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Siciak’s extremal function via Bernstein and Markov constants for compact sets in ℂ N

Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set E ⊂ ℂ N . We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function Φ E . Moreover, we show that one of these extremal-like functions is equal to Φ E if E is a nonpluripolar set with l i m n → ∞ M ₙ ( E ) 1 / n = 1 where M ₙ ( E ) : = s u p | | | g r a d P | | | E / | | P | | E , the supremum is taken over all polynomials P of N variables of total...

Siciak-Zahariuta extremal functions and polynomial hulls

Finnur Lárusson, Ragnar Sigurdsson (2007)

Annales Polonici Mathematici

We use our disc formula for the Siciak-Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs.

Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

Let E be a compact set in the complex plane, g E be the Green function of the unbounded component of ℂ ∞ ∖ E with pole at infinity and M ₙ ( E ) = s u p ( | | P ' | | E ) / ( | | P | | E ) where the supremum is taken over all polynomials P | E ≢ 0 of degree at most n, and | | f | | E = s u p | f ( z ) | : z ∈ E . The paper deals with recent results concerning a connection between the smoothness of g E (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence M ₙ ( E ) n = 1 , 2 , . . . . Some additional conditions are given for special classes of sets.

Some applications of the trace condition for pluriharmonic functions in Cn.

Alessandro Perotti (2000)

Publicacions Matemàtiques

In this paper we investigate some applications of the trace condition for pluriharmonic functions on a smooth, bounded domain in Cn. This condition, related to the normal component on ∂D of the ∂-operator, permits us to study the Neumann problem for pluriharmonic functions and the ∂-problem for (0,1)-forms on D with solutions having assigned real part on the boundary.

Some characterizations of the class m ( Ω ) and applications

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)

Annales Polonici Mathematici

We give some characterizations of the class m ( Ω ) and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

Stochastic characterization of plurisubharmonicity and convexity of functions

Maciej Klimek (2015)

Banach Center Publications

It is described how both plurisubharmonicity and convexity of functions can be characterized in terms of simple to work with classes of holomorphic martingales, namely a class of driftless Itô processes satisfying a skew-symmetry property and a family of linear modifications of Brownian motion parametrized by a compact set.

Currently displaying 241 – 260 of 338