Weighted inequalities for discrete iterated kernel operators

We develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a positive constant C such that ∑n∈Z∑i=−∞nU(i,n)aiqwn1/q≤C∑n∈Zan...

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Bibliographic Details
Published in:Mathematische Nachrichten 2022-11, Vol.295 (11), p.2171-2196
Main Authors: Gogatishvili, Amiran, Pick, Luboš, Ünver, Tuğçe
Format: Article
Language:English
Subjects:
Inequalities
kernel operator
Kernels
Operators
supremum operator
weighted inequality
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Summary:We develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a positive constant C such that ∑n∈Z∑i=−∞nU(i,n)aiqwn1/q≤C∑n∈Zanpvn1/p$$\begin{equation*}\hskip4pc {\left (\sum _{n\in \operatorname{\mathbb {Z}}}{\left (\sum _{i=-\infty }^n{U}(i,n)a_i\right )}^{q} {w}_n\right )}^{1/q} \le C {\left (\sum _{n\in \operatorname{\mathbb {Z}}}a_n^p{v}_n\right )}^{1/p} \end{equation*}$$holds for every sequence of nonnegative numbers {an}n∈Z$\lbrace a_n\rbrace _{n\in \operatorname{\mathbb {Z}}}$ where U is a kernel satisfying certain regularity condition, 0
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.202000144