The research work done can be divided in three major areas namely:

1. Harmonic analysis on locally compact groups and hypergroups.

2. Operator spaces tensor product of C*-algebras.

3. Complex analytic methods in partial differential equations.

Detailed Summary of the Research Work

1. The research work in harmonic analysis mainly deals with following problems:

a) Ideal structure of group algebras.

b) Spectral synthesis on convolution spaces of probability measures (hypergroups).

c) Uncertainty principle on Nilpotent Lie groups.

d) Potential theory on Nilpotent Lie groups.

In part (a) Wiener property, *-regularity of the Beurling algebras on [FC]^- groups have been studied.

In part (b), spectral synthesis of commutative hypergroup algebras including Wiener Tauberian theorem have been studied. Representation theory for hypergroups has been developed including Plancherel theorem, inversion theorem and Wiener property.

In part (c), Hardy’s theorem, qualitative uncertainty principle, Heisenberg inequality are proved for a variety of locally compact groups including several general classes of connected nilpotent Lie groups.

In part (d), representation formula for the higher powers of the sub-Laplacian by defining Green’s function and Poisson kernel of higher order for H-type groups, which include the Heisenberg group have been given. Boundary value problems like Dirichlet and Neumann have been studied in Heisenberg group.

The proofs exploit induced representations, Mackey theory, Kirillov’s representation theory and harmonic analysis of nilpotent Lie group . The work done has been published in Journal of Functional Analysis; Pacific J. Math.; Proc.Amer.Math. Soc.; Mathematische Zeitschirft.; Forum Mathematicum; Math. Proc. Camb. Phil. Soc.; Complex Variables and Elliptic Equations; J.Aust.Math Soc.; Analysis., Boundary Value Problems, Journal of Geometric Analysis, Journal of Inequalities and Applications, Journal of Mathematical Inequalities and Potential Analysis.

2. The theory of operator spaces grew out of the analysis of completely positive and completely bounded mappings. These maps were first studied on C*-algebras, and later on suitable subspaces of C*-algebras. The extension and representation theorems for completely bounded maps show that subspaces of C*-algebras carry an intrinsic metric structure which is preserved by completely isometries. Just as the theory of C*-algebras can be viewed as noncommutative topology and the theory of von Neumann algebras as noncommutative measure theory, one can think of the theory of operator spaces as noncommutative functional analysis. The operator space tensor product of C*-algebras particularly the Haagerup norm, operator space projective norm and Banach space projective norm have been investigated in the research work done. The ideal structure of *- Banach algebra of operator space tensor product of C*-algebras and of the Banach algebra of Haagerup tensor product have been studied. In particular, it has been shown that the Haagerup norm and Banach space projective norms are equivalent on the algebraic tensor product of two C*-algebras A and B if,and only if , either A or B is finite dimensional, or A and B are infinite dimensional and subhomogeneous. The same conclusion is proved if the Banach space projective norm is replaced by operator space projective norm. Involution on the Haagerup tensor product have been studied. It is shown that the centre of the operator space tensor product of two C*-algebras is the operator space tensor products of their centres. A characterization of isometric automorphisms of the two Banach algebras have been obtained. Methods involve the commutative and non-commutative Grothendieck inequalities, lifting maps to second duals and to tensor product of second duals and the structure of C*-algebras and von Neumann algebras. We have developed a systematic study on the schur tensor product of operator systems. The λ-theory of operator spaces (based on the tensor norms obtained from homogeneous polynomials), which generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces, has been extended to matrix ordered spaces and Operator Systems. This work has been published in Trans.Amer.Math.Soc.; Mathematische Zeitschrift.; Proc.Edinburgh.Math.Soc.; Forum Mathematicum; Positivity; J. Aust. Math. Soc.; Archiv Mathematik etc.

3. A systematic investigation of basic boundary value problems for complex partial differential of arbitrary order has been initiated. Integral representations are one of the important tools in analysis. Well-known representation the Cauchy’s formula for analytic functions and Green’s representation for harmonic functions. A variety of boundary value problems arising from Schwarz, Dirichlet, Neumann, Riemann Hilbert problems for the inhomogeneous polyanalytic equations have been investigated. Mixed boundary arising from a combination of Schwarz, Dirichlet and Neumann boundary value conditions have been studied for the inhomogeneous polyanalytic equations. Explicit representation formulas are obtained on several regions. Boundary value problems for bi-polyanalytic functions on different regions have been investigated. These investigations are based on Gauss theorem, Cauchy’s Pompeiu representation, hierarchy of integral representations and iteration process leading from lower order equations to higher ones. The work has been published in Applicable analysis; Complex variables and Elliptic equations: Arch.Math.; Analysis; altogether there are more than 382 citations including 23 citations in books and Several Ph.D. thesis all over the world have cited the work done . Important Books from Academic Press, Cambridge Univ. Press, World Scientific, Gryter Berlin, Longman, Kulwer Academic Publishers, Birkhauser, Springer Monograph have cited the work done.

The work has stimulated several researchers and is well-cited in several reputed journals, books and Ph.D. theses.