Characterization of eigenfunctions of the Laplace-Beltrami operator through radial averages on rank one symmetric spaces/ Muna Naik

Thesis (Ph.D.) - Indian Statistical Institute, 2019

Preliminaries -- Characterization of eigenfunctions from the equation f ∗ µ = f -- Characterization of eigenfunctions via Roe–Strichartz type theorems -- Mean value property in limit, a result of Plancherel–P´olya and Benyamini–Weit -- Large and small time behaviour of heat propagation

Guided by Prof. Rudra Sarkar

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 7→ 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 7→ 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 ∆.