# Weighted Hardy inequalities and Hardy transforms of weights

Studia Mathematica (2000)

• Volume: 139, Issue: 2, page 189-196
• ISSN: 0039-3223

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## Abstract

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Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as ${A}_{p}$-weights of Muckenhoupt and ${B}_{p}$-weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family ${M}_{p}$ of weights w for which the Hardy transform is ${L}_{p}\left(w\right)$-bounded. A ${B}_{p}$-weight is precisely one for which its Hardy transform is in ${M}_{p}$, and also a weight whose indefinite integral is in ${A}_{p+1}$

## How to cite

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Cerdà, Joan, and Martín, Joaquim. "Weighted Hardy inequalities and Hardy transforms of weights." Studia Mathematica 139.2 (2000): 189-196. <http://eudml.org/doc/216718>.

@article{Cerdà2000,
abstract = {Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as $A_p$-weights of Muckenhoupt and $B_p$-weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family $M_p$ of weights w for which the Hardy transform is $L_p(w)$-bounded. A $B_p$-weight is precisely one for which its Hardy transform is in $M_p$, and also a weight whose indefinite integral is in $A_\{p+1\}$},
author = {Cerdà, Joan, Martín, Joaquim},
journal = {Studia Mathematica},
keywords = {Hardy's inequalities; Hardy transform; weights; Hardy inequalities; Hardy-Littlewood maximal operator; weighted Lebesgue spaces; Hardy operator; Muckenhoupt class},
language = {eng},
number = {2},
pages = {189-196},
title = {Weighted Hardy inequalities and Hardy transforms of weights},
url = {http://eudml.org/doc/216718},
volume = {139},
year = {2000},
}

TY - JOUR
AU - Cerdà, Joan
AU - Martín, Joaquim
TI - Weighted Hardy inequalities and Hardy transforms of weights
JO - Studia Mathematica
PY - 2000
VL - 139
IS - 2
SP - 189
EP - 196
AB - Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as $A_p$-weights of Muckenhoupt and $B_p$-weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family $M_p$ of weights w for which the Hardy transform is $L_p(w)$-bounded. A $B_p$-weight is precisely one for which its Hardy transform is in $M_p$, and also a weight whose indefinite integral is in $A_{p+1}$
LA - eng
KW - Hardy's inequalities; Hardy transform; weights; Hardy inequalities; Hardy-Littlewood maximal operator; weighted Lebesgue spaces; Hardy operator; Muckenhoupt class
UR - http://eudml.org/doc/216718
ER -

## References

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2. [ArM] M. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), 727-735. Zbl0716.42016
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8. [Ma] L. Maligranda, Weighted estimates of integral operators decreasing functions, in: Proc. Internat. Conf. dedicated to F. D. Gakhov, Minsk, 1996, 226-236.
9. [Mu1] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. Zbl0236.26016
10. [Mu2] B. Muckenhoupt, Hardy's inequalities with weights, Studia Math. 44 (1972), 31-38. Zbl0236.26015
11. [Ne1] C. J. Neugebauer, Weighted norm inequalities for averaging operators of monotone functions, Publ. Mat. 35 (1991), 429-447. Zbl0746.42014
12. [Ne2] C. J. Neugebauer, Some classical operators on Lorentz space, Forum Math. 4 (1992), 135-146. Zbl0824.42014
13. [So] J. Soria, Lorentz spaces of weak-type, Quart. J. Math. 49 (1998), 93-103. Zbl0943.42010
14. [Wi] I. Wik, On Muckenhoupt's classes of weight functions, Studia Math. 94 (1989), 245-255. Zbl0686.42012

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