The endpoint Fefferman-Stein inequality for the strong maximal function

Let Mnf denote the strong maximal function of f on Rn, that is the maximal average of f with respect to n-dimensional rectangles with sides parallel to the coordinate axes. For any dimension n⩾2 we prove the natural endpoint Fefferman-Stein inequality for Mn and any strong Muckenhoupt weight w:w({x∈Rn:Mnf(x)>λ})≲w,n∫Rn|f(x)|λ(1+(log+|f(x)|λ)n−1)Mnw(x)dx. This extends the corresponding two-dimensional result of T. Mitsis.