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The Size of Plurisubharmonic Lemniscates in Terms of Hausdorff–Riesz Measures and Capacities
Author(s) -
Zeriahi Ahmed
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504014698
Subject(s) - mathematics , pure mathematics , hausdorff space , bounded function , riesz potential , riesz representation theorem , m. riesz extension theorem , class (philosophy) , mathematical analysis , artificial intelligence , computer science
The main goal of this paper is to establish new uniform estimates on the size of sublevel sets of plurisubharmonic functions (called plurisubharmonic lemniscates) in terms of Hausdorff–Riesz measures and capacities of certain orders. We first prove a new uniform version of Skoda's integrability theorem for a given class of plurisubharmonic functions in terms of Borel measures of Hausdorff–Riesz type of certain orders with a precise estimate of the integrability exponent in terms of Lelong numbers of the class and the order of the measures. Then we present several applications of this result. We first deduce uniform estimates on the size of plurisubharmonic lemniscates associated to functions from some important classes of plurisubharmonic functions in terms of Hausdorff–Riesz measures. We also derive a new comparison inequality between certain Hausdorff–Riesz capacities and the pluricomplex logarithmic capacity for borelean sets of a fixed bounded domain in C n or more generally in an affine algebraic manifold. Furthermore, using results from classical potential theory, we finally deduce from this comparison inequality new estimates of the size of polynomial lemniscates in terms of Hausdorff contents in the spirit of the famous lemma of H. Cartan. 2000 Mathematics Subject Classification 31C10, 31C15, 32F05, 32F99, 32U05, 32U99.