 
Summary: Nonconformal Loewner type estimates for modulus
of curve families
Tomasz Adamowicz
and Nageswari Shanmugalingam
February 1, 2010
Abstract
We develop various upper and lower estimates for pmodulus 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 QAhlfors regular metric measure spaces supporting
(1, p)Poincar´e inequality for the situations of 1 p Q and p > Q. We also
study pmodulus estimates with respect to Riesz potentials.
Keywords: pmodulus of curve family, Loewner type theorem, metric measure
spaces, conformal mappings, pcapacity, pharmonic 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 pharmonic world.
If p = n, the relations between nharmonics and conformal and quasiconformal maps
allow us to discover a variety of properties of nharmonic functions and mappings (see
e.g. Chapters 14, 15 in [HKM] or [MV1, MV2] and references therein). The situation
