Rademacher's Theorem for Calderon-Zygmund-type Spaces

Rademacher’s theorem can be interpreted as an almost-everywhere little-o improvement principle: when a function satisfies a uniform first-order Lipschitz-type control at every point, this control actually improves to a vanishing one at almost every point. In the framework of Calderón–Zygmund pointwise regularity spaces, this expresses the idea that a uniform first-order approximation property everywhere automatically strengthens to a finer, asymptotically vanishing approximation property almost everywhere.

The aim of this paper is to establish a similar almost-everywhere improvement principle in a refined L-p integrability setting. We study pointwise Calderón–Zygmund spaces defined through polynomial approximation measured in an L-p sense and governed by a functional parameter. This framework allows one to treat fractional regularity orders as well as logarithmic corrections described by Boyd-type functions.

Our main result shows that, under natural assumptions on this functional parameter, if a function satisfies such an approximation property uniformly on a measurable set, then the approximation rate improves almost everywhere on that set to a stronger, vanishing form.

The proof combines measurability considerations, a generalized Whitney-type extension theorem, and fine structural properties of Sobolev spaces. We also demonstrate that the result is essentially sharp: in general, one cannot expect a stronger almost-everywhere improvement for fractional regularity orders, and explicit counterexamples illustrate this limitation.