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 Calderón--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 Calderón-Zygmund spaces $T_{\varphi }^p(x)$ defined via polynomial approximation in $L^p$ with a function parameter $\varphi$, allowing for fractional regularity indices and logarithmic corrections through Boyd functions. We prove that, under natural assumptions on $\varphi$, the uniform membership \[ f \in T^p_ϕ(x) \quad \forall x \in E \] on a measurable set $E\subset {\mathbb{R}}^d$ implies an almost-everywhere improvement to a vanishing approximation rate, namely \[ f \in t^p_{ϕ,n+1}(x) \quad \text{for a.e. } x \in E, \] where $n<\underline{b}(\varphi )\le \overline{b}(\varphi )<n+1$. 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 $t_{\varphi ,n}^p(x)$ for fractional indices, and explicit counterexamples are provided.

arXiv commentaccepted in Analysis Mathematica