Annales Academić Scientiarum Fennicć
Mathematica
Volumen 38, 2013, 677-689

BEURLING'S CRITERION AND EXTREMAL METRICS FOR FUGLEDE MODULUS

Matthew Badger

Stony Brook University, Department of Mathematics
Stony Brook, NY 11794-3651, U.S.A.; badger 'at' math.sunysb.edu

Abstract. For each 1 \leq p < \infty, we formulate a necessary and sufficient condition for an admissible metric to be extremal for the Fuglede p-modulus of a system of measures. When p = 2, this characterization generalizes Beurling's criterion, a sufficient condition for an admissible metric to be extremal for the extremal length of a planar curve family. In addition, we prove that every Borel function \varphi : Rⁿ -> [0,\infty] satisfying 0 < \int \varphi^p < \infty is extremal for the p-modulus of some curve family in Rⁿ.

2010 Mathematics Subject Classification: Primary 31B15; Secondary 28A33, 49K27.

Key words: Beurling's criterion, extremal metric, modulus, extremal length.

Reference to this article: M. Badger: Beurling's criterion and extremal metrics for Fuglede modulus. Ann. Acad. Sci. Fenn. Math. 38 (2013), 677-689.

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doi:10.5186/aasfm.2013.3826