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Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

Author(s): Jesús Bastero; Mario Milman; Francisco J. Ruiz
Journal: Proc. Amer. Math. Soc. 128 (2000), 65-74.
MSC (1991): Primary 42B25, 46E30
Posted: June 30, 1999
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Abstract: For the classical Hardy-Littlewood maximal function $Mf$, a well known and important estimate due to Herz and Stein gives the equivalence $(Mf)^{*}(t)\sim f^{**}(t)$. In the present note, we study the validity of analogous estimates for maximal operators of the form

\begin{equation*}M_{p,q}f(x)= \sup _{x\in Q}{\frac{\Vert f\chi _{Q} \Vert _{p,q} }{\Vert \chi _{Q} \Vert _{p,q}}}, \end{equation*}

where $\Vert . \Vert _{p,q}$ denotes the Lorentz space $L(p,q)$-norm.

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Additional Information:

Jesús Bastero
Affiliation: Department of Mathematics, University of Zaragoza, 50009-Zaragoza, Spain
Email: bastero@posta.unizar.es

Mario Milman
Affiliation: Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431
Email: milman@acc.fau.edu

Francisco J. Ruiz
Affiliation: Department of Mathematics, University of Zaragoza, 50009-Zaragoza, Spain
Email: fjruiz@posta.unizar.es

DOI: 10.1090/S0002-9939-99-05128-X
PII: S 0002-9939(99)05128-X
Keywords: Maximal functions, rearrangement inequalities, Lorentz spaces
Received by editor(s): March 2, 1998
Posted: June 30, 1999
Additional Notes: The first author was partially supported by DGICYT PB94-1185.
The third author was partially supported by DGICYT and IER
Communicated by: Frederick W. Gehring
Copyright of article: Copyright 1999, American Mathematical Society

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