Journal of Geometric Analysis

, Volume 21, Issue 1, pp 118–149 | Cite as

The Multilinear Strong Maximal Function

  • Loukas Grafakos
  • Liguang Liu
  • Carlos Pérez
  • Rodolfo H. Torres
Article

Abstract

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Bilinear interpolation results between distributional estimates, such as those satisfied by the multivariable strong maximal function, are also proved.

Keywords

Maximal operators Weighted norm inequalities Multilinear singular integrals Calderón–Zygmund theory Commutators 

Mathematics Subject Classification (2000)

42B20 42B25 46B70 47B38 

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Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  • Loukas Grafakos
    • 1
  • Liguang Liu
    • 2
  • Carlos Pérez
    • 3
  • Rodolfo H. Torres
    • 4
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of Mathematics, School of InformationRenmin University of ChinaBeijingChina
  3. 3.Departamento De Análisis Matemático, Facultad de MatemáticasUniversidad De SevillaSevillaSpain
  4. 4.Department of MathematicsUniversity of KansasLawrenceUSA

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