QUASIADDITIVITY AND MEASURE PROPERTY OF CAPACITY AND THE TANGENTIAL BOUNDARY BEHAVIOR OF HARMONIC FUNCTIONS

摘要

We show that ii a set E is dispersely decomposed into subsets, then the capacity of E is comparable to the summation of tile capacities of the subsets. From this fact it is derived that the Lebesgue measure of a certain expanded set is estimated by the capacity of E. These properties hold for classical capacities. L(P)-capacities and energy capacities of general kernels. The estimation is applied to the boundary behavior of harmonic functions. We introduce a boundary thin set and show a fine limit type boundary behavior of harmonic functions. We show that; a thin set does not meet essentially Nagel-Stein and Nagel-Rudin-Shapiro type approaching regions at almost all bounary points. [References: 9]