A weighted inequality for the Hardy operator involving suprema

Hofmanová, Pavla. "A weighted inequality for the Hardy operator involving suprema." Commentationes Mathematicae Universitatis Carolinae 57.3 (2016): 317-326. <http://eudml.org/doc/286801>.

@article{Hofmanová2016,
abstract = {Let $u$ be a weight on $(0, \infty )$. Assume that $u$ is continuous on $(0, \infty )$. Let the operator $S_\{u\}$ be given at measurable non-negative function $\varphi $ on $(0, \infty )$ by \[ S\_\{u\}\varphi (t)= \sup \_\{0< \tau \le t\}u(\tau )\varphi (\tau ). \] We characterize weights $v,w$ on $(0, \infty )$ for which there exists a positive constant $C$ such that the inequality \[ \left( \int \_\{0\}^\{\infty \}[S\_\{u\}\varphi (t)]^\{q\}w(t)\,dt\right)^\{\frac\{1\}\{q\}\} \lesssim \left( \int \_\{0\}^\{\infty \}[\varphi (t)]^\{p\}v(t)\,dt\right)^\{\frac\{1\}\{p\}\} \] holds for every $0<p, q<\infty $. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.},
author = {Hofmanová, Pavla},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Hardy operators involving suprema; weighted inequalities},
language = {eng},
number = {3},
pages = {317-326},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A weighted inequality for the Hardy operator involving suprema},
url = {http://eudml.org/doc/286801},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Hofmanová, Pavla
TI - A weighted inequality for the Hardy operator involving suprema
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 3
SP - 317
EP - 326
AB - Let $u$ be a weight on $(0, \infty )$. Assume that $u$ is continuous on $(0, \infty )$. Let the operator $S_{u}$ be given at measurable non-negative function $\varphi $ on $(0, \infty )$ by \[ S_{u}\varphi (t)= \sup _{0< \tau \le t}u(\tau )\varphi (\tau ). \] We characterize weights $v,w$ on $(0, \infty )$ for which there exists a positive constant $C$ such that the inequality \[ \left( \int _{0}^{\infty }[S_{u}\varphi (t)]^{q}w(t)\,dt\right)^{\frac{1}{q}} \lesssim \left( \int _{0}^{\infty }[\varphi (t)]^{p}v(t)\,dt\right)^{\frac{1}{p}} \] holds for every $0<p, q<\infty $. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.
LA - eng
KW - Hardy operators involving suprema; weighted inequalities
UR - http://eudml.org/doc/286801
ER -