Certain geometric maximal functions in harmonic analysis

Abstract

The broad theme of the thesis is of geometric maximal functions associated to curved surfaces. We produce novel results about two maximal functions of different types, presented in two parts of the thesis.

In the first part (Chapter 2), we study the Lᵖ → Lᵖ boundedness of a lacunary maximal function on a graded homogeneous group. The main theorem of this part generalises the existing maximal results in specific homogeneous groups, such as the Euclidean space and the Heisenberg group. Using an iteration scheme, we estimate the maximal function, assuming that the measure associated to the maximal function satisfies a curvature condition.

This second part of this thesis (Chapters 3 and 4) deals with the problem of Lᵖ → Lᵖ boundedness of a Nikodym maximal function in the Euclidean space. The maximal function is defined using a one-parameter family of tubes in Rᵈ⁺¹, whose directions are determined by a non-degenerate curve in Rᵈ. These operators naturally arise in the analysis of maximal averages over space curves. The main theorem generalises the known results for d = 2 and d = 3 to general dimensions.