Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characterizations of Sobolev functions via Besov-type energy functionals in fractals

Ryosuke Shimizu

TL;DR

This work develops BBM-type characterizations of Sobolev spaces on metric measure spaces, including fractals, by linking Korevaar–Schoen p-energy to Besov-type energies via a general scale function $\Psi$ under doubling, Poincaré, and capacity upper bounds. It introduces a unified $p$-energy framework with energy measures $\Gamma_p\langle\cdot\rangle$, proving energy equivalences through kernel-based limit functionals, and identifies the $\mathcal{F}_p$ space with Besov spaces $B^{\beta/p}_{p,\infty}$ under suitable conditions. The paper establishes BBM-type and Korevaar–Schoen-type characterizations, derives Besov-critical exponents, and provides extension theory on uniform domains via Whitney covers and a scale-invariant extension operator, with applications to fractals such as the Vicsek set, the Sierpiński gasket, and the Sierpiński carpet. Overall, it connects Sobolev, Korevaar–Schoen, and Besov notions in fractal metric spaces under minimal geometric hypotheses, advancing analysis on irregular spaces. The results yield localized, robust characterizations and extension tools essential for non-smooth geometric analysis on fractals and related spaces.

Abstract

In the spirit of the ground-breaking result of Bourgain--Brezis--Mironescu, we establish some characterizations of Sobolev functions in metric measure spaces including fractals like the Vicsek set, the Sierpiński gasket and the Sierpiński carpet. As corollaries of our characterizations, we present equivalent norms on the Korevaar--Schoen--Sobolev space, and show that the domain of a $p$-energy form is identified with a Besov-type function space under a suitable $(p,p)$-Poincaré inequality, capacity upper bound and the volume doubling property.

Characterizations of Sobolev functions via Besov-type energy functionals in fractals

TL;DR

This work develops BBM-type characterizations of Sobolev spaces on metric measure spaces, including fractals, by linking Korevaar–Schoen p-energy to Besov-type energies via a general scale function under doubling, Poincaré, and capacity upper bounds. It introduces a unified -energy framework with energy measures , proving energy equivalences through kernel-based limit functionals, and identifies the space with Besov spaces under suitable conditions. The paper establishes BBM-type and Korevaar–Schoen-type characterizations, derives Besov-critical exponents, and provides extension theory on uniform domains via Whitney covers and a scale-invariant extension operator, with applications to fractals such as the Vicsek set, the Sierpiński gasket, and the Sierpiński carpet. Overall, it connects Sobolev, Korevaar–Schoen, and Besov notions in fractal metric spaces under minimal geometric hypotheses, advancing analysis on irregular spaces. The results yield localized, robust characterizations and extension tools essential for non-smooth geometric analysis on fractals and related spaces.

Abstract

In the spirit of the ground-breaking result of Bourgain--Brezis--Mironescu, we establish some characterizations of Sobolev functions in metric measure spaces including fractals like the Vicsek set, the Sierpiński gasket and the Sierpiński carpet. As corollaries of our characterizations, we present equivalent norms on the Korevaar--Schoen--Sobolev space, and show that the domain of a -energy form is identified with a Besov-type function space under a suitable -Poincaré inequality, capacity upper bound and the volume doubling property.

Paper Structure

This paper contains 14 sections, 39 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Framework
  3. Main results
  4. Preliminary results
  5. BBM type characterizations
  6. Upper bound of Theorem \ref{['thm:main1']}
  7. Lower bounds of Theorem \ref{['thm:main1']}
  8. Proof of Theorem \ref{['thm:main1']} and its corollaries
  9. $L^{p}$-Besov critical exponents
  10. The extension property \ref{['defn:sextdomain']} for uniform domains
  11. Whitney covering, uniform domain and their properties
  12. (p,p)-Poincaré inequality on uniform domains
  13. Extension map and its scale-invariant boundedness
  14. Connectedness of the underlying metric space

Key Result

Theorem 1.12

Assume that e:VD, e:PI and e:capu hold. Let $\{ \rho_{\varepsilon} \}_{\varepsilon > 0}$ satisfy Assumption assum:kernel.1 or assum:kernel.2. Then there exists $C \in [1,\infty)$ such that for any open subset $\Omega$ of $X$ satisfying defn:sextdomain and any $u \in \mathcal{F}_{p}(\Omega)$, Moreover, if $\{ \rho_{\varepsilon} \}_{\varepsilon > 0}$ satisfies Assumption assum:kernel.2, then for an

Theorems & Definitions (86)

  • Definition 1.1
  • Remark 1.3
  • Definition 1.4: Doubling properties
  • Definition 1.5: $(p,p)$-Poincaré inequality and $p$-capacity upper estimate
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Remark 1.11
  • Theorem 1.12
  • Corollary 1.13: BBM type characterization
  • ...and 76 more