Open AccessThe problem of trigonometric Fourier series multipliers of classes in λp,q spaces
Author(s) -
Amangul Bakhyt,
Nazerke Tleukhanova
Publication year - 2020
Publication title -
ķaraġandy universitetìnìn̦ habaršysy. matematika seriâsy
Language(s) - English
Resource type - Journals
eISSN - 2663-5011
pISSN - 2518-7929
DOI - 10.31489/2020m4/17-25
Subject(s) - mathematics , interpolation space , lorentz space , hardy space , norm (philosophy) , space (punctuation) , interpolation (computer graphics) , lp space , fourier series , pure mathematics , mathematical analysis , lorentz transformation , banach space , combinatorics , discrete mathematics , functional analysis , animation , biochemistry , chemistry , physics , linguistics , philosophy , computer graphics (images) , classical mechanics , political science , computer science , law , gene
In this article, we consider weighted spaces of numerical sequences λp,q, which are defined as sets of sequences a = {ak}^∞_k=1, for which the norm ||a||λp,q :=\sum^∞_k=1|ak|^q k^(q/p −1)^1/q<∞ is finite. In the case of non-increasing sequences, the norm of the space λp,q coincides with the norm of the classical Lorentz space lp,q. Necessary and sufficient conditions are obtained for embeddings of the space λp,q into the space λp1,q1. The interpolation properties of these spaces with respect to the real interpolation method are studied. It is shown that the scale of spaces λp,q is closed in the relative real interpolation method, as well as in relative to the complex interpolation method. A description of the dual space to the weighted space λp,q is obtained. Specifically, it is shown that the space is reflective, where p', q' are conjugate to the parameters p and q. The paper also studies the properties of the convolution operator in these spaces. The main result of this work is an O’Neil type inequality. The resulting inequality generalizes the classical Young-O’Neil inequality. The research methods are based on the interpolation theorems proved in this paper for the spaces λp,q.