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Fractal Geometry and Stochastics VIAssouad Type Dimensions in Geometric Analysis

Fractal Geometry and Stochastics VI: Assouad Type Dimensions in Geometric Analysis [We consider applications of the dual pair of the (upper) Assouad dimension and the lower (Assouad) dimension in analysis. We relate these notions to other dimensional conditions such as a Hausdorff content density condition and an integrability condition for the distance function. The latter condition leads to a characterization of the Muckenhoupt Ap properties of distance functions in terms of the (upper) Assouad dimension. It is also possible to give natural formulations for the validity of Hardy–Sobolev inequalities using these dual Assouad dimensions, and this helps to understand the previously observed dual nature of certain cases of these inequalities.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Fractal Geometry and Stochastics VIAssouad Type Dimensions in Geometric Analysis

Part of the Progress in Probability Book Series (volume 76)
Editors: Freiberg, Uta; Hambly, Ben; Hinz, Michael; Winter, Steffen
Lehrbäck, Juha
Fractal Geometry and Stochastics VI — Mar 24, 2021
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References (36)

Publisher
Springer International Publishing
Copyright
© Springer Nature Switzerland AG 2021
ISBN
978-3-030-59648-4
Pages
25 –46
DOI
10.1007/978-3-030-59649-1_2
Publisher site
See Chapter on Publisher Site

Abstract

[We consider applications of the dual pair of the (upper) Assouad dimension and the lower (Assouad) dimension in analysis. We relate these notions to other dimensional conditions such as a Hausdorff content density condition and an integrability condition for the distance function. The latter condition leads to a characterization of the Muckenhoupt Ap properties of distance functions in terms of the (upper) Assouad dimension. It is also possible to give natural formulations for the validity of Hardy–Sobolev inequalities using these dual Assouad dimensions, and this helps to understand the previously observed dual nature of certain cases of these inequalities.]

Published: Mar 24, 2021

Keywords: Assouad dimension; Lower dimension; Aikawa condition; Muckenhoupt weight; Hardy–Sobolev inequality; Primary: 28A75; Secondary: 28A80; 35A23

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