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Weighted Hardy inequalities and Hardy transforms of weights

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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 $L_p(w)$-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 $A_{p+1}$
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  • Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, E-08071 Barcelona, Spain, cerda@mat.ub.es
  • Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, E-08071 Barcelona, Spain, jmartin@mat.ub.es
Bibliografia
  • [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.
  • [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.
  • [BMR] J. Bastero, M. Milman and F. J. Ruiz, On the connection between weighted norm inequalities, commutators and real interpolation, preprint.
  • [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.
  • [CM] J. Cerdà and J. Martín, Conjugate Hardy's inequalities with decreasing weights, ibid. 126 (1998), 2341-2344.
  • [CU] D. Cruz-Uribe, Piecewise monotonic doubling measures, Rocky Mountain J. Math. 26 (1996), 1-39.
  • [GR] J. García-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, 1985.
  • [Ma] L. Maligranda, Weighted estimates of integral operators decreasing functions, in: Proc. Internat. Conf. dedicated to F. D. Gakhov, Minsk, 1996, 226-236.
  • [Mu1] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.
  • [Mu2] B. Muckenhoupt, Hardy's inequalities with weights, Studia Math. 44 (1972), 31-38.
  • [Ne1] C. J. Neugebauer, Weighted norm inequalities for averaging operators of monotone functions, Publ. Mat. 35 (1991), 429-447.
  • [Ne2] C. J. Neugebauer, Some classical operators on Lorentz space, Forum Math. 4 (1992), 135-146.
  • [So] J. Soria, Lorentz spaces of weak-type, Quart. J. Math. 49 (1998), 93-103.
  • [Wi] I. Wik, On Muckenhoupt's classes of weight functions, Studia Math. 94 (1989), 245-255.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv139i2p189bwm
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