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Square Functions Controlling Smoothness and Higher-Order Rectifiability

John Hoffman

TL;DR

The paper develops square-function characterizations of $I_\alpha(BMO)$ for $0<\alpha<2$, distinguishing between multiscale approximation by constants ($0<\alpha<1$) and by affine functions ($1\le\alpha<2$). It proves both directions: (i) $f\in I_\alpha(BMO)$ implies precise $L^2$-type square-function bounds tied to $\nu_0^f$ or $\nu_1^f$, and (ii) those square-function bounds imply $f\in I_\alpha(BMO)$, using Strichartz’s criteria, approximate Taylor tools, and a Dorronsoro-type framework. The work connects $BMO$-Sobolev spaces with geometric measure theory notions of higher-order rectifiability, and it provides rigorous foundations for the behavior and representation of distributions in $I_\alpha(BMO)$, including duality and regularity properties. The results yield a quantitative, end-point analogue of Dorronsoro's theorem and establish a concrete link between analytic square-function control and geometric approximation, with potential implications for parabolic uniform rectifiability and related areas.

Abstract

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

Square Functions Controlling Smoothness and Higher-Order Rectifiability

TL;DR

The paper develops square-function characterizations of for , distinguishing between multiscale approximation by constants () and by affine functions (). It proves both directions: (i) implies precise -type square-function bounds tied to or , and (ii) those square-function bounds imply , using Strichartz’s criteria, approximate Taylor tools, and a Dorronsoro-type framework. The work connects -Sobolev spaces with geometric measure theory notions of higher-order rectifiability, and it provides rigorous foundations for the behavior and representation of distributions in , including duality and regularity properties. The results yield a quantitative, end-point analogue of Dorronsoro's theorem and establish a concrete link between analytic square-function control and geometric approximation, with potential implications for parabolic uniform rectifiability and related areas.

Abstract

We provide new characterizations of the -Sobolev space for the range . When , our characterizations are in terms of square functions measuring multiscale approximation of constants, and when our characterizations are in terms of square functions measuring multiscale approximation by linear functions.

Paper Structure

This paper contains 5 sections, 18 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Notation, Definitions
  3. $I_\alpha(BMO)$ Implies Square Function Bounds
  4. From Square Function Bounds to $I_{\alpha}(BMO)$.
  5. Appendix

Key Result

Theorem 1.3

Suppose $\mu$ is a measure on $\mathbb R^d$ such that the upper density $\theta^{k,*}(\mu)(X)$ of a measure $\mu$ satisfies $0 < \theta^{k,*}(\mu)(X) < \infty$ for $\mu$-a.e. $X \in \mathbb R^d$ for some $k \in \{1,...,d-1\}$, and one has the following qualitative version of the aforementioned Carle then $\mu$ is a $k$-rectifiable measure.

Theorems & Definitions (34)

  • Theorem 1.3: AT
  • Theorem 1.4: T
  • Theorem 1.5: ENV
  • Theorem 1.6: Ghi
  • Theorem 1.8: Ghi
  • Theorem 1.10: DI
  • Theorem 1.12
  • Remark 1.16
  • Definition 2.1
  • Definition 2.2
  • ...and 24 more