| In the theory of geometric analysis on metric measure spaces, two properties of a metric measure space make the theory richer. These two properties are the doubling property of the measure, and the support of a Poincare inequality by the metric measure space. The focus of this dissertation is to show that the doubling property of the measure and the support of a Poincare inequality are preserved by two transformations of the metric measure space: sphericalization (to obtain a bounded space from an unbounded space), and flattening (to obtain an unbounded space from a bounded space). We will show that if the given metric measure space is equipped with an Ahlfors Q-regular measure, then so are the spaces obtained by the sphericalization/flattening transformations. We then show that even if the measure is not Ahlfors regular, if it is doubling, then the transformed measure is still doubling. We then show that if the given metric space satisfies an annular quaisconvexity property and the measure is doubling, and in addition if the metric measure space supports a p-Poincare inequality in the sense of Heinonen and Koskela's theory, then so does the transformed metric measure space (under the sphericalization/flattening procedure). Finally, we show that if we relax the annular quasiconvexity condition to an analog of the starlike condition for the metric measure space, then if the metric measure space also satisfies a p-Poincare inequality, the transformed space also must satisfy a q-Poincare inequality for some p ≤ q < infinity. We also show that under a weaker version of the starlikeness hypothesis, support of infinity-Poincare inequality is preserved under the sphericalization/flattening procedure. We also provide some examples to show that the assumptions of annular quasiconvexity and the various versions of starlikeness conditions are needed in the respective results. |
