Completion of the Kernel of the Differentiation Operator

Abstract

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 \(W_1^1\)
$$
\Lambda~:~W_1^1~\mapsto~L_1,\; \Lambda f = f^{\;'}
$$.
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.