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

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 \[ \lVert f\rVert_{L^{p}(σ)}\sim \left\lVert\left(\int_{0}^{\infty} \left(α_{fσ}(x,r)+|f|_{x,r}α_σ(x,r)\right)^2\ \frac{dr}{r} \right)^{\frac{1}{2}}\right\rVert_{L^{p}(σ)}. \]

23 pages