Weighted Hardy inequalities and Hardy transforms of weights

Cerdà, Joan; Martín, Joaquim

Studia Mathematica, Tome 141 (2000), p. 189-196 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $A_{p + 1}$

Publié le : 2000-01-01

Zbl 0979.26008

EUDML-ID : urn:eudml:doc:216718

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     title = {Weighted Hardy inequalities and Hardy transforms of weights},
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     year = {2000},
     pages = {189-196},
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Cerdà, Joan; Martín, Joaquim. Weighted Hardy inequalities and Hardy transforms of weights. Studia Mathematica, Tome 141 (2000) pp. 189-196. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv139i2p189bwm/

Bibliographie

[00000] [AnM] K. F. Anderssen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9-26.

[00001] [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. | Zbl 0716.42016

[00002] [BMR] J. Bastero, M. Milman and F. J. Ruiz, On the connection between weighted norm inequalities, commutators and real interpolation, preprint. | Zbl 0992.46021

[00003] [CGS] M. J. Carro, A. García del Amo and J. Soria, Weak-type weights and normable Lorentz spaces, Proc. Amer. Math. Soc. 124 (1996), 849-857. | Zbl 0853.42016

[00004] [CM] J. Cerdà and J. Martín, Conjugate Hardy's inequalities with decreasing weights, ibid. 126 (1998), 2341-2344. | Zbl 0907.26010

[00005] [CU] D. Cruz-Uribe, Piecewise monotonic doubling measures, Rocky Mountain J. Math. 26 (1996), 1-39.

[00006] [GR] J. García-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, 1985.

[00007] [Ma] L. Maligranda, Weighted estimates of integral operators decreasing functions, in: Proc. Internat. Conf. dedicated to F. D. Gakhov, Minsk, 1996, 226-236.

[00008] [Mu1] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. | Zbl 0236.26016

[00009] [Mu2] B. Muckenhoupt, Hardy's inequalities with weights, Studia Math. 44 (1972), 31-38. | Zbl 0236.26015

[00010] [Ne1] C. J. Neugebauer, Weighted norm inequalities for averaging operators of monotone functions, Publ. Mat. 35 (1991), 429-447. | Zbl 0746.42014

[00011] [Ne2] C. J. Neugebauer, Some classical operators on Lorentz space, Forum Math. 4 (1992), 135-146. | Zbl 0824.42014

[00012] [So] J. Soria, Lorentz spaces of weak-type, Quart. J. Math. 49 (1998), 93-103. | Zbl 0943.42010

[00013] [Wi] I. Wik, On Muckenhoupt's classes of weight functions, Studia Math. 94 (1989), 245-255. | Zbl 0686.42012