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Non-conformal Loewner type estimates for modulus of curve families

Summary: Non-conformal Loewner type estimates for modulus
of curve families
Tomasz Adamowicz
and Nageswari Shanmugalingam
February 1, 2010
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)-Poincar´e inequality for the situations of 1 p Q and p > Q. We also
study p-modulus estimates with respect to Riesz potentials.
Keywords: p-modulus of curve family, Loewner type theorem, metric measure
spaces, conformal mappings, p-capacity, p-harmonic functions.
Mathematics Subject Classification (2000): 30C65, 28A75, 28A78, 31C15, 46E35.
1 Introduction and preliminaries
Recently there has been increasing interest in the geometry of the p-harmonic world.
If p = n, the relations between n-harmonics and conformal and quasiconformal maps
allow us to discover a variety of properties of n-harmonic functions and mappings (see
e.g. Chapters 14, 15 in [HKM] or [MV1, MV2] and references therein). The situation


Source: Adamowicz, Tomasz - Matematiska Institutionen, Linköpings Universitet


Collections: Mathematics