A T(P) theorem for Sobolev spaces on domains
Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given 0 < s 1, 1 p ∞ with sp 2 and a Lipschitz domain Ω ⊂ C, the Beurling transform Bf = -p.v. 1 πz2 ∗ f is bounded in the Sobolev space Ws,p(Ω) if and only if Bχ...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2015 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:287586 |
| Acceso en línea: | https://ddd.uab.cat/record/287586 https://dx.doi.org/urn:doi:10.1016/j.jfa.2015.01.007 |
| Access Level: | acceso abierto |
| Palabra clave: | Harmonic analysis Sobolev spaces Calderón-Zygmund operators Carleson measures |
| Sumario: | Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given 0 < s 1, 1 p ∞ with sp 2 and a Lipschitz domain Ω ⊂ C, the Beurling transform Bf = -p.v. 1 πz2 ∗ f is bounded in the Sobolev space Ws,p(Ω) if and only if BχΩ ∈ Ws,p(Ω). In this paper we obtain a generalized version of the former result valid for any s ∈ N and for a larger family of Calderón-Zygmund operators in any ambient space Rd as long as p d. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally wefind a sufficient condition in terms of Carleson measures for p d. In the particular case s = 1, this condition is in fact necessary, which yields a complete characterization. |
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