Title: k-plane Transforms, related operators and their properties on Constant Curvature Spaces
Description: We will be looking at the X-Ray transform, its generalization, the Radon transform, and its further generalization, the k-plane transforms. A lot of research has been done in the field of reconstructing a function using the k-plane transforms on the Euclidean spaces. We will be interested in some properties of these transforms and their behaviour on (smooth) manifolds and spaces of constant curvature.
Title: Boundedness of singular and hypersingular Integrals along Hypersurfaces
Description: We are interested in knowing the L^p-boundedness of hypersingular integrals defined along hypersurfaces. Starting from the codimension 1 case, we would like to study the problem when the said hypersurface is of arbitrary codimension in a Euclidean space. The project also deals with understanding the curvature properties of the surfaces that allow L^p-boundedness. Further, we would also like to investigate whether we have boundedness of such integrals in other function spaces (such as L^p-L^q or Lorentz spaces).
We are also interested in understanding the boundedness of singular integral operators along hyperplanes, which are flat submanifolds. Due to flatness, the results expected here are vastly different from the curved surfaces case.
Title: Harmonic Analysis in the setting of capacities
Description: Capacities occur in the study of the structure of rough and exceptional sets arising in partial differential equations. As a mathematical object, a capacity mimics an outer measure on the Euclidean space and therefore gives a notion of integration that is not linear. Recently, certain aspects of Harmonic Analysis, such as the study of maximal operators, Lebesgue and BMO spaces, was initiated by certain groups of Mathematicians. In this project, we wish to continue looking at the Harmonic Analysis aspects of capacities (particularly Hausdorff capacities, which play an important role in Geometric Measure Theory and the study of fractals). We wish to study weighted boundedness of capacitary maximal functions, singular integrals related to capacities and many other directions.