Sparse bounds for various spherical maximal functions

Abstract

Harmonic analysis mainly deals with the qualitative and quantitative properties of functions and transforms of those functions. It has applications in various areas of Mathematics like PDE, Differential geometry, Ergodic theory etc and also in several areas of Physics like Classical and Quantum mechanics etc and this makes it a very attractive area of study. The theory of spherical means plays a very crucial role in the field of Classical harmonic analysis. In 1976, E.M.Stein first studied the boundedness properties of maximal function associated to spherical means taken over the Euclidean sphere. Theory of spherical means taken over geodesic spheres in different Lie groups and Symmetric spaces has received considerable attention in the last few decades. In this thesis, we consider various versions of spherical maximal function, mainly on Euclidean space and its non-commutative neighbour Heisenberg group