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A metric counterpart of the Gu-Yung formula

Stefano Buccheri, Wojciech Górny

Abstract

In this note we consider a generalisation to the metric setting of the recent work [Gu-Yung, JFA 281 (2021), 109075]. In particular, we show that under relatively weak conditions on a metric measure space $(X,d,ν)$, it holds true that \[ \bigg[ \frac{u(x)-u(y)}{d(x,y)^{\frac{s}{p}}} \bigg]_{L^p_w(X \times X, ν\otimes ν)} \approx \| u \|_{L^p(X,ν)}, \] where $s$ is a generalised dimension associated to $X$ and $[\cdot]_{L^p_w}$ is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.

A metric counterpart of the Gu-Yung formula

Abstract

In this note we consider a generalisation to the metric setting of the recent work [Gu-Yung, JFA 281 (2021), 109075]. In particular, we show that under relatively weak conditions on a metric measure space , it holds true that \[ \bigg[ \frac{u(x)-u(y)}{d(x,y)^{\frac{s}{p}}} \bigg]_{L^p_w(X \times X, ν\otimes ν)} \approx \| u \|_{L^p(X,ν)}, \] where is a generalised dimension associated to and is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.
Paper Structure (10 sections, 4 theorems, 148 equations)

This paper contains 10 sections, 4 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Particular cases
  5. Euclidean space
  6. Anisotropic norms on $\mathbb{R}^N$
  7. Weighted Euclidean spaces
  8. Carnot groups
  9. Riemannian manifolds
  10. Spaces with synthetic Ricci curvature bounds

Key Result

Theorem 3.1

Suppose that $\nu(X) = + \infty$. Assume that $\nu$ is upper Ahlfors regular, namely there exists $C_A > 0$ such that Furthermore, we require that its asymptotic value ratio ${\rm AVR}$ (see Definition dfn:avr) corresponding to the exponent $s$ is finite, and that the measure of balls with a fixed center is continuous as a function of radius, i.e., Then, there exist constants $c_1, c_2, c_3 > 0$

Theorems & Definitions (12)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Remark 3.3
  • proof : Proof of Theorem \ref{['thm:maintheorem2']}
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • Example 3.6
  • ...and 2 more