Local homogeneity and dimensions of measures
Antti K\"aenm\"aki, Tapio Rajala, Ville Suomala
TL;DR
This paper introduces new concepts of local homogeneity and local L^q-spectrum to analyze the local structure and dimensions of measures, providing new estimates in conical densities and porous measures within doubling metric spaces.
Contribution
It presents novel tools for studying local measure structures and extends the analysis of local dimensions to more general settings with new estimates.
Findings
Introduction of local homogeneity and local L^q-spectrum concepts
Application to local dimensions in doubling metric spaces
New estimates for conical densities and porous measures
Abstract
We introduce two new concepts, local homogeneity and local L^q-spectrum, both of which are tools that can be used in studying the local structure of measures. The main emphasis is given to the examination of local dimensions of measures in doubling metric spaces. As an application, we reach a new level of generality and obtain new estimates in the study of conical densities and porous measures.
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