#### Date of Award

Summer 7-16-2014

#### Document Type

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Markus Pflaum

#### Second Advisor

Florian Sobieczky

#### Third Advisor

Judith Packer

#### Abstract

Starting with the normal symbold of M. Pflaum, 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.

#### Recommended Citation

Hower, John Christian, "A Global Symbol for the Small b-Calculus on Manifolds with Boundary" (2014). *Mathematics Graduate Theses & Dissertations*. 1.

http://scholar.colorado.edu/math_gradetds/1