Approximation in Hölder Spaces

TL;DR

This work develops a comprehensive framework for approximating Hölder-type functions in $\dot{C}^{0,\omega}(X,Y)$ by smoother, compactly supported, or Lipschitz functions. It introduces vanishing subspaces—small, large, and far—that precisely capture when such approximations are possible, and proves full characterizations across diverse spaces including separable, super-reflexive, and $X=c_0$. The results connect pointwise vanishing criteria to mean-oscillation notions like $\mathrm{BMO}^{\omega}$ and $\mathrm{VMO}^{\omega}$, with Bochner-integrable mollification techniques underpinning the constructions. Applications to bi-parameter harmonic analysis are highlighted, notably in the compactness theory for bi-commutators of Calderón-Zygmund operators, via product $\mathrm{BMO}$/VMO spaces. Collectively, the paper provides both constructive approximation schemes and deep structural insights into Hölder-type function spaces in finite and infinite dimensions, broadening the toolbox for PDE regularity and harmonic analysis problems.

Abstract

We introduce new vanishing subspaces of the homogeneous Hölder space $\dot{C}^{0,ω}(X,Y)$ in the generality of a doubling modulus $ω$ and normed spaces $X$ and $Y.$ For many couples $X,Y,$ we show these vanishing subspaces to completely characterize those Hölder functions that admit approximations, in the Hölder seminorm, by smooth, Lipschitz and boundedly supported functions. We present connections to bi-parameter harmonic analysis on the Euclidean space by providing applications to the compactness of the bi-commutator of two Calderón-Zygmund operators

Theorems & Definitions (48)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 1.7
• Definition 1.8
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• Theorem 1.10