A Global Symbol for the Small b-Calculus on Manifolds with Boundary

Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. A Global Symbol for the Small b-Calculus on Manifolds With Boundary Thesis directed by Prof. Markus Pflaum Starting with the normal symbold of M. Pflaum ([Pfl98]), we generalize the machinery of global symbol calculi from the familiar setting of a compact Riemannian manifold to the b-Calculus on a compact manifold-with-boundary, endowed with an exact b-metric. We define a notion of b-linearization, crucial to the symbol calculus. and show that an exact b-metric can be used to construct a b-linearization. Using the b-linearization, we define the global symbol of a b-pseudodifferential operator as a fiberwise Fourier transform over the b-tangent bundle. We prove that the global symbol is truly a symbol, of the same order of its operator, and define a quantization map with which one can recover the operator (modulo b-smoothing operators.) We compute the global symbol for a b-Laplacian, and give a formula for the b-trace in terms of the global symbol. Acknowledgements Special thanks to my advisor, Markus Pflaum, for finding a thesis topic that was perfect for me, for supporting me through the years, and for laying the foundations of this work in the boundaryless setting. To Florian Sobieczky, for his service on my defense committee and his help proofreading this work. And to the rest of my committee, Additional thanks to Paul Loya and Dan Grieser, whose clear, intuitive expositions of the b-calculus helped me see the big picture.