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

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