Lipschitz space with mixed logarithmic smoothness and embedding theorems

TL;DR

This work develops a theory of Lipschitz spaces with mixed logarithmic smoothness for $2\pi$-periodic functions on $\mathbb{T}^{m}$ under Lorentz norms. It introduces $\text{Lip}_{p,\tau,\theta}^{(\boldsymbol{\alpha}, -\mathbf{b})}(\mathbb{T}^{m})$ via a mixed modulus of smoothness with logarithmic weighting and proves a dyadic norm-equivalence $\|f\|_{\text{Lip}} \asymp \|f\|_{p,\tau} + (J_{p,\tau,\theta}^{\boldsymbol{\alpha}, -\mathbf{b}}(f))^{1/\theta}$, where $J_{p,\tau,\theta}^{\boldsymbol{\alpha}, -\mathbf{b}}(f)$ aggregates Fourier-block contributions. The paper establishes embeddings between these Lip-spaces and generalized Nikol\'skii–Besov spaces $S_{p,\tau,\theta}^{\mathbf{r},\mathbf{b}}B(\mathbb{T}^{m})$, including sharp (optimal) results across ranges of $p,\tau,\theta$, and logarithmic exponents. It provides a comprehensive set of auxiliary tools (modulus properties, Littlewood–Paley-type bounds, and dyadic decompositions) to justify the embeddings and their optimality. Overall, the results extend Besov–Lip theory to the setting of mixed smoothness with logarithmic refinements, yielding precise norm descriptions and sharp embeddings for multivariate periodic functions.

Abstract

This article considers the Lipschitz space with mixed logarithmic smoothness of $2π$ periodic functions of several variables. We obtain equivalent descriptions of the norm of the Lipschitz space and prove embedding theorems between Besov and Lipschitz spaces.

Theorems & Definitions (29)

• Remark 1
• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Remark 1.2
• Theorem 1.4
• Remark 1.3
• Theorem 1.5
• Remark 1.4