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Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness

Jae-Hwan Choi, Jin Bong Lee, Jinsol Seo, Kwan Woo

TL;DR

This work extends the theory of Besov and Triebel–Lizorkin spaces to weighted settings with variable smoothness, establishing that LP-difference and LP-based norms remain equivalent under Muckenhoupt weights by leveraging shifted maximal functions and related operator estimates. The authors develop a robust framework using the classes $\mathcal{I}_o(a,b)$ and $A_p$ weights to obtain precise norm equivalences and provide concrete applications: weighted regularity results for time-fractional evolution equations and a weight-free generalized Sobolev embedding that yields Hölder-type control. The results unify harmonic-analytic representations and PDE-analytic norms in the weighted, variable-smoothness regime, enabling sharp trace theorems and optimal regularity theory. Overall, the paper supplies a comprehensive toolkit for analyzing weighted function spaces with spatially varying smoothness and demonstrates their relevance to evolution PDEs and embedding phenomena.

Abstract

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse Hölder inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights.

Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness

TL;DR

This work extends the theory of Besov and Triebel–Lizorkin spaces to weighted settings with variable smoothness, establishing that LP-difference and LP-based norms remain equivalent under Muckenhoupt weights by leveraging shifted maximal functions and related operator estimates. The authors develop a robust framework using the classes and weights to obtain precise norm equivalences and provide concrete applications: weighted regularity results for time-fractional evolution equations and a weight-free generalized Sobolev embedding that yields Hölder-type control. The results unify harmonic-analytic representations and PDE-analytic norms in the weighted, variable-smoothness regime, enabling sharp trace theorems and optimal regularity theory. Overall, the paper supplies a comprehensive toolkit for analyzing weighted function spaces with spatially varying smoothness and demonstrates their relevance to evolution PDEs and embedding phenomena.

Abstract

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse Hölder inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights.
Paper Structure (17 sections, 21 theorems, 233 equations)

This paper contains 17 sections, 21 theorems, 233 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Self-improving and the reverse Hölder properties of the Muckenthoupt $A_p$ class
  4. Properties of functions of type $\mathcal{I}(a, b)$
  5. Shifted maximal functions
  6. Peetre's maximal functions
  7. Vector-valued inequalities
  8. Equivalent norms for $B_{p,q}^{\phi}(w)$ and $F_{p,q}^{\phi}(w)$
  9. Proof of Theorem \ref{['thm_lp_diff_equiv']}
  10. Proof of Lemma \ref{['23.02.08.15.58']}
  11. Proof of Lemma \ref{['23.02.09.14.46']}
  12. Case of $(X,\mathcal{X})=(B,\mathcal{B})$
  13. Case of $(X,\mathcal{X})=(F,\mathcal{F})$
  14. Proof of Proposition \ref{['prop_wBesov_equiv']}
  15. Applications
  16. ...and 2 more sections

Key Result

Theorem 1.5

Let $w \in A_p(\mathbb{R}^d)$ with $p \in (1, \infty]$, and we assume that Then Here, the constants $R_w$ and $\Gamma_w$ are defined by

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 1.8
  • Proposition 2.1
  • proof
  • ...and 32 more