AN α-NUMBER CHARACTERIZATION OF Lp 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 and $f\in L^p(\sigma )$, that

$$\Vert f{\Vert }_{L^p(\sigma )}~{\Vert ({\displaystyle {\int }_0^{\infty }({\alpha }_{f\sigma }(}x,r)+|f{|}_{x,r}{\alpha }_{\sigma }(x,r){)}^2\frac{dr}{r}{)}^{\frac{1}{2}}\Vert }_{L^p(\sigma )}.$$