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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.

Strict fine maxima.

Fitzsimmons, P.J. (2000)

Electronic Communications in Probability [electronic only]

Sur la théorie du potentiel dans les domaines de John.

Alano Ancona (2007)

Publicacions Matemàtiques

Using rather elementary and direct methods, we first recover and add on some results of Aikawa-Hirata-Lundh about the Martin boundary of a John domain. In particular we answer a question raised by these authors. Some applications are given and the case of more general second order elliptic operators is also investigated. In the last parts of the paper two potential theoretic results are shown in the framework of uniform domains or the framework of hyperbolic manifolds.

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