Rademacher's Theorem for Calderon-Zygmund-type Spaces

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 $$f\in T_1^{\mathrm{\infty }}(x)\mathrm{\forall }x\in {\mathbb{R}}^d⟹f\in t_1^{\mathrm{\infty }}(x)\text{for a.e. }x\in {\mathbb{R}}^d\mathrm{.}$$ 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 $T_{\phi }^p(x)$ 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 $$f\in T_{\phi }^p(x)\mathrm{\forall }x\in E$$ on a measurable set $E\subset {\mathbb{R}}^d$ implies an almost-everywhere improvement to a vanishing approximation rate, namely $$f\in t_{\phi ,n+1}^p(x)\text{for a.e. }x\in E,$$ where $n<\underset{‾}{b}(\phi )\le \bar{b}(\phi )<n+1$. 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 $t_{\phi ,n}^p(x)$ for fractional indices, and explicit counterexamples are provided.