Article

Journal of Geometric Analysis

, Volume 21, Issue 1, pp 118-149

First online:

The Multilinear Strong Maximal Function

  • Loukas GrafakosAffiliated withDepartment of Mathematics, University of Missouri Email author 
  • , Liguang LiuAffiliated withDepartment of Mathematics, School of Information, Renmin University of China
  • , Carlos PérezAffiliated withDepartamento De Análisis Matemático, Facultad de Matemáticas, Universidad De Sevilla
  • , Rodolfo H. TorresAffiliated withDepartment of Mathematics, University of Kansas

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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