Topics in harmonic analysis and partial differential equations: extension theorems and geometric maximum principles

Abstract

The present thesis consists of two main parts. In the first part, we prove that a function defined on a closed subset of a geometrically doubling quasi-metric space which satisfies a Hölder-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. The results presented in this part are based upon work done in collaboration M. Mitrea. In the second 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, at any location, in between the two rounded components of a certain fixed region, whose shape resembles that of an ordinary hour-glass, suitably re-positioned. The limiting cases of the result are as follows: Lipschitz domains may be characterized by a uniform double cone condition, whereas domains of class ℂ1,1 may be characterized by a uniform two-sided ball condition. 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. This, in turn, is used to obtain a sharp version of Hopf's Strong Maximum Principle for second-order, non-divergence form differential operators with singular drift. This part of my thesis originates from a recent paper in collaboration with D. Brigham, V. Maz'ya, M. Mitrea, and E. Ziad e.