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Geometric topics related to Besov type spaces on the Grushin setting

Nan Zhao, Zhiyong Wang, Pengtao Li, Yu Liu

Abstract

The Grushin spaces, as one of the most important models in the Carnot-Carathéodory space, are a class of locally compact and geodesic metric spaces which admit a dilation. Function spaces on Grushin spaces and some related geometric problems are always the research hotspots in this field. Firstly, we investigate two classes of Besov type spaces based on the Grushin semigroup and the fractional Grushin semigroup, respectively, and prove some important properties of these two Besov type spaces. Moreover, we also reveal the relationship between them. Secondly, we establish the isoperimetric inequality for the fractional perimeter, which is defined by the Grushin-Laplace operator on Grushin spaces. Finally, we combine the semigroup theory with a nonlocal calculus for the Grushin-Laplace operator to obtain the Sobolev type inequality. As a corollary, we also obtain the embedding theorem for Besov type spaces.

Geometric topics related to Besov type spaces on the Grushin setting

Abstract

The Grushin spaces, as one of the most important models in the Carnot-Carathéodory space, are a class of locally compact and geodesic metric spaces which admit a dilation. Function spaces on Grushin spaces and some related geometric problems are always the research hotspots in this field. Firstly, we investigate two classes of Besov type spaces based on the Grushin semigroup and the fractional Grushin semigroup, respectively, and prove some important properties of these two Besov type spaces. Moreover, we also reveal the relationship between them. Secondly, we establish the isoperimetric inequality for the fractional perimeter, which is defined by the Grushin-Laplace operator on Grushin spaces. Finally, we combine the semigroup theory with a nonlocal calculus for the Grushin-Laplace operator to obtain the Sobolev type inequality. As a corollary, we also obtain the embedding theorem for Besov type spaces.
Paper Structure (14 sections, 35 theorems, 336 equations)

This paper contains 14 sections, 35 theorems, 336 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation
  4. Preliminaries
  5. Grushin spaces and their metrics
  6. The Grushin semigroup $\{e^{-t\mathcal{L}}\}_{t>0}$
  7. Fractional powers of the Grushin-Laplace operator
  8. Besov type spaces on Grushin settings
  9. Besov type spaces associated with $\{e^{-t\mathcal{L}}\}_{t>0}$
  10. Besov type spaces associated with $\{e^{-t\mathcal{L}^{s}}\}_{t>0}$
  11. Some limiting behaviour of Besov semi-norms
  12. Fractional perimeters on Grushin spaces
  13. Proofs of Theorems \ref{['main1']} and \ref{['main2']}
  14. Proofs of Theorem \ref{['Sobolev']} and Corollary \ref{['propo2.6']}

Key Result

Theorem 1.1

Let $\mathcal{L}$ be as in (K) and $s\in(0,1/2).$ For any $E\subset \mathbb{G}_{\alpha}^{n}$ with finite $s$-perimeter, there exists a positive constant $C(Q,s),$ depending on $Q$ and $s,$ such that where $Q$ is the homogeneous dimension of $\mathbb{G}_{\alpha}^{n}$ and $\mathscr{P}^{\mathcal{L}}_{s}$ is $P^{\mathcal{L}}_{s}, P^{\mathcal{L},*}_{s}$ or $P^{\mathcal{L}}_{s,\infty}.$

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • ...and 73 more