The weighted weak type inequality for the strong maximal function
Let \(f\) be a locally integrable function on \(\mathbb R^2\). The strong maximal function \(M\) is defined by \(Mf(x)=\sup_{R}| R| ^{-1}\int_{R}| f(y)| \,dy\), where the supremum is tsken over all rectangles \(R\subset \mathbb R^2\) with sides parallel to the coordinate axes, such that \(x\in R\). A nonnegative valued function \(w\) is said to be a strong \(A_p\) weight, \(p>1\), if there is a constant \(C\) such that for all rectangles \(R\) it holds \[ (| R| ^{-1}\int_{R}| w(y)| \,dy)(| R| ^{-1}\int_{R}| w(y)| ^{-1/(p-1)}\,dy)^{p-1}\leq C. \] The author states the following: Let \(w\) be a strong \(A_r\) weight for some \(r>1\). Then there exists \(C_1>0\) such that \(w(\{Mf>\lambda\})\leq C_1 \int \frac{| f| }{\lambda}\bigl(1+\log^+\frac{| f| }{\lambda}\bigr)Mw\, dx\), \(\lambda>0\). This is a weighted version of a classical result of \textit{B. Jessen, J. Marcinkiewicz} and \textit{A. Zygmund} [Fundam. Math. 25, 217--234 (1935; Zbl 0012.05901 and JFM 61.0255.01)]. And this extends a weighted result of \textit{R. J. Bagby} and \textit{D. S. Kurtz} for \(A_1\) weight [J. Anal. Math. 44, 21--31 (1985; Zbl 0583.42008)], and by interpolation this implies the same result for weighted \(L^p\) inequalities by \textit{B.~Jawerth} [Am. J. Math. 108, 361--414 (1986; Zbl 0608.42012)].
scientific article; zbMATH DE number 5148895