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Weighted Inequalities for Hardy–Steklov Operators

Published online by Cambridge University Press:  20 November 2018

A. L. Bernardis ,
F. J. Martín-Reyes  and
P. Ortega Salvador
A. L. Bernardis
Affiliation:
IMAL-CONICET, Güemes 3450, (3000) Santa Fe, Argentina email: [email protected]
F. J. Martín-Reyes
Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain email: [email protected], [email protected]
P. Ortega Salvador
Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain email: [email protected], [email protected]
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Abstract

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We characterize the pairs of weights $\left( v,\,w \right)$ for which the operator $Tf\left( x \right)=g\left( x \right)\int{_{s\left( x \right)}^{h\left( x \right)}}\,f\,$ with $s$ and $h$ increasing and continuous functions is of strong type $\left( p,\,q \right)$ or weak type $\left( p,\,q \right)$ with respect to the pair $\left( v,\,w \right)$ in the case $0\,<\,q\,<\,p$ and $1\,<\,p\,<\,\infty$. The result for the weak type is new while the characterizations for the strong type improve the ones given by H. P. Heinig and G. Sinnamon. In particular, we do not assume differentiability properties on $s$ and $h$ and we obtain that the strong type inequality $\left( p,q \right),q\,<p$, is characterized by the fact that the function

$$\Phi \left( x \right)\,=\,\sup \,{{\left( \int_{c}^{d}{{{g}^{q}}w} \right)}^{1/p}}\left( \int_{s\left( d \right)}^{h\left( c \right)}{{{v}^{1-{p}'}}} \right){{\,}^{1/{p}'}}$$

belongs to ${{L}^{r}}\left( {{g}^{q}}w \right)$, where ${1}/{r\,=\,{\,1}/{q\,-\,{1}/{p}\;}\;}\;$ and the supremum is taken over all $c$ and $d$ such that $c\le x\le d$ and $s\left( d \right)\,\le \,h\left( c \right)$.

Keywords

26D1546E3042B25Hardy–Steklov operatorweightsinequalities
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 2 , 01 April 2007 , pp. 276 - 295
Copyright
Copyright © Canadian Mathematical Society 2007

References

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