Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Sarkar, Rudra (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

Let X be a rank one Riemannian symmetric space of noncompact type and ∆ be the Laplace–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 Θ on functions f on X defined by, Θ : f → f ∗ µ where µ is a radial (possibly complex) measure on X. These operators will be called multipliers. In particular Θ is a radial average when µ is a radial probability measure. Notable examples of radial averages are ball, sphere and annular averages. Another well known example is f → f ∗ ht , where ht is the heat kernel on X. This will be called heat propagator and will be denoted by e t∆. In this thesis we shall study the questions of the following genre. Below by eigenfunction we mean eigenfunction of ∆.(i) Characterization of eigenfunctions from the equation f ∗ µ = f, which generalizes the classical question: Is a µ-harmonic function harmonic?(ii) Fix a multiplier, in particular an averaging operator Θ. Suppose that {fk}k∈Z is a bi-infinite sequence of functions on X such that for all k ∈ Z, Θfk = Afk+1 and kfkk < M for some constants A ∈ C, M > 0 and a suitable norm k · 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. Plancherel–P´olya (1931) and Benyamini–Weit (1989) proved that for continuous functions f, g on R n , if Btf → g uniformly on compact sets as t → ∞, 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 ∞ and that of the the heat propagator as time goes to ∞ and use this relation for the characterization of eigenfunctions.


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


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