Date of Graduation


Document Type


Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level



Mathematical Sciences


John Ryan

Committee Member

Maria Tjani

Second Committee Member

Daniel Luecking


Ahlfors-Beurling, Beltrami Equation, Clifford Analysis, Singular Integral Operator, Spectrum


In this dissertation, we studies Π-operators in different spaces using Clifford algebras. This approach generalizes the Π-operator theory on the complex plane to higher dimensional spaces. It also allows us to investigate the existence of the solutions to Beltrami equations in different spaces.

Motivated by the form of the Π-operator on the complex plane, we first construct a Π-operator on a general Clifford-Hilbert module. It is shown that this operator is an L^2 isometry. Further, this can also be used for solving certain Beltrami equations when the Hilbert space is the L^2 space of a measure space. This idea is applied to examples of some conformally flat manifolds, the real projective space, cylinders, Hopf manifolds and n-dimensional hyperbolic upper half space.

It is worth pointing out that the proof for the L^2 isometry of Π-operator on the unit sphere is different from the idea mentioned above. In that idea, it requires the Dirac operator and its dual operator commute to prove the L^2 isometry of the Π-operator. However, this is no longer true for the spherical Dirac operator. Hence, we use the spectrum of spherical Dirac operator to overcome this problem. Since the real projective space can be defined as a projection from the unit sphere, Π-operator theory in the real projective space can be induced from the one on the unit sphere. Similarly, Π-operator theory on cylinders (Hopf manifolds) is derived from the one on n-dimensional Euclidean space via a projection map.

Classical Clifford analysis is centered at the study of functions on n-dimensional Euclidean space taking values in Clifford numbers. In contrast, Clifford analysis in higher spin spaces is the study of functions on n-dimensional Euclidean space taking values in arbitrary irreducible representations of the Spin group. At the end of this thesis, we construct an L^2 isometric Π-operator in higher spin spaces.

Further, we provide an Ahlfors-Beurling type inequality in higher spin spaces to conclude the thesis.