On Regularity and Irregularity of Certain Holomorphic Singular Integral Operators

Lanzani, Loredana; Stein, Elias M.

doi:10.1007/978-3-030-72058-2_14

Cited by 3 publications

(4 citation statements)

References 56 publications

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“…A similar phenomenon, where the geometry of the domain forces an irregular behavior of the Bergman projection, were already observed in the setting of the worm and model worm domains [3, 5, 6, 30–32]. Analogous results, but for the Szegő projection, were discussed and proved in the setting of model worm domains in [35, 37–41] and in other setting in [7, 34, 42].…”

Section: Introductionsupporting

confidence: 67%

Holomorphic function spaces on the Hartogs triangle

Monguzzi

2021

Mathematische Nachrichten

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The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the Lp mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.

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Section: Introductionsupporting

confidence: 67%

Holomorphic function spaces on the Hartogs triangle

Monguzzi

2021

Mathematische Nachrichten

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“…A similar phenomenon, where the geometry of the domain forces an irregular behavior of the Bergman projection, were already observed in the setting of the worm and model worm domains [Bar92, KP07, KP08, BS ¸12, BEP15,KPS16]. Analogous results, but for the Szegő projection, were discussed and proved in the setting of model worm domains in [Mon16b,Mon16c,Mon16a,MP17a,MP17b,LS19].…”

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confidence: 63%

Holomorphic function spaces on the Hartogs triangle

Monguzzi¹

2019

Preprint

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The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle H is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the L p mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.

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“…The Szegő projection is the orthogonal projection S Ω : L 2 (∂Ω, dσ) → H 2 (∂Ω, dσ); see [30] for the case of bounded domains. The regularity of S Ω when Ω is a (model) worm domain was studied in a series of papers [20,[22][23][24]26,27]. In particular, in [20], it was announced that S Wμ does not preserve L p (∂W μ ) when 1 2 − 1 p ≥ π μ , in analogy to the case of the Bergman projection.…”

Section: Final Remarks and Open Questionsmentioning

confidence: 99%

“…The regularity of S Ω when Ω is a (model) worm domain was studied in a series of papers [20,[22][23][24]26,27]. In particular, in [20], it was announced that S Wμ does not preserve L p (∂W μ ) when 1 2 − 1 p ≥ π μ , in analogy to the case of the Bergman projection. L. Lanzani and E. Stein also studied the L p -regularity of the Szegő and other projections on the boundary on bounded domains under minimal smoothness conditions [18,19], whereas a definition of Hardy spaces and associated Szegő projection for singular domains was studied, for instance, in [9,25].…”

Section: Final Remarks and Open Questionsmentioning

confidence: 99%

Irregularity of the Bergman Projection on Smooth Unbounded Worm Domains

et al. 2023

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In this work, we consider smooth unbounded worm domains $${\mathcal {Z}}_\lambda $$ Z λ in $${\mathbb {C}}^2$$ C 2 and show that the Bergman projection, densely defined on the Sobolev spaces $$H^{s,p}({\mathcal {Z}}_\lambda ),$$ H s , p ( Z λ ) , $$p\in (1,\infty ),$$ p ∈ ( 1 , ∞ ) , $$s\ge 0,$$ s ≥ 0 , does not extend to a bounded operator $$P_\lambda :H^{s,p}({\mathcal {Z}}_\lambda )\rightarrow H^{s,p}({\mathcal {Z}}_\lambda )$$ P λ : H s , p ( Z λ ) → H s , p ( Z λ ) when $$s>0$$ s > 0 or $$p\ne 2.$$ p ≠ 2 . The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.

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On Regularity and Irregularity of Certain Holomorphic Singular Integral Operators

Cited by 3 publications

References 56 publications

Holomorphic function spaces on the Hartogs triangle

Holomorphic function spaces on the Hartogs triangle

Holomorphic function spaces on the Hartogs triangle

Irregularity of the Bergman Projection on Smooth Unbounded Worm Domains

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