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Let Mf denote the strong maximal function of f on R^n, that is the maximalaverage of f with respect to n-dimensional rectangles with sides parallel tothe coordinate axes. For any dimension n>1 we prove the natural endpointFefferman-Stein inequality for M and any strong Muckenhoupt weight w: w({x \inR^n: M f (x) > t}) \lesssim_{w,n} \int_{R^n} |f|/t [1 + (log^+ |f|/t)^{n-1}]Mw.  This extends the corresponding two-dimensional result of T. Mitsis.

The endpoint Fefferman-Stein inequality for the strong maximal function