Rademacher's Theorem for Calderon-Zygmund-type Spaces
Thomas Lamby
TL;DR
The paper extends Rademacher's differentiability theorem to weighted Calderón-Zygmund spaces with a function parameter $\phi$, formulating and analyzing the spaces $T^p_\phi$ and $t^p_{\phi}$. Under the index condition $n<\underline{b}(\phi)\le \overline{b}(\phi)<n+1$, it proves that if $f\in T^p_\phi(x)$ for every $x\in E$, then $f\in t^p_{\phi,n+1}(x)$ for almost every $x\in E$, by reducing the problem to the integer-index case via Whitney extension and Sobolev-type estimates and employing Lusin's theorem. The work provides a Rademacher-type theorem within the broader Calderón-Zygmund framework, includes a generalized Whitney-extension approach for $T^p_\phi$ spaces, and discusses sharpness and limitations through illustrative examples, including Brownian motion. These results broaden pointwise regularity analysis to refined function spaces and have potential applications in elliptic PDE estimates and fine-scale regularity theory.
Abstract
Rademacher's Theorem is a classical result stating that a Lipschitz function on $\mathbb{R}^d$ possesses a total differential almost everywhere. This implies that if $f$ is a function defined on $\mathbb{R}^d$ belonging to the Calderón--Zygmund space $T^\infty_1(x)$ for every $x \in \mathbb{R}^d$, then $f \in t^\infty_1(x)$ for almost every $x \in \mathbb{R}^d$. The main purpose of this paper is to extend this property to a broader functional framework. More precisely, we replace the Lipschitz condition by the assumption that $f$ belongs to a general weighted Calderón--Zygmund space $T^p_φ(x)$ for $x \in E$, where $E$ is a measurable subset of $\mathbb{R}^d$ and $φ$ is a weight with fractional indices. We then show that, under suitable assumptions, the function belongs to $t^p_φ(x)$ for almost every $x \in E$. The result can be obtained either by imposing additional hypotheses or by working within a suitably adapted $t^p_φ$ space. Whenever relevant, we also provide counterexamples illustrating the sharpness and the limitations of the statement.