L(log L) spaces and weights for the strong maximal function (Q1070153)

Relations between weighted integral inequalities for the strong maximal function \(M_ nf\) on \({\mathbb{R}}^ m\) and the corresponding weight function are given. The main result shows that one has \[ w(\{x\in {\mathbb{R}}^ m: M_ nf(x)>\lambda \})\leq \int \Phi (| f(x)| /\lambda)w(x)dx \] for all \(\lambda >0\), whenever w belongs to the \(A_ p\)-class with respect to the system of all rectangles parallel to the axes, and \(\Phi (t):=Ct^ p(1+\log_+t)^{n-1},\) for a suitable constant C. For \(1<p<\infty\) the \(A_ p\)-condition is also necessary and sufficient for \(M_ n\) to be bounded on \(L^ p(w)\), however, unlike the classical case of the Hardy-Littlewood maximal function there is a gap between the necessary and the sufficient conditions in the case \(p=1\).