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 $1<p<\mathrm{\infty }$ and $f\in L^p(\sigma )$, that $$||f|{|}_{L^p(\sigma )}\sim ||\left({\int }_0^{\mathrm{\infty }}{({\alpha }_{f\sigma }(x,r)+|f{|}_{x,r}{\alpha }_{\sigma }(x,r))}^2\frac{dr}{r}\right)|{|}_{L^p(\sigma )}^{\frac{1}{2}}.$$