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We quantify the Sobolev space norm of the Beltrami resolvent \((I- \mu S)^{-1}\), where \(S\) is the Beurling–Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation \(\mu\) in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in \(W^{1,p}\), \(p \ge 2\). Our proof strategy is then adapted to yield quantitative estimates for the resolvent \((I-\mu S_\Omega)^{-1}\) of the Beltrami equation on a sufficiently regular domain \(\Omega\), with \(\mu\in W^{1,p}(\Omega)\). Here, \(S_\Omega\) is the compression of \(S\) to a domain \(\Omega\). Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón–Zygmund operators to domains, recently obtained by the authors, to extend the Astala–Iwaniec–Saksman technique to higher regularities.
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This content will become publicly available on March 1, 2026
Multilinear paraproducts on Sobolev spaces
Paraproducts are a special subclass of the multilinear Calderón-Zygmund operators, and their Lebesgue space estimates in the full multilinear range are characterized by the norm of the symbol. In this note, we characterize the Sobolev space boundedness properties of multilinear paraproducts in terms of a suitable family of Triebel-Lizorkin type norms of the symbol. Coupled with a suitable wavelet representation theorem, this characterization leads to a new family of Sobolev space T(1)-type theorems for multilinear Calderón-Zygmund operators.
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- PAR ID:
- 10582453
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Bollettino dell'Unione Matematica Italiana
- Volume:
- 18
- Issue:
- 1
- ISSN:
- 1972-6724
- Page Range / eLocation ID:
- 167 to 183
- Subject(s) / Keyword(s):
- Wavelet representation theorem Paraproducts Triebel-Lizorkin norms Sparse domination Multilinear Calderón-Zygmund theory Sobolev spaces
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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