Poincaré Inequalities for Quasi-Doubling Measures in Metric Spaces with Fractal Boundaries

Abstract

This parer investigates the validity of poincare inequalities in metric measure spaces that fail to stisfy the classical doubling condition .We construct a specific metric space X by taking the open unit disk in R² and replacing its boundary with Koch curve ,a well-known fractal. We define a measure m on X that combines the 2D Lebesgue measure on the interior with the s-dimensional Hausdorff measure (where s=log 4/log 3)on the fractal boundary . We first demonstrate that this measure m is not a doubling measure due to the dimensional mismatch at the interface between the disk and its boundary. Our main result shows that m is,however, a quasi-doubling measure.