Around Fatou Theorem and Its Converse on Certain Lie Groups

Date of Submission

July 2021

Date of Award

7-1-2022

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Mathematics

Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Supervisor

Ray, Swagato Kumar (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

A classical result due to Fatou relates the radial and nontangential behaviour of the Poisson integral of suitable measures on the real line with certain differentiability properties of the measure. Loomis proved the converse of Fatou's theorem for positive measures on the real line. Rudin and Ramey-Ullrich later extended the results of Loomis in higher dimensions. In the first part of the thesis, we have proved generalizations of the result of Rudin, involving a large class of approximate identities generalizing the Poisson kernel. We have then used it to show that the analogue of Rudin's result holds for certain positive eigenfunctions of the Laplace-Beltrami operator on real hyperbolic spaces. In the second part of the thesis, we have proved the analogues of the result of Ramey-Ullrich, regarding nontangential convergence of Poisson integrals, for certain positive eigenfunctions of the Laplace-Beltrami operator of Harmonic NA groups. We have also proved similar results for positive solutions of the heat equation on stratified Lie groups

Comments

ProQuest Collection ID: https://www.proquest.com/pqdtlocal1010185/dissertations/fromDatabasesLayer?accountid=27563

Control Number

ISILib-TH537

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

Link to Full Text

Share

COinS