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Oscillations and differences in Besov-Morrey and Besov-type spaces

Marc Hovemann, Markus Weimar

TL;DR

This work develops intrinsic oscillation and higher-order difference characterizations for Besov-Morrey spaces $\mathcal{N}^{s}_{u,p,q}(\Omega)$ and Besov-type spaces $B^{s,\tau}_{p,q}(\Omega)$ on Lipschitz domains and $\mathbb{R}^d$. By integrating the Hedberg–Netrusov framework with Triebel’s kernel representations, the authors derive new equivalent quasi-norms based on local oscillations and higher-order differences, including domain-specific variants and special Lipschitz/convex domains. The results unify and extend classical Besov characterizations, provide tools for PDE regularity analysis, and connect function-space theory with concrete approximation measures. The methods—local oscillations, higher-order differences, Littlewood–Paley decompositions, and domain localization—offer a robust framework for analyzing regularity in Morrey-scale Besov spaces with sharp parameter ranges and domain restrictions.

Abstract

In this paper we investigate Besov-Morrey spaces $\mathcal{N}^{s}_{u,p,q}(Ω)$ and Besov-type spaces $B^{s,τ}_{p,q}(Ω)$ of positive smoothness defined on Lipschitz domains $Ω\subset \mathbb{R}^d$ as well as on $\mathbb{R}^d$. We combine the Hedberg-Netrusov approach to function spaces with distinguished kernel representations due to Triebel, in order to derive novel characterizations of these scales in terms of local oscillations provided that some standard conditions concerning the parameters are fulfilled. In connection with that we also obtain new characterizations of $\mathcal{N}^{s}_{u,p,q}(Ω)$ and $B^{s,τ}_{p,q}(Ω)$ via differences of higher order. By the way we recover and extend corresponding results for the scale of classical Besov spaces $B^{s}_{p,q}(Ω)$. Key words: Besov-Morrey space, Besov-type space, Morrey space, Lipschitz domain, oscillations, higher order differences

Oscillations and differences in Besov-Morrey and Besov-type spaces

TL;DR

This work develops intrinsic oscillation and higher-order difference characterizations for Besov-Morrey spaces and Besov-type spaces on Lipschitz domains and . By integrating the Hedberg–Netrusov framework with Triebel’s kernel representations, the authors derive new equivalent quasi-norms based on local oscillations and higher-order differences, including domain-specific variants and special Lipschitz/convex domains. The results unify and extend classical Besov characterizations, provide tools for PDE regularity analysis, and connect function-space theory with concrete approximation measures. The methods—local oscillations, higher-order differences, Littlewood–Paley decompositions, and domain localization—offer a robust framework for analyzing regularity in Morrey-scale Besov spaces with sharp parameter ranges and domain restrictions.

Abstract

In this paper we investigate Besov-Morrey spaces and Besov-type spaces of positive smoothness defined on Lipschitz domains as well as on . We combine the Hedberg-Netrusov approach to function spaces with distinguished kernel representations due to Triebel, in order to derive novel characterizations of these scales in terms of local oscillations provided that some standard conditions concerning the parameters are fulfilled. In connection with that we also obtain new characterizations of and via differences of higher order. By the way we recover and extend corresponding results for the scale of classical Besov spaces . Key words: Besov-Morrey space, Besov-type space, Morrey space, Lipschitz domain, oscillations, higher order differences
Paper Structure (19 sections, 28 theorems, 195 equations)

This paper contains 19 sections, 28 theorems, 195 equations.

Table of Contents

  1. Introduction and Main Results
  2. Function Spaces
  3. Morrey Spaces Mup(Rd) and Further Basic Definitions
  4. Besov-Morrey Spaces Nsupq(Rd)
  5. Besov-Type Spaces Bstaupq(Rd)
  6. Function Spaces on Domains
  7. Quasi-Norms of Interest
  8. Characterizations on Rd
  9. Characterizations via Local Oscillations for Nsupq(Rd)
  10. Difference Characterizations of Nsupq(Rd)
  11. Characterizations via Local Oscillations for Bstaupq(Rd)
  12. Difference Characterizations of Bstaupq(Rd)
  13. Characterizations on Domains
  14. General Lower Bounds
  15. Differences on Special Lipschitz Domains
  16. ...and 4 more sections

Key Result

Theorem 1.1

For $d\in\mathbb{N}$ let $\Omega\subseteq{\mathbb{R}}^d$ be either $\mathbb{R}^d$ or a bounded $C^{\infty}$-domain. Let $0 < p \leq \infty$, $0 < q \leq \infty$, $1 \leq v \leq \infty$, $N \in \mathbb{N}$, and $s \in \mathbb{R}$ with Then $f \in L_{\max\{p,v\}}(\Omega)$ belongs to $B^{s}_{p,q}(\Omega)$ if and only if the equivalent quasi-norm is finite, where for $q = \infty$ the usual modificat

Theorems & Definitions (57)

  • Theorem 1.1: Triebel Tr92
  • Theorem 1.2: Oscillation Characterizations for Besov-Morrey Spaces
  • Theorem 1.3: Oscillation Characterizations for Besov-Type Spaces
  • Theorem 1.4: Difference Characterizations for Besov-Morrey Spaces
  • Theorem 1.5: Difference Characterizations for Besov-Type Spaces
  • Definition 1: Morrey Spaces
  • Definition 2: Besov-Morrey Spaces
  • Lemma 1: HaMoSk
  • Lemma 2
  • proof
  • ...and 47 more