Date of Defense
11-11-2024 10:00 AM
Location
F1-2007
Document Type
Thesis Defense
Degree Name
Master of Science in Mathematics
College
COS
Department
Mathematical Sciences
First Advisor
Prof. Salem Ben Said
Keywords
Dunkl Laplacian, Fractional Laplacian, Generalized Fourier transform, The Heat semigroup, Pointwise convergence, Spherical mean operator, Extension Theorem.
Abstract
This thesis provides a comprehensive study of the fractional Dunkl-type Laplacian operator (—llxllΔk) σ for 0 < σ < 1, focusing on four key equivalent characterizations: the heat semigroup approach, the pointwise formulation, the spherical mean representation, and through an extension theorem. A significant part of the thesis is devoted to the extension theorem, where, following the approach of Caffarelli and Silvestre for the Euclidean Laplacian, we prove that (—llxllΔk) σ can be characterized as an operator that maps a Dirichlet boundary condition to a Neumann-type condition via an extension PDE problem. Further, a Poisson formula for the extension was established. These four characterizations provide distinct perspectives on the operator llxllΔk, extending Euclidean Laplacian results to include reflection symmetries and revealing connections between the fractional operator (—llxllΔk) σ, harmonic analysis, and applications in mathematical physics and PDEs.
Included in
ON FRACTIONAL DUNKL-TYPE LAPLACIAN
F1-2007
This thesis provides a comprehensive study of the fractional Dunkl-type Laplacian operator (—llxllΔk) σ for 0 < σ < 1, focusing on four key equivalent characterizations: the heat semigroup approach, the pointwise formulation, the spherical mean representation, and through an extension theorem. A significant part of the thesis is devoted to the extension theorem, where, following the approach of Caffarelli and Silvestre for the Euclidean Laplacian, we prove that (—llxllΔk) σ can be characterized as an operator that maps a Dirichlet boundary condition to a Neumann-type condition via an extension PDE problem. Further, a Poisson formula for the extension was established. These four characterizations provide distinct perspectives on the operator llxllΔk, extending Euclidean Laplacian results to include reflection symmetries and revealing connections between the fractional operator (—llxllΔk) σ, harmonic analysis, and applications in mathematical physics and PDEs.