Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces

Abstract

We prove the Lorentz-Shimogaki and Boyd theorems for the spaces ${\Lambda }_u^p(w)$. As a consequence, we give the complete characterization of the strong boundedness of $H$ on these spaces in terms of some geometric conditions on the weights $u$ and $w$, whenever $p>1$. For these values of $p$, we also give the complete solution of the weak-type boundedness of the Hardy-Littlewood operator on ${\Lambda }_u^p(w)$.

20 pages, 2 figures

References in corpus (2)

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

Cited by in corpus (1)

• Weak Type Boundedness of the Hardy Littlewood Maximal Operator on Weighted Lorentz Spaces