Clifford Analysis Techniques for Spherical PDE

Abstract

For α ∈ R, the class of α−order spherical harmonic functions in an open setis the spherical Laplace-Beltrami operator of order α and s is the radially independent part of the Laplace operator. We obtain a Green's integral formula for the functions in H α ( ) with kernel expressed as a Gegenbauer function. As generalizations, higher order spherical iterated Dirac operators are defined in a polynomial form. Integral representations of the null solutions to these operators and an intertwining formula relating these operators on the sphere and their analogues in Euclidean space are presented.

Clifford and Harmonic Analysis on Cylinders and Tori

“…In this way one constructs other examples of conformally flat manifolds, see for instance [23]. One can now ask if the constructions developed here and in [16] extend to this more general context. This analysis will be developed elsewhere.…”

“…It should though be pointed out that Dirac operators and associated Cauchy integral formulas have been introduced in a very general setting in [5,18,7]. The approach taken here and in [16] afford a more concrete viewpoint. The general intention is to find as explicit an approach to Clifford analysis as possible for reasonable choices of manifolds.…”

“…For the case of the sphere and hyperbolas this has been partially developed in [16,27] and elsewhere. Here we will deal with the simpler cases of cylinders and tori.…”

Some Sharp L2 Inequalities for Dirac Type Operators

“…In [13] it is shown that C 1 (w, y) = 1 ωn y−w y−w n where ω n is the surface area of the unit sphere in R n . See also [11].…”

Dirac type operators for spin manifolds associated to congruence subgroups of generalized modular groups

“…Basic aspects of Clifford analysis over spin manifolds have been developed in [6,7,37]. Further in [25,26,27,33,35,39] and elsewhere it is illustrated that the context of conformally flat manifolds provide a useful setting for developing Clifford analysis. Conformally flat manifolds are those manifolds which possess an atlas whose transition functions are Möbius transformations.…”