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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.13909 |
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| _version_ | 1866911777185333248 |
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| author | Di Plinio, Francesco Green, A. Walton Wick, Brett D. |
| author_facet | Di Plinio, Francesco Green, A. Walton Wick, Brett D. |
| contents | 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 $W^{1,p}(Ω)$, $p>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_13909 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wavelet resolution and Sobolev regularity of Calderón-Zygmund operators on domains Di Plinio, Francesco Green, A. Walton Wick, Brett D. Classical Analysis and ODEs Analysis of PDEs Complex Variables 42B20 (Primary) 42B25, 30C62 (Secondary) 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 $W^{1,p}(Ω)$, $p>2$. |
| title | Wavelet resolution and Sobolev regularity of Calderón-Zygmund operators on domains |
| topic | Classical Analysis and ODEs Analysis of PDEs Complex Variables 42B20 (Primary) 42B25, 30C62 (Secondary) |
| url | https://arxiv.org/abs/2304.13909 |