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Wavelet resolution and Sobolev regularity of Calderón-Zygmund operators on domains

Francesco Di Plinio, A. Walton Green, Brett D. Wick

Abstract

Given a uniform domain $Ω\subset {\mathbb R}^d$, we resolve each element of a suitably defined class of Calderòn-Zygmund (CZ) singular integrals on $Ω$ as the linear combination of Triebel wavelet operators and paraproduct terms. Our resolution formula entails a testing type characterization, loosely in the vein of the David-Journé theorem, of weighted Sobolev space bounds in terms of Triebel-Lizorkin and tree Carleson measure norms of the paraproduct symbols, which is new already in the case $Ω={\mathbb R}^d$ with Lebesgue measure. Our characterization covers the case of compressions to $Ω$ of global CZ operators, extending and sharpening past results of Prats and Tolsa for the convolution case. The weighted estimates we obtain, particularized to the Beurling operator on a Lipschitz domain with normal to the boundary in the corresponding sharp Besov class, may be used to deduce quantitative estimates for quasiregular mappings with dilatation in the Sobolev space $W^{1,p}(Ω)$, $p>2$.

Wavelet resolution and Sobolev regularity of Calderón-Zygmund operators on domains

Abstract

Given a uniform domain , we resolve each element of a suitably defined class of Calderòn-Zygmund (CZ) singular integrals on as the linear combination of Triebel wavelet operators and paraproduct terms. Our resolution formula entails a testing type characterization, loosely in the vein of the David-Journé theorem, of weighted Sobolev space bounds in terms of Triebel-Lizorkin and tree Carleson measure norms of the paraproduct symbols, which is new already in the case with Lebesgue measure. Our characterization covers the case of compressions to of global CZ operators, extending and sharpening past results of Prats and Tolsa for the convolution case. The weighted estimates we obtain, particularized to the Beurling operator on a Lipschitz domain with normal to the boundary in the corresponding sharp Besov class, may be used to deduce quantitative estimates for quasiregular mappings with dilatation in the Sobolev space , .
Paper Structure (32 sections, 38 theorems, 324 equations)

This paper contains 32 sections, 38 theorems, 324 equations.

Table of Contents

  1. Introduction
  2. Summary of the main results
  3. Techniques and comparison with past literature
  4. Weighted estimates on domains and Beltrami regularity
  5. The basic toolbox of singular integrals on domains
  6. CZ localization on domains
  7. Triebel-Lizorkin norms
  8. Tree Structure
  9. Packing estimates for uniform Whitney coverings
  10. Boundary windows
  11. Averaging Wavelets
  12. Anti-Integration by Parts
  13. Sparse and weighted domination of wavelet forms
  14. Sparse operators and multilinear maximal functions
  15. Sparse domination of wavelet forms
  16. ...and 17 more sections

Key Result

Theorem 3

Let $\Omega\subset \mathbb{C}$ be a bounded Lipschitz domain and $2<p<\infty$. For each $2<r<p$ there is a positive increasing function $\mathcal{G}$ such that, for each weight $w$ Furthermore, for each $q>p$ there is a function $\tilde{\mathcal{G}}$ such that if the boundary normal vector $N_\Omega$ to $\Omega$ lies in the Besov space $B^{1-\frac{1}{q}}_{q,q}(\partial \Omega)$, then

Theorems & Definitions (83)

  • Theorem 3
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5: Projections
  • Remark 2.6: Dense class
  • Definition 2.7: Wavelet forms
  • Remark 2.8
  • Remark 2.10
  • Definition 2.12
  • ...and 73 more