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Quantitative harmonic approximations and Dorronsoro's Theorem in metric measure spaces

Matthew Hyde

Abstract

Suppose $X$ is an $\rm{RCD}(K,N)$ space with $K \in \mathbb{R}$ and $N \in (1,\infty)$. We obtain a characterisation of the Newtonian-Sobolev space $N^{1,2}(X)$ in terms of a quantity which measures to what extent a function is locally (across all scales and locations) well-approximated by harmonic functions. A similar characterisation is obtained which further takes into account the local oscillations of the approximating harmonic functions. The first characterisation is new even when $X = \mathbb{R}^n$; the second characterisation is a version of Dorronsoro's Theorem in RCD spaces and gives a new proof of (a special case) of this theorem in Euclidean space.

Quantitative harmonic approximations and Dorronsoro's Theorem in metric measure spaces

Abstract

Suppose is an space with and . We obtain a characterisation of the Newtonian-Sobolev space in terms of a quantity which measures to what extent a function is locally (across all scales and locations) well-approximated by harmonic functions. A similar characterisation is obtained which further takes into account the local oscillations of the approximating harmonic functions. The first characterisation is new even when ; the second characterisation is a version of Dorronsoro's Theorem in RCD spaces and gives a new proof of (a special case) of this theorem in Euclidean space.
Paper Structure (7 sections, 44 theorems, 173 equations)

This paper contains 7 sections, 44 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of the $H$-coefficients
  4. Estimating the $H$-coefficients
  5. Estimating the oscillatory $H$-coefficients in $\mathrm{RCD}(K,N)$ spaces
  6. Estimating the gradient
  7. Proof of Lemma \ref{['l:continuous-to-discrete']}

Key Result

Theorem 1.1

Let $f \in L^2(\mathbb{R}^n)$. Then $f \in W^{1,2}(\mathbb{R}^n)$ if and only if Here, where the infimum is taken over all affine functions $A \colon \mathbb{R}^n \to \mathbb{R}$. In this case, it follows that $\| f \|_{W^{1,2}(X)}^2 \sim \| f \|_{L^2(X)}^2 + \Omega(f)$.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Corollary 2.5
  • Theorem 2.6
  • ...and 68 more