Countable Additivity of Spreading the Differentiation Operator

Abstract

In this article, we continue the study of the properties acquired by the differentiation operator Λ with spreading beyond the space W₁¹. The study is conducted by introducing the family of spaces Yp¹ , 0 < p < 1, having analogy with the family Wp¹, 1 ≤ p < ∞. Spaces Yp¹ are equiped with quasinorms constructed on quasinorms spaces Lp as the basis; Λ : Yp¹ → Lp. We have given a sufficient condition for a function, piecewise belonging to the space Yp¹ to be in this space (if f ∈ Yp¹ [x_i-1; x_i ], i ∈ N, 0 = x0 < x₁ < · · · < x_i < · · · < 1, then f ∈ Yp¹ [0; 1]). In other words, it is the sign when the equality: Λ(S fi) = S Λ(fi) 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 W₁¹ and has a bounded variation, f belongs to each space Yp¹ , 0 < p < 1.