scholarly journals Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of Cn

2006 ◽  
Vol 88 (1) ◽  
pp. 59-82
Author(s):  
Kuzman Adzievski
Keyword(s):  
Unit Ball ◽  
Exceptional Sets ◽  
Hausdorff Capacity ◽  
Green Potentials

Exceptional Sets of Slices for Functions From the Bergman Space in the Ball

2001 ◽  
Vol 44 (2) ◽  
pp. 150-159 ◽  
Author(s):  
Piotr Jakóbczak
Keyword(s):  
Unit Ball ◽  
Lebesgue Measure ◽  
Bergman Space ◽  
Dimensional Subspace ◽  
Natural Topology ◽  
Exceptional Set ◽  
One Dimensional ◽  
Exceptional Sets

AbstractLet BN be the unit ball in and let f be a function holomorphic and L2-integrable in BN. Denote by E(BN, f) the set of all slices of the form , where L is a complex one-dimensional subspace of , for which is not L2-integrable (with respect to the Lebesgue measure on L). Call this set the exceptional set for f. We give a characterization of exceptional sets which are closed in the natural topology of slices.

scholarly journals Radial growth of harmonic functions in the unit ball

2012 ◽  
Vol 110 (2) ◽  
pp. 273 ◽  
Author(s):  
Kjersti Solberg Eikrem ◽  
Eugenia Malinnikova
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Harmonic Functions ◽  
Radial Growth ◽  
Measure Zero ◽  
Hausdorff Measures ◽  
Doubling Condition ◽  
Exceptional Sets

Let $\Psi_v$ be the class of harmonic functions in the unit disk or unit ball in ${\mathsf R}^m$ which admit a radial majorant $v(r)$. We prove that a function in $\Psi_v$ may grow or decay as fast as $v$ only along a set of radii of measure zero. For the case when $v$ fulfills a doubling condition, we give precise estimates of these exceptional sets in terms of Hausdorff measures.

Holomorphically embedded discs with rapidly growing area

1999 ◽  
Vol 129 (2) ◽  
pp. 343-349
Author(s):  
Josip Globevnik
Keyword(s):  
Unit Ball ◽  
Open Unit ◽  
Open Unit Ball

It is shown that if V is a closed submanifold of the open unit ball of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ 1. It is also shown that if V is a closed submanifold of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ ∞.

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