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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.09159 |
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| _version_ | 1866912889454985216 |
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| author | Lamby, Thomas |
| author_facet | Lamby, Thomas |
| contents | 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 $T^p_ϕ(x)$ defined via polynomial approximation in $L^p$ with a function parameter $ϕ$, allowing for fractional regularity indices and logarithmic corrections through Boyd functions. We prove that, under natural assumptions on $ϕ$, 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 $n < \underline{b}(ϕ) \leq \overline{b}(ϕ) < n+1$.
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 $t^p_{ϕ,n}(x)$ for fractional indices, and explicit counterexamples are provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rademacher's Theorem for Calderon-Zygmund-type Spaces Lamby, Thomas Functional Analysis Analysis of PDEs Classical Analysis and ODEs 46E35, 26A16 (Primary), 42B20, 35J60 (Secondary) 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 $T^p_ϕ(x)$ defined via polynomial approximation in $L^p$ with a function parameter $ϕ$, allowing for fractional regularity indices and logarithmic corrections through Boyd functions. We prove that, under natural assumptions on $ϕ$, 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 $n < \underline{b}(ϕ) \leq \overline{b}(ϕ) < n+1$. 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 $t^p_{ϕ,n}(x)$ for fractional indices, and explicit counterexamples are provided. |
| title | Rademacher's Theorem for Calderon-Zygmund-type Spaces |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs 46E35, 26A16 (Primary), 42B20, 35J60 (Secondary) |
| url | https://arxiv.org/abs/2511.09159 |