N}_0^k)$.This talk is based on a joint work with Prof. Gestur Olafsson.

On the boundedness of oscillating singular integrals

Speaker:               Duván Cardona Sánchez - Ghent University
Time:                     Friday, February 11, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
https://drive.google.com/file/d/1xf7gKTQ47cUgCSP4923eI4mrnbN9KCST/view?invite=CIeN7d8O&ts=6206ab74
Abstract:
It was proved by Fefferman [1] and Fefferman and Stein [2], the weak (1,1) boundedness of oscillating singular integrals on Rn, and the boundedness from the Hardy space H1 into L1, respectively. The aim of this talk is to discuss the recent extension of these results in the Euclidean setting (in view of the paradigm introduced by Grafakos and Stockdale in [3]) and on Lie groups of polynomial growth. In view of the solution of the Rockland conjecture by Helffer and Nourrigat [4], and of the Hormander theorem of sums of squares [5], our criteria are presented in terms of the analysis of sub-Laplacians and of Rockland operators.  This talk is based on my joint works [CR1,CR2] with Michael Ruzhansky on the subject.

References.
[1] Fefferman, C. Inequalities for strongly singular integral operators, Acta Math. 24, 9–36, (1970).
[2] Fefferman, C., Stein, E. Hp spaces of several variables, Acta Math., 129, 137-193, (1972).
[3] Grafakos, L., Stockdale, C. B. A limited-range Calderón-Zygmund theorem. Bull. Hellenic Math. Soc. 63, 54–63, (2019).
[4] Helffer, B. Nourrigat, J. Caracterisation des operateurs hypoelliptiques homogenes invariantsa gauche sur un groupe de Lie nilpotent gradue, Comm. Partial Differential Equations, 4(8),899–958, (1979)
[5] Hörmander, L. Hypoelliptic second order differential equations, Acta Math., 119, 147–171,(1967).
[CR1] Cardona, D. Ruzhansky, M. Boundedness of oscillating singular integrals on Lie groups of polynomial growth,  arXiv:2201.12883
[CR2] Cardona, D. Ruzhansky, M. Weak (1,1) continuity and Lp-theory for oscillating singular integral operators, arXiv:2201.12881

Space-time sampling for the functions with bounded spectrum

Speaker:                Ilia Zlotnikov - University of Stavanger
Time:                     Monday, Februrary 28, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           https://clemson.zoom.us/rec/share/H1WVDSGPwuSOF-KtRw4FvNerWtC3WofP5V9ghmJDDtEMcKII1NDE1dbwBRkFS2nf.7yMX9bBfUgtXLgnZ?startTime=1646064943000
Abstract:
Abstract Link

Characterizations of the Toeplitz algebra on the Fock space

Speaker:               Raffael Hagger - Christian Albrechts University in Kiel
Time:                     Wednesday, March 9, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:          
https://clemson.zoom.us/rec/share/EF1zhHTRXI6RimLp4Ykj932cGDB3IiYfchwLXYuq-PJ35UsHMJ6Vd81Fms1svg1q.VL9r5mAjpI_wnUL6?startTime=1646842520000
Abstract:
Abstract Link

On Asymptotic Moments of Patterned Random Matrices

Speaker:                Tapesh Yadav - University of Florida
Time:                      Friday, March 18, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           https://clemson.zoom.us/rec/share/FFjieQfKsSzTK3JbCAtUU4XbRcub3QiNxQVtUzIIFkDChHTpCxHhlUnbA7BpoeDG.atiAH9ADZ2LYfojc?startTime=1647616527000
Abstract:
For a sufficiently nice 2 dimensional shape, we define its approximating matrix (or patterned matrix) as a random matrix with iid entries arranged according to the given pattern. For large approximating matrices, we observe that the eigenvalues roughly follow an underlying distribution. This phenomenon is similar to the classical observation on Wigner matrices. We prove that the moments of such matrices converge asymptotically as the size increases and equals to the integral of a combinatorially-defined function which counts certain paths on a finite grid.

Quantitative estimates in the matrix weighted setting

Speaker:                Israel Rivera-RÍos - University of Málaga
Time:                      Friday, April 1, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
https://clemson.zoom.us/rec/share/9e1cdmo7P-8edcZjhAqQhuNFe-Uvfz3d_cjsbT159Q0l3lODbk9NcyPkE1Qdb83i.QeQEbImDRP9V62ys?startTime=1648826148000
Abstract:
In this talk we will provide an overview from the quantitative theory of Muckenhoupt weights to the quantitative matrix weights theory. The plan of the talk will consist in revisiting sparse domination ideas, which have been a fruitful field of development in the scalar setting, and providing some insight on how they have been adapted to the vector valued setting, making them useful to obtain quantitative weighted estimates in that setting.

Integral geometry problems in Lorentzian geometry

Speaker:                Yiran Wang - Emory University
Time:                      Friday, April 8, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
https://clemson.zoom.us/rec/share/B3w9BygG_H-Hk5Eh8k-9hHRY28-8ZmSvzA5t-CTLDA5xpE8wdAzHRHKwiWahARHn.HeFJOWUMdp5AcD1a?startTime=1649430970000
Abstract: 
We consider the light ray transform on Lorentzian manifolds, which concerns the integral of functions along light-like (or null) geodesics. The transform is related to the Radon transform or geodesic ray transform in the Riemannian setting. An outstanding question is what information regarding the function can be recovered from the light ray transform. In this talk, we discuss recent developments, including injectivity, stability results and microlocal properties of the transform. Also, we discuss the application to some inverse problems in cosmology. In particular, we will show how to recover space-time structures from the Cosmic Microwave Background (CMB).

Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidty is fixed, via rearrangements

Speaker:                Orli Herscovici - Georgia Institute of Technology
Time:                      Friday, April 15, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           https://clemson.zoom.us/rec/share/1rRx_jxazzn17stYHrAC8iB8wEco_dQlgLmLxHKBdXP7UIdfmzhByBrrRHMUMF03.nOmms9m4PIFwlnyf?startTime=1650035766000
Abstract:
In this talk, we show an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we present the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Polya-Szego: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a ``modified''  torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure.

We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle.

Based on joint work with Galyna Livshyts.

Inversion, convexity, and realization problems in free noncommutative analysis

Speaker:                Mark Mancuso - Lafayette College
Time:                      Wednesday, April 27, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:            https://clemson.zoom.us/rec/share/--kAltOo36rcYUNbASLvmlYTbnuYoqcjRh1VoGOCwso-QCXYKi16VJYwj1uBIB3Z.izJ9S5LpnLxsnLKA?startTime=1651072661000
Abstract:
This talk will provide an overview of some central problems in free analysis related to noncommutative inversion, matrix and operator convexity, and obtaining realizations. Free analysis, or noncommutative function theory, finds its origin in J. L. Taylor's work on the functional calculus of several noncommuting operators. Loosely speaking, noncommutative function theory can be thought of as an analogue of several complex variables where the holomorphic functions act on domains consisting of either tuples of bounded operators on a Hilbert space or tuples of square complex matrices of all sizes. The simplest examples of noncommutative functions are given by free polynomials with complex coefficients; $p(X,Y)=XY^2+XY-YX$ is a free polynomial and can be evaluated at pairs of matrices or operators. Well-known results along with recent progress will be shared throughout this talk.

Clemson Analysis Seminar Archive