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 $, 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 $, 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 $.
@article{MAIS_2014_21_3_a4,
author = {A. N. Morozov},
title = {Countable {Additivity} of {Spreading} the {Differentiation} {Operator}},
journal = {Modelirovanie i analiz informacionnyh sistem},
pages = {81--90},
year = {2014},
volume = {21},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MAIS_2014_21_3_a4/}
}
TY - JOUR
AU - A. N. Morozov
TI - Countable Additivity of Spreading the Differentiation Operator
JO - Modelirovanie i analiz informacionnyh sistem
PY - 2014
SP - 81
EP - 90
VL - 21
IS - 3
UR - http://geodesic.mathdoc.fr/item/MAIS_2014_21_3_a4/
LA - ru
ID - MAIS_2014_21_3_a4
ER -
%0 Journal Article
%A A. N. Morozov
%T Countable Additivity of Spreading the Differentiation Operator
%J Modelirovanie i analiz informacionnyh sistem
%D 2014
%P 81-90
%V 21
%N 3
%U http://geodesic.mathdoc.fr/item/MAIS_2014_21_3_a4/
%G ru
%F MAIS_2014_21_3_a4
A. N. Morozov. Countable Additivity of Spreading the Differentiation Operator. Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 3, pp. 81-90. http://geodesic.mathdoc.fr/item/MAIS_2014_21_3_a4/
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[2] Morozov A. N., “On Smoothness in $L_p, \; 0 p 1$”, Modeling and analysis of inform. systems, 19:3 (2012), 97—104 (in Russian)
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