Completion of the kernel of the differentiation operator

When investigating piecewise polynomial approximations in spaces $L_p, \; 0 < p < 1,$ the author considered the spreading of k-th derivative (of the operator) from Sobolev spaces $W_1 ^ k$ on spaces that are, in a sense, their successors with a low index less than one. In this article, we continue the study of the properties acquired by the differentiation operator $\Lambda$ with spreading beyond the space $W_1^1$ $\big/ \Lambda : W_1^1 \mapsto L_1,\; \Lambda f = f^{\;'} \big/$. The study is conducted by introducing the family of spaces $Y_p^1, \; 0 <p < 1,$ which have analogy with the family $W_p^1, \; 1 \le p <\infty.$ This approach gives a new perspective for the properties of the derivative. It has been shown, for example, the additivity property relative to the interval of the spreading differentiation operator:
$$ \bigcup_{n=1}^{m} \Lambda (f_n) = \Lambda (\bigcup_{n=1}^{m} f_n).$$
Here, for a function $f_n$ defined on $[x_{n-1}; x_n], \; a = x_0 < x_1 < \cdots <x_m = b$, $\Lambda (f_n)$ was defined. One of the most important characteristics of a linear operator is the composition of the kernel. During the spreading of the differentiation operator from the space $ C ^ 1 $ on the space $ W_p ^ 1 $ the kernel does not change. In the article, it is constructively shown that jump functions and singular functions $f$ belong to all spaces $ Y_p ^ 1 $ and $\Lambda f = 0.$ Consequently, the space of the functions of the bounded variation $H_1 ^ 1 $ is contained in each $ Y_p ^ 1 ,$ and the differentiation operator on $H_1^1$ satisfies the relation $\Lambda f = f^{\; '}.$ Also, we come to the conclusion that every function from the added part of the kernel can be logically named singular.