Rademacher's Theorem for Calderon-Zygmund-type Spaces

TL;DR

This paper extends Rademacher's Theorem to Calderón-Zygmund-type spaces with fractional and logarithmic regularity, establishing an almost-everywhere improvement principle in a refined $L^p$ setting.

Contribution

It introduces a new almost-everywhere improvement principle for pointwise Calderón-Zygmund spaces with fractional regularity and logarithmic corrections, generalizing classical results.

Findings

Proves that uniform membership in $T^p_{\phi}(x)$ implies almost-everywhere vanishing approximation rates.

Establishes the sharpness of the result with explicit counterexamples for fractional indices.

Combines measurability, Whitney extension, and Sobolev space properties in the proof.

Abstract

Rademacher's Theorem can be interpreted as an almost-everywhere \emph{little-$o$ improvement principle}: if a function admits a uniform pointwise first-order Lipschitz control at every point, then this control improves to a vanishing one at almost every point. In the language of Calder\'on--Zygmund pointwise spaces, this means that \[ f \in T^\infty_1(x) \quad \forall x \in \mathbb{R}^d \qquad \Longrightarrow \qquad f \in t^\infty_1(x) \quad \text{for a.e. } x \in \mathbb{R}^d. \] The purpose of this paper is to establish an analogous almost-everywhere improvement principle in a refined $L^p$ setting. We consider pointwise Calder\'on-Zygmund spaces $T^p_{\phi}(x)$ defined via polynomial approximation in $L^p$ with a function parameter $\phi$, allowing for fractional regularity indices and logarithmic corrections through Boyd functions. We prove that, under natural assumptions on $\phi$, the uniform membership \[ f \in T^p_{\phi}(x) \quad \forall x \in E \] on a measurable set $E \subset \mathbb{R}^d$ implies an almost-everywhere improvement to a vanishing approximation rate, namely \[ f \in t^p_{\phi,n+1}(x) \quad \text{for a.e. } x \in E, \] where $n < \underline{b}(\phi) \leq \overline{b}(\phi) < n+1$. The proof combines measurability arguments, a generalized Whitney extension theorem, and fine properties of Sobolev spaces. We also show that this result is essentially sharp: in general, one cannot expect almost-everywhere membership in $t^p_{\phi,n}(x)$ for fractional indices, and explicit counterexamples are provided.