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. Certain bilinear interpolation results between distributional
estimates, such as that obtained for the multivariable strong maximal function,
are also proved.Comment: appeared in J. of Geometric Ana
