Weighted reverse weak type inequalities for the Hardy-Littlewood maximal function
The author investigates weighted weak type inequalities. Let M denote the Hardy-Littlewood maximal operator on \({\mathbb{R}}^ n\) (with respect to the n-dimensional Lebesgue measure \(| \cdot |)\) defined by non- centered cubes with sides parallel to the coordinate axes. The main result is as follows. Theorem. Let U(x) and V(x) be nonnegative functions on a cube \(Q_ 0\) such that, for any cube \(Q\subset Q_ 0\), \((1/| Q|)\int_{2Q\cap Q_ 0}U(x) dx\geq A \quad \sup_{x\in Q}\quad V(x)\) for some \(A\geq 0\) (independent of Q). Then \[ \int_{Q_ 0\cap \{x;\quad Mf(x)>\lambda \}}U(x) dx\geq \{A(300n)^{-n}/\lambda \}\int_{\{x;| f(x)| >\lambda \}}| \quad f(x)| V(x) dx \] as long as f(x) is supported on \(Q_ 0\) and \(\lambda \geq (1/| Q_ 0|)\int_{Q_ 0}| f(x) dx.\) This theorem shows that a necessary condition given by Anderson and Young is also sufficient [cf. \textit{K. F. Andersen} and \textit{W.-S. Young}, Pac. J. Math. 112, 257-264 (1984; Zbl 0494.42011)].
scientific article; zbMATH DE number 3925501