Open AccessInterpolation of nonlinear integral Urysohn operators in net spaces
Author(s) -
A.H. Kalidolday,
E.D. Nursultanov
Publication year - 2022
Publication title -
ķaraġandy universitetìnìn̦ habaršysy. matematika seriâsy
Language(s) - English
Resource type - Journals
eISSN - 2663-5011
pISSN - 2518-7929
DOI - 10.31489/2022m1/66-73
Subject(s) - mathematics , net (polyhedron) , interpolation (computer graphics) , operator (biology) , pure mathematics , interpolation space , finite rank operator , shift operator , operator theory , mathematical analysis , compact operator , discrete mathematics , banach space , functional analysis , extension (predicate logic) , computer science , programming language , animation , biochemistry , chemistry , computer graphics (images) , geometry , repressor , transcription factor , gene
In this paper, we study the interpolation properties of the net spaces N_p,q(M), in the case when M is a sufficiently general arbitrary system of measurable subsets from R^n. The integral Urysohn operator is considered. This operator generalizes all linear, integral operators, and non-linear integral operators. The Urysohn operator is not a quasilinear or subadditive operator. Therefore, the classical interpolation theorems for these operators do not hold. A certain analogue of the Marcinkiewicz-type interpolation theorem for this class of operators is obtained. This theorem allows to obtain, in a sense, a strong estimate for Urysohn operators in net spaces from weak estimates for these operators in net spaces with local nets. For example, in order for the Urysohn integral operator in a net space, where the net is the set of all balls in R^n, it is sufficient for it to be of weak type for net spaces, where the net is concentric balls.