Topics in Geometric Analysis and Harmonic Analysis on Spaces of Homogeneous Type

Monday, April 20, 2015 - 3:00pm
MSB 111
Ryan Alvarado (Advisor: Professor Marius Mitrea)

The present thesis consists of three main parts. One theme underscoring the work carried out in this thesis concerns the relationship between analysis and geometry. As a first illustration of the interplay between these two branches of mathematics we will prove that, in the setting of d-dimensional Ahlfors-regular quasi-metric spaces, a satisfactory theory of Hardy spaces (Hp spaces) exists for an optimal range of p's. A fascinating feature of this range is that it is intimately linked to both the geometric and measure theoretic aspects of the ambient. Many facets of this theory will be discussed including sharp versions of several tools used in the area of analysis on quasi-metric spaces such as a sharp Lebesgue differentiation theorem. The presented work is in collaboration with M. Mitrea.
In the second part, we prove that a function defined on a subset of a geometrically doubling quasi-metric space which satisfies a Holder-type condition may be extended to the entire space with preservation of regularity. The proof proceeds along the lines of the original work of Whitney in 1934 and yields a linear extension operator. A similar extension result is also proved in the absence of the geometrically doubling hypothesis, albeit the resulting extension procedure is nonlinear in this case. This work is done in collaboration I. Mitrea and M. Mitrea.
In the final part of the thesis we prove that an open, proper, nonempty subset of Rn is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The latter is a property of a purely geometrical nature, which amounts to the ability of threading the boundary in between the two rounded components of a certain fixed hour-glass-type region. Additionally, we discuss a sharp generalization of the Hopf-Oleinik boundary point principle for domains satisfying a one-sided, interior pseudo-ball condition, for semi-elliptic operators with singular drift. These results have been obtained in collaboration with D. Brigham, V. Maz'ya, M. Mitrea, and E. Ziade.