The Bi-dimensional space of Korenblum and composition operator

In this paper we present the concept of total κ-variation in the sense of Hardy-Vitali-Korenblum for a real function defined in the rectangle I_a^b⊂R². We show that the space κBV(I_a^b, R) of real functions of two variables with finite total κ-variation is a Banach space endowed with the norm ||f||κ = |f (a)| + κTV( f, I_a^b). Also, we characterize the Nemytskij composition operator H that maps the space of functions of two real variables of bounded κ-variation κBV(I_a^b, R) into another space of a similar type and is uniformly bounded (or Lipschitzian or uniformly continuous).