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Interpolation d'opérateurs entre espaces de fonctions holomorphes

Patrice Lassere — 1991

Annales Polonici Mathematici

Let K be a compact subset of an hyperconvex open set D n , forming with D a Runge pair and such that the extremal p.s.h. function ω(·,K,D) is continuous. Let H(D) and H(K) be the spaces of holomorphic functions respectively on D and K equipped with their usual topologies. The main result of this paper contains as a particular case the following statement: if T is a continuous linear map of H(K) into H(K) whose restriction to H(D) is continuous into H(D), then the restriction of T to H ( D α ) is a continuous...

Convexity of sublevel sets of plurisubharmonic extremal functions

Finnur LárussonPatrice LassereRagnar Sigurdsson — 1998

Annales Polonici Mathematici

Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function u E , X for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of u E , X are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

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