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 $1 \leq \alpha <2$ our characterizations are in terms of square functions measuring multiscale approximation by linear functions.