scholarly journals Zero varieties for the Nevanlinna class on all convex domains of finite type

2001 ◽  
Vol 163 ◽  
pp. 215-227 ◽  
Author(s):  
Klas Diederich ◽  
Emmanuel Mazzilli
Keyword(s):  
Finite Type ◽  
Convex Domain ◽  
Main Tool ◽  
Smooth Convex ◽  
Convex Domains ◽  
Zero Sets ◽  
Nevanlinna Class ◽  
Cauchy Riemann ◽  
Bounded Smooth

It is shown, that the so-called Blaschke condition characterizes in any bounded smooth convex domain of finite type exactly the divisors which are zero sets of functions of the Nevanlinna class on the domain. The main tool is a non-isotropic L1 estimate for solutions of the Cauchy-Riemann equations on such domains, which are obtained by estimating suitable kernels of Berndtsson-Andersson type.

Zero Varieties for the Nevanlinna Class in Convex Domains of Finite Type in \Bbb C n

1998 ◽  
Vol 147 (2) ◽  
pp. 391 ◽  
Author(s):  
Joaquim Bruna ◽  
Philippe Charpentier ◽  
Yves Dupain
Keyword(s):  
Finite Type ◽  
Convex Domains ◽  
Nevanlinna Class

scholarly journals Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains

2018 ◽  
Vol 30 (1) ◽  
pp. 159-170
Author(s):  
Peter Pflug ◽  
Włodzimierz Zwonek
Keyword(s):  
Convex Domain ◽  
Gromov Hyperbolicity ◽  
Smooth Convex ◽  
Tube Domain ◽  
Convex Domains ◽  
Hilbert Metric ◽  
Kobayashi Distance ◽  
Continuity Properties ◽  
Gromov Hyperbolic ◽  
Tube Domains

Abstract We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains. The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain Ω in {\mathbb{R}^{n}} with the Kobayashi distance of the tube domain over the domain Ω is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tube domains with a smooth convex bounded base are also studied in detail.

scholarly journals A-priori bounds for quasilinear problems in critical dimension

2019 ◽  
Vol 9 (1) ◽  
pp. 788-802 ◽  
Author(s):  
Giulio Romani
Keyword(s):  
Convex Domain ◽  
Blow Up ◽  
A Priori ◽  
Critical Dimension ◽  
Smooth Convex ◽  
A Priori Bounds ◽  
A Priori Bound ◽  
Bounded Smooth ◽  
Quasilinear Problems

Abstract We establish uniform a-priori bounds for solutions of the quasilinear problems $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta}_Nu=f(u)\quad&\mbox{in }{\it\Omega},\\ u=0\quad&\mbox{on }{\partial{\it\Omega}}, \end{cases} \end{array}$$ where Ω ⊂ ℝN is a bounded smooth convex domain and f is positive, superlinear and subcritical in the sense of the Trudinger-Moser inequality. The typical growth of f is thus exponential. Finally, a generalisation of the result for nonhomogeneous nonlinearities is given. Using a blow-up approach, this paper completes the results in [1, 2], extending the class of nonlinearities for which the uniform a-priori bound applies.

Remark on Zero Sets of Holomorphic Functions in Convex Domains of Finite Type

2010 ◽  
Vol 53 (2) ◽  
pp. 311-320 ◽  
Author(s):  
Michał Jasiczak
Keyword(s):  
Holomorphic Function ◽  
Finite Type ◽  
Holomorphic Functions ◽  
Bounded Mean Oscillation ◽  
Carleson Measures ◽  
Zero Set ◽  
Convex Domains ◽  
Zero Sets ◽  
Analytic Subvariety ◽  
Mean Oscillation

AbstractWe prove that if the (1, 1)-current of integration on an analytic subvariety V ⊂ D satisfies the uniform Blaschke condition, then V is the zero set of a holomorphic function ƒ such that log |ƒ| is a function of bounded mean oscillation in bD. The domain D is assumed to be smoothly bounded and of finite d’Angelo type. The proof amounts to non-isotropic estimates for a solution to the -equation for Carleson measures.

scholarly journals Gain of Regularity in Extension Problem on Convex Domains

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
M. Jasiczak
Keyword(s):  
Finite Type ◽  
Bergman Space ◽  
Bergman Spaces ◽  
Extension Problem ◽  
Convex Domains

We investigate the extension problem from higher codimensional linear subvarieties on convex domains of finite type. We prove that there exists a constantdsuch that on Bergman spacesHp(D)with1≤p<dthere appears the so-called “gain regularity.” The constantddepends on the minimum of the dimension and the codimension of the subvariety. This means that the space of functions which admit an extension to a function in the Bergman spaceHp(D)is strictly larger thanHp(D∩A), whereAis a subvariety.

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