The present dissertation consists of three main parts. One theme underscoring the work carried out in this dissertation concerns the relationship between analysis and geometry. As a first illustration of the interplay between these two branches of mathematics we develop a sharp theory of Hardy spaces in the setting of spaces of homogeneous type. 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 dissertation we prove that an open, proper, nonempty subset of R n is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. Additionally, we prove 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. Ziad ́e.