Characterization of the weak-type boundedness of the Hilbert transform on weighted Lorentz spaces

Abstract

We characterize the weak-type boundedness of the Hilbert transform $H$ on weighted Lorentz spaces ${\Lambda }_u^p(w)$, 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 ${\Lambda }^p(w)$ and Ariño Muckenhoupt weights $B_p$.

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References in corpus (1)

• Boundedness of the Hilbert transform on weighted Lorentz spaces

Cited by in corpus (1)

• Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces