Square Functions Controlling Smoothness and Higher-Order Rectifiability

Abstract

We provide new characterizations of the $BMO$-Sobolev space $I_{\hat{I}}\pm (BMO)$ for the range $0<\hat{I}\pm <2$. When $0<\hat{I}\pm <1$, our characterizations are in terms of square functions measuring multiscale approximation of constants, and when $1\le \hat{I}\pm <2$ our characterizations are in terms of square functions measuring multiscale approximation by linear functions.

25 pages. Fixed various typos and made other minor improvements