Characterization of n-rectifiability in terms of Jones’ square function: Part II

By Jonas Azzam, Xavier Tolsa. Geometric and Functional Analysis, 2015

103 citations recorded by the source corpus.

We show that a Radon measure μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document} in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^d}$$\end{document} which is absolutely continuous with respect to the n-dimensional Hausdorff measure Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}^n}$$\end{document} is n-rectifiable if the so called Jones’ square function is finite μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document}-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely continuous with respect to Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}^{n}}$$\end{document}. Further, in this paper we also investigate the relationship between the Jones’ square function and the so called Menger curvature of a measure with linear growth, and we show an application to the study of analytic capacity.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fc8db5735d8042b2724e3459d5109521f45c7c5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Xavier Tolsa
2. Tuomas Hytönen
3. Dachun Yang
4. Wen Yuan
5. Xing Fu
6. Yoshihiro Sawano
7. Hitoshi Tanaka
8. Dongyong Yang

In Geometry and Topology

1. Xavier Tolsa
2. Tuomas Hytönen
3. Dachun Yang
4. Alexander Volberg
5. Brett D. Wick

In Mathematical Physics

1. Xavier Tolsa
2. Tuomas Hytönen
3. Dachun Yang
4. Wen Yuan
5. Xing Fu
6. Yoshihiro Sawano
7. Hitoshi Tanaka
8. Alexander Volberg

Citations in the corpus

• Characterization of n-rectifiability in terms of Jones’ square function: part I (2015, methodology): “In the sequel [3] of the present work, by Azzam and the author of the present paper, it is shown that the finiteness of Jones’ square function for p = 2 implies n-rectifiability.”