# 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
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### Authors

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

### Abstract

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 $L^{p}(\omega d\mu)$, $1<p<\infty$.

### Keywords

• Fefferman-Stein inequality
• strong maximal function
• nondoubling measure
• $A^\infty$ weights
• reverse Holder's inequality

•  42B25
•  42B37

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