Annales Academić Scientiarum Fennicć
Mathematica
Volumen 35, 2010, 609-626

NON-CONFORMAL LOEWNER TYPE ESTIMATES FOR MODULUS OF CURVE FAMILIES

Tomasz Adamowicz and Nageswari Shanmugalingam

University of Cincinnati, Department of Mathematical Sciences
P.O. Box 210025, Cincinnati, OH 45221-0025, U.S.A.; tadamowi 'at' gmail.com

University of Cincinnati, Department of Mathematical Sciences
P.O. Box 210025, Cincinnati, OH 45221-0025, U.S.A.; nages 'at' math.uc.edu

Abstract. We develop various upper and lower estimates for p-modulus of curve families on ring domains in the setting of abstract metric measure spaces and derive p-Loewner type estimates for continua. These estimates are obtained for doubling metric measure spaces or Q-Ahlfors regular metric measure spaces supporting (1,p)-Poincaré inequality for the situations of 1 \leq p \leq Q and p > Q. We also study p-modulus estimates with respect to Riesz potentials.

2000 Mathematics Subject Classification: Primary 30C65, 28A75, 28A78, 31C15, 46E35.

Key words: p-modulus of curve family, Loewner type theorem, metric measure spaces, conformal mappings, p-capacity, p-harmonic functions.

Reference to this article: T. Adamowicz and N. Shanmugalingam: Non-conformal Loewner type estimates for modulus of curve families. Ann. Acad. Sci. Fenn. Math. 35 (2010), 609-626.

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doi:10.5186/aasfm.2010.3538