The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure

Volume 11, Issue 11, pp 1271--1281 Publication Date: September 05, 2018       Article History


Wei Ding - School of Sciences, Nantong University, Nantong 226007, P. R. China.


Let \(d\mu(x_1, \ldots, x_n)=d\mu_1(x_1)\cdots d\mu_n(x_n)\) be a product measure which is not necessarily doubling in \(\mathbb{R}^n\) (only assuming \(d\mu_i\) is doubling on \(\mathbb{R}\) for \(i=2, \ldots, n\)), and \(M_{d\mu}^n\) be the strong maximal function defined by \[ M_{d\mu}^n f(x)=\sup_{x\in R\in \mathcal{R}}\frac{1}{\mu(R)}\int_{R}|f(y)|d\mu(y),\] where \(\mathcal{R}\) is the collection of rectangles with sides parallel to the coordinate axes in \(\mathbb{R}^n\), and \(\omega,\nu\) are two nonnegative functions. We give a sufficient condition on \(\omega,\nu\) for which the operator \(M_{d\mu}^n\) is bounded from \(L(1+(\log^{+})^{n-1})(\nu d\mu)\) to \(L^{1,\infty}(\omega d\mu)\). By interpolation, \(M^{n}_{d\mu}\) is bounded from \(L^{p}(\nu d\mu)\) to \(L^{p}(\omega d\mu)\), \(1<p<\infty\).