L^s(\mu)-Averaging Domains

Shusen Ding and Craig A. Nolder

In this paper we first introduce L^s(\mu)-averaging domains which are generalizations of L^s-averaging domains introduced by Susan G. Staples. We characterize L^s(\mu)-averaging domains using the quasihyperbolic metric. As applications, we prove norm inequalities for conjugate A-harmonic tensors in L^s(\mu)-averaging domains which can be considered as generalizations of the Hardy and Littlewood theorem for conjugate harmonic functions. Finally, we give applications to quasiconformal and quasiregular mappings.