Countable Additivity of Spreading the Differentiation Operator

In this article, we continue the study of the properties acquired by the differentiation operator $ \Lambda $ with spreading beyond the space $ W_1^1 $. The study is conducted by introducing the family of spaces $ Y_p^1$, $0 <p < 1$, having analogy with the family $ W_p^1$, $1 \le p <\infty.$ Spaces $ Y_p^1 $ are equiped with quasinorms constructed on quasinorms spaces $ L_p $ as the basis; $ \Lambda: Y_p^1 \mapsto L_p $. We have given a sufficient condition for a function, piecewise belonging to the space $ Y_p^1 $ to be in this space (if $ f \in Y_p^1 [x_{i-1}; x_i]$, $i \in N$, $0 = x_0 <x_1 < \cdots <x_i < \cdots < 1 $, then $ f\in Y_p^1[0;1] $). In other words, it is the sign when the equality: $ \Lambda (\bigcup f_i) = \bigcup \Lambda (f_i)$ is true. The bounded variation in the Jordan sense is closest to the sufficient condition among the classic characteristics of functions. As a corollary, it comes out that, if a function $ f $ piecewise belongs to the space of $ W_1^1 $ and has a bounded variation, $ f $ belongs to each space $ Y_p^1$, $0 <p < 1$.