Date of Graduation

8-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor

John Ryan

Committee Member

Daniel Luecking

Second Committee Member

Phillip Harrington

Abstract

In this dissertation, we complete the work of constructing arbitrary order conformally invariant operators in higher spin spaces, where functions take values in irreducible representations of Spin groups. We provide explicit formulas for them.

We first construct the Dirac operator and Rarita-Schwinger operator as Stein Weiss type operators. This motivates us to consider representation theory in higher spin spaces. We provide corrections to the proof of conformal invariance of the Rarita-Schwinger operator in [15]. With the techniques used in the second order case [7, 18], we construct conformally invariant differential operators of arbitrary order with the target space being degree-1 homogeneous polynomial spaces. Meanwhile, we generalize these operators and their fundamental solutions to some conformally flat manifolds, such as cylinders and Hopf manifolds. To generalize our results to the case where the target space is a degree k homogeneous polynomial space, we first construct third order and fourth order conformally invariant differential operators by similar techniques. To complete this work, we notice that the techniques we used previously are computationally infeasible for higher order (≥ 5) cases. Fortunately, we found a different approach to conquer this problem. This approach relies heavily on fundamental solutions of these differential operators. We also define a large class of conformally invariant convolution type operators associated to fundamental solutions. Further, their inverses, when they exist, are conformally invariant analogues of pseudo-differential operators.

We also point out that these conformally invariant differential operators with their fundamental solutions can be generalized to some conformally flat manifolds, for instance, cylinders and Hopf manifolds. This can be done with the help of Eisenstein series as in [31].

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