Square Functions Controlling Smoothness and Higher-Order Rectifiability

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