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 L p(u) and Muckenhoupt weights A p , and the theory on classical Lorentz spaces Λ p(w) and Ariño-Muckenhoupt weights B p .
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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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Communicated by Loukas Grafakos.
This work was partially supported by the Spanish Government Grant MTM2010-14946.
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Agora, E., Carro, M.J. & Soria, J. Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces. J Fourier Anal Appl 19, 712–730 (2013). https://doi.org/10.1007/s00041-013-9278-1
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DOI: https://doi.org/10.1007/s00041-013-9278-1