Square Functions Controlling Smoothness and Higher-Order Rectifiability

TL;DR

This paper introduces new characterizations of the $BMO$-Sobolev space $I_{eta}(BMO)$ for $0<eta<2$, using square functions that measure multiscale approximation by constants or linear functions.

Contribution

It provides novel characterizations of $I_{eta}(BMO)$ spaces based on square functions, extending understanding of smoothness and rectifiability in harmonic analysis.

Findings

Characterizations for $0<eta<1$ using square functions measuring approximation by constants.

Characterizations for $1 eqeta<2$ using square functions measuring approximation by linear functions.

Enhanced understanding of the relationship between square functions and higher-order smoothness.

Abstract

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.