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- Title
Weighted inequalities for discrete iterated kernel operators.
- Authors
Gogatishvili, Amiran; Pick, Luboš; Ünver, Tuğçe
- Abstract
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<p,q≤∞$0 < p,q \le \infty$ and {vn}n∈Z${\big \lbrace v_n\big \rbrace} _{n\in \operatorname{\mathbb {Z}}}$ and {wn}n∈Z${\big \lbrace w_n\big \rbrace} _{n\in \operatorname{\mathbb {Z}}}$ are fixed weight sequences. We do the same for the inequality ∑n∈Zwnsup−∞<i≤nU(i,n)∑j=−∞iajq1/q≤C∑n∈Zanpvn1/p.$$\begin{equation*}\hskip3pc {\left (\sum _{n\in \operatorname{\mathbb {Z}}}w_n {\left[ \sup _{-\infty <i\le n} U(i,n) \sum _{j=-\infty }^{i} a_j \right]}^q \right)}^{1/q} \le C {\left (\sum _{n\in \operatorname{\mathbb {Z}}}a_n^p v_n \right)}^{1/p}. \end{equation*}$$We characterize these inequalities by conditions of both discrete and continuous nature.
- Subjects
PROBLEM solving
- Publication
Mathematische Nachrichten, 2022, Vol 295, Issue 11, p2171
- ISSN
0025-584X
- Publication type
Article
- DOI
10.1002/mana.202000144