Rademacher's Theorem for Calderon-Zygmund-type Spaces

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.