An $\alpha$-number characterization of $L^{p}$ spaces on uniformly rectifiable sets

TL;DR

This paper provides a new characterization of $L^{p}$ spaces on uniformly rectifiable sets using Tolsa's $oldsymbol{ extalpha}$-numbers, linking geometric measure theory with functional analysis.

Contribution

It introduces an $oldsymbol{ extalpha}$-number based criterion for $L^{p}$ spaces on uniformly rectifiable measures, extending previous geometric characterizations.

Findings

Characterization of $L^{p}(\sigma)$ via $oldsymbol{ extalpha}$-numbers

Equivalence of $L^{p}$ norm and $oldsymbol{ extalpha}$-number integral expression

Applicable for $1<p< exists$

Abstract

We give a characterization of $L^{p}(\sigma)$ for uniformly rectifiable measures $\sigma$ using Tolsa's $\alpha$-numbers, by showing, for $1<p<\infty$ and $f\in L^{p}(\sigma)$, that \[ \lVert f\rVert_{L^{p}(\sigma)}\sim \left\lVert\left(\int_{0}^{\infty} \left(\alpha_{f\sigma}(x,r)+|f|_{x,r}\alpha_{\sigma}(x,r)\right)^2\ \frac{dr}{r} \right)^{\frac{1}{2}}\right\rVert_{L^{p}(\sigma)}. \]