On the Neumann Operator of the Arithmetical Mean

The symbol λk (k ∈ {1, 2}) will denote the k-dimensional Hausdorff measure (with the usual normalization, so that λk([0, 1] ) = 1). For M ⊂ R we use the symbols ∂M , intM and clM to denote the boundary, the interior and the closure of M , respectively. For M 6= ∅ we denote by C(M) the Banach space of all bounded continuous functions on M with the supremum norm, by 1M the constant function equal to 1 on M , by Const (M) = {α1M ;α ∈ R} the class of all constant functions on M . C 0 will stand for the class of all continuously differentiable functions with a compact support in R, for bounded M we write C(M) = {φ ∣∣ M : φ ∈ C (1) 0 } for the class of all restrictions to M of functions in C 0 . Throughout, K ⊂ R will be a fixed non-void compact set which is massive at each z ∈ K in the sense that each disk Br(z) = {x ∈ R; |x− z| < r}