On weighted iterated Hardy-type operators
type: Item
instance of: scholarly article
title: On weighted iterated Hardy-type operators
zbMATH Open document ID: 1424.26041
published in: Analysis Mathematica
publication date: 2018-07-23T00:00:00Z
review text: Let $\mathfrak{M}$ be the set of all Lebesgue measurable functions on $[0,\infty)$, and $\mathfrak{M}^+$ be the subset of $\mathfrak{M}$ consisting of all non-negative functions. Fix $\rho,v,u\in \mathfrak{M}^+$ and $0 < r,q < \infty$. The paper characterizes the estimate $$\Vert Tf\Vert_{L^r_\rho}\le C_T\Vert f\Vert_{L^1_v}\quad \left(f\in \mathfrak{M}^+\right),$$ where $T$ is the weighted iterated Hardy-type operator defined by $$Tf(x)=\left(\int_x^\infty u(y)\left(\int_0^yf(z)\,dz\right)^q\,dy\right)^{1/q}\quad (f\in \mathfrak{M}^+,\; 0 \le x< \infty),$$ $\Vert \cdot \Vert_{L^s_w}$ represents the (quasi)-norm of the $w$-weighted Lebesgue space $L^s_w$ $(w\in \mathfrak{M}^+, \; 0 < s < \infty)$, and $C_T$ is an optimal constant independent of $f$. To this end, the above estimate is reduced to weighted inequalities on the cone of all non-decreasing functions in $\mathfrak{M}^+$.
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 46E20
zbMATH DE Number: 6909542
zbMATH Keywords: weighted Lebesgue space
zbMATH Keywords: Hardy-type operator
zbMATH Keywords: cone of non-decreasing functions
author: V. D. Stepanov
author: Guldarya E. Shambilova
reviewed by: Isabel Marrero
MaRDI profile type: Publication
full work available at URL: s10476-018-0211-3
OpenAlex ID: W2807780485
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Wikidata QID: Q129685827
DOI: 10.1007/S10476-018-0211-3
description: scientific article; zbMATH DE number 6909542