#### Date of Submission

2-28-2019

#### Date of Award

2-28-2020

#### 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

Sarkar, Rudra (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

Let X be a rank one Riemannian symmetric space of noncompact type and âˆ† be the Laplaceâ€“Beltrami operator of X. The space X can be identified with the quotient space G/K where G is a connected noncompact semisimple Lie group of real rank one with finite centre and K is a maximal compact subgroup of G. Thus G acts naturally on X by left translations. Through this identification, a function or measure on X is radial (i.e. depends only on the distance from eK), when it is invariant under the left-action of K. We consider right-convolution operators Î˜ on functions f on X defined by, Î˜ : f â†’ f âˆ— Âµ where Âµ is a radial (possibly complex) measure on X. These operators will be called multipliers. In particular Î˜ is a radial average when Âµ is a radial probability measure. Notable examples of radial averages are ball, sphere and annular averages. Another well known example is f â†’ f âˆ— ht , where ht is the heat kernel on X. This will be called heat propagator and will be denoted by e tâˆ†. In this thesis we shall study the questions of the following genre. Below by eigenfunction we mean eigenfunction of âˆ†.(i) Characterization of eigenfunctions from the equation f âˆ— Âµ = f, which generalizes the classical question: Is a Âµ-harmonic function harmonic?(ii) Fix a multiplier, in particular an averaging operator Î˜. Suppose that {fk}kâˆˆZ is a bi-infinite sequence of functions on X such that for all k âˆˆ Z, Î˜fk = Afk+1 and kfkk < M for some constants A âˆˆ C, M > 0 and a suitable norm k Â· k. We try to infer that then f0, hence every fk, is an eigenfunction(iii) Let Btf be the ball (of radius t) average of f. Plancherelâ€“PÂ´olya (1931) and Benyaminiâ€“Weit (1989) proved that for continuous functions f, g on R n , if Btf â†’ g uniformly on compact sets as t â†’ âˆž, then g is harmonic. We endeavour to generalize this result for eigenfunctions on X.(iv) We explore the behaviour of heat propagator in X in large and small time to illustrate the differences with the corresponding results in R n . In particular we study the relation between the limiting behaviour of the ball-averages as radius tends to âˆž and that of the the heat propagator as time goes to âˆž and use this relation for the characterization of eigenfunctions.

#### Control Number

ISILib-TH486

#### Creative Commons 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

#### Recommended Citation

Naik, Muna Dr., "Characterization of Eigenfunctions of the Laplace-Beltrami Operator Through Radial Averages on Rank One Symmetric Spaces." (2020). *Doctoral Theses*. 442.

https://digitalcommons.isical.ac.in/doctoral-theses/442

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843862