Cauchy–Riemann meet Monge–Ampère

Błocki, Zbigniew

doi:10.1007/s13373-014-0058-2

Cited by 22 publications

(21 citation statements)

References 102 publications

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“…That example is in dimension one, and has three connected components. We refer to [6,7] for further details on these conjectures and their applications, to for example the multidimensional Suita conjecture. In Theorem 2.2, it is shown in a straight forward manner that these two conjectures are also valid for domains that are biholomorphic to a bounded, balanced, and pseudoconvex Vol.…”

Section: The B Locki-zwonek Conjecturesmentioning

confidence: 99%

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On the Błocki–Zwonek conjectures and beyond

Åhag

Czyż

2015

Arch. Math.

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Let Ω be a bounded pseudoconvex domain in C n , and let gΩ(z, a) be the pluricomplex Green function with pole at a in Ω. B locki and Zwonek conjectured that the function given byis nondecreasing, and that the function given byis convex. Here λn is the Lebesgue measure in C n . In this note we give an affirmative answer to these conjectures when Ω is biholomorphic to a bounded, balanced, and pseudoconvex domain in C n , n ≥ 1. The aim of this note is to consider generalizations of the functions α, β defined by the Green function with two poles in D ⊂ C. We prove that α is not nondecreasing, and β is not convex. By using the product property for pluricomplex Green functions, we then generalize this to n-dimensions. Finally, we end this note by considering two other possibilities generalizing the B locki-Zwonek conjectures.Mathematics Subject Classification. Primary 32A25; Secondary 32U35.

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Section: The B Locki-zwonek Conjecturesmentioning

confidence: 99%

“…It is interesting to know whether the right-hand side of (1.1) is monotone in t (see e.g. [6]). Therefore B locki and Zwonek [7] made the following statements.…”

mentioning

confidence: 99%

On the Błocki–Zwonek conjectures and beyond

Åhag

Czyż

2015

Arch. Math.

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“…After this paper was written, we received the interesting expository paper [7], which has some overlap with our article. In both papers, the aim is to explain L 2 estimates on∂ that have been derived after the work of Andreotti-Hörmander-Vesentini and to apply them to certain problems.…”

Section: Introductionmentioning

confidence: 99%

“…5.2.4. There is no doubt, however, that such a level of generality could also have been achieved in [7], had the author wanted to include it. A more significant difference between the two papers is conceptual: we view the idea of twisting the∂ complex as a basic method, one that has led to new L 2 estimates and provides a framework to obtain additional estimates.…”

Section: Introductionmentioning

confidence: 99%

$$L^2$$ L 2 estimates for the $$\bar{\partial }$$ ∂ ¯ operator

McNeal

Varolin

2015

Bull. Math. Sci.

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We present the theory of twisted L 2 estimates for the Cauchy-Riemann operator and give a number of recent applications of these estimates. Among the applications: extension theorem of Ohsawa-Takegoshi type, size estimates on the Bergman kernel, quantitative information on the classical invariant metrics of Kobayshi, Caratheodory, and Bergman, and sub elliptic estimates on the∂-Neumann problem. We endeavor to explain the flexibility inherent to the twisted method, through examples and new computations, in order to suggest further applications.

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“…The recent survey of the theory of Bergman functions -writing Bergman function we mean one of three discussed objects: Bergman kernel, Bergman metric, or Bergman distance (the latter two to be defined later) -especially with the stress put on its links with the pluripotential theory, may be found in [8,11], and [28]. The topics contained below reflect the interest of the authors and refer to different aspects of the interactions of the theory of Bergman functions and classical complex analysis.…”

Section: Introductionmentioning

confidence: 99%

Bergman Kernel in Complex Analysis

Kosiński¹

2014

Operator Theory

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In this survey a brief review of results on the Bergman kernel and Bergman distance concentrating on those fields of complex analysis which remain in the focus of the research interest of the authors is presented. The topics discussed contain general discussion of L 2 h spaces, behavior of the Bergman distance, regularity of extension of proper holomorphic mappings, and recent development in the theory of Bergman distance stemming from the pluripotential theory and very short discussion of the Lu Qi Keng problem.

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Cauchy–Riemann meet Monge–Ampère

Cited by 22 publications

References 102 publications

On the Błocki–Zwonek conjectures and beyond

On the Błocki–Zwonek conjectures and beyond

$$L^2$$ L 2 estimates for the $$\bar{\partial }$$ ∂ ¯ operator

Bergman Kernel in Complex Analysis

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