Pseudo-differential operators

I developed these operators on the affine group and obtained a characterization of the following: a) compact pseudo-differential operators on compact lie groups, b) Hilbert-Schimdt pseudo-differential operators and trace class operators on the Heisenberg group and Heisenberg (H-type) group. I developed the complete spectral theory of the SG pseudo-differential operators on Euclidean space (Rn), computed the spectrum of the operators and established the equivalence of the ellipticity and Fredholm property for SG pseudo-differential operators in Euclidean space.

Harmonic Analysis

  • Ultra-differentiable functions - I obtained the global characterizations of classes of ultradifferentiable functions and corresponding ultra-distributions using the eigenvalues of the Laplace Betrami operator (Casimir element) on the compact Lie group groups and extended the characterizations on the compact manifolds. I also proved universality theorem of the Komatsu class of ultradistributions on compact manifolds.
  • H-type Groups - I studied the heat kernel transform on the nilmanifold associated with a H-type group. The problem was reduced to the study of a family of Heisenberg nilmanifolds using partial Radon transform.

Partial differential equations

  • Partial differential operators - I studied the properties of the twisted Laplacian, sub-Laplacians and Grushin operator on Rn, Heisenberg group and R2, respectively. I obtained estimates for the inverses and the heat semigroups of these operators. I also obtained Liouville‚Äôs theorems for harmonic functions for the sub-Laplacian and the Laplacian on the Heisenberg group. I proved a version of the Liouville theorem for the entire solutions and obtained the Laurent series expansion of a solution of the heat equation with an isolated singularity.
  • Non-linear partial differential equations - I proved a much improved global well posedness result for nonlinear waves on the curved background.

Presently, I am working on the wave equation for the Laplacian on the discrete lattice trying to establish the sharp well-posedness results and describe the regularity phenomenon.