An extension theorem for separately holomorphic functions with pluripolar singularities

Let be a pseudoconvex domain and let be a locally pluriregular set, . Put

Let be an open neighborhood of and let be a relatively closed subset of . For let be the set of all for which the fiber is not pluripolar. Assume that are pluripolar. Put

Then there exists a relatively closed pluripolar subset of the ``envelope of holomorphy' of such that:

,

for every function separately holomorphic on there exists exactly one function holomorphic on with on , and

is singular with respect to the family of all functions .

     

Traces of pluriharmonic functions on curves

We prove that, if γ is a simple smooth curve in the unit sphere inC ⁿ, the space o pluriharmonic functions in the unit ball, continuous up to the boundary, has a trace of finite cof dimension in the space of all continuous functions on the curve.     

Traces of pluriharmonic functions on curves

We prove that, if γ is a simple smooth curve in the unit sphere inC ⁿ, the space o pluriharmonic functions in the unit ball, continuous up to the boundary, has a trace of finite cof dimension in the space of all continuous functions on the curve. First author partially supported by the Swedish Natural Science Research Council. Second author partially supported by CICYT grant PB85-0374.     

On the extension of L~2 holomorphic functions VII:Hypersurfaces with isolated singularities

An extension theorem for holomorphic functions with L~2 growth condition is strengthened for the case of the extension from hypersurfaces with isolated singularities.     

Noncommutative transforms and free pluriharmonic functions

In this paper, we study free pluriharmonic functions on noncommutative balls γ[Bn(H)], γ>0, and their boundary behavior. These functions have the form

     

Bohr's inequality for holomorphic and pluriharmonic mappings with values in complex Hilbert spaces

Let ${\mathbb{B}}_H$ be the unit ball of a complex Hilbert space H. First, we give a Bohr's inequality for the holomorphic mappings with lacunary series with values in complex Hilbert balls. Next, we give several results on Bohr's inequality for pluriharmonic mappings with values in ℓ₂. Note that the Bohr phenomenons that we have obtained are completely different from those in the case with values in $\mathbb{C}$ and are sharp in the case with values in ℓ₂.     

On subextension of pluriharmonic and plurisubharmonic functions

The problem of subextension of plurisubharmonic functions is considered. Recently it was shown by Cegrell–Zeriahi, that subextension is always possible for negative plurisubharmonic functions in the energy class ℱ. In this paper we construct, for every hyperconvex domain Ω, a negative plurisubharmonic function in the class ℰ which cannot be subextended. Given any pseudoconvex domain we construct a pluriharmonic function that cannot be subextended.     

Bounded pluriharmonic functions on symmetric irreducible Siegel domains

Let be an irreducible, symmetric Siegel domain and let S be a solvable group which acts simply transitively on . We exhibit three S-invariant, real, second order, degenerate elliptic operators such that a bounded function F on is pluriharmonic, if and only if . The three operators are the same as in our former paper [DHMP], however there we needed a considerably stronger condition on F to derive the same conclusion. Received: 26 October 2000; in final form: 10 September 2001 / Published online: 28 February 2002     

Separation of boundary singularities for holomorphic generators

We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates rather strikingly the particular role of the binary operation ? given by 1/ f ? g = 1/f + 1/g on generators.     

Residues of holomorphic sections and lelong currents

LetZ be the zero set of a holomorphic sectionf of a Hermitian vector bundle. It is proved that the current of integration over the irreducible components ofZ of top degree, counted with multiplicities, is a product of a residue factorR ^f and a “Jacobian factor”. There is also a relation to the Monge-Ampère expressions (dd ^c log|f|) ^k , which we define for all positive powersk. The author was partially supported by the Swedish Research Council.