Journal of Fourier Analysis and Applications

, Volume 19, Issue 4, pp 712–730 | Cite as

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

Article

Abstract

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces \(\varLambda^{p}_{u}(w)\), with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces Lp(u) and Muckenhoupt weights Ap, and the theory on classical Lorentz spaces Λp(w) and Ariño-Muckenhoupt weights Bp.

Keywords

Weighted Lorentz spaces Hilbert transform Muckenhoupt weights Bp weights 

Mathematics Subject Classification (2010)

26D10 42A50 

Notes

Acknowledgements

We would like to thank the referee for some useful comments which have improved the final version of this paper.

The first author would also like to thank the State Scholarship Foundation I.K.Y., of Greece. H oλoϰλήρωση της εργασίας αυτής έγιϛε στo πλαίσιo της υλoπoίησης τoυ μεταπτυχιαϰoύ πρoγράμματoς πoυ συγχρηματoδoτήϑηϰε μέσω της Πράξης “Πρóγραμμα χoρήγησης υπoτρoφιώϛ I.K.ϒ. με διαδιϰασία εξατoμιϰευμέϛης αξιoλóγησης αϰαδ. έτoυς 2011–2012” απó πóρoυς τoυ E.Π. “Eϰπαίδευση ϰαι δια βίoυ μάϑηση” τoυ Eυρωπαϊϰoύ ϰoιϛωϛιϰoύ ταμείoυ (EKT) ϰαι τoυ EΣΠA, τoυ 2007–2013.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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