The limiting weak-type behaviors of the strong maximal operators

Abstract

It is well known that the weak (1, 1) estimate does not hold for the strong maximal operator, but it still enjoys certain weak $L\log L$-type norm inequality. Let $\Phi _n(t)=t(1+(\log ^+t)^{n-1})$ and the space $L_{\Phi _n}({\mathbb {R}^{n}})$ be the set of all measurable functions on ${\mathbb {R}^{n}}$ such that

$$\begin{aligned} \Vert f\Vert _{L_{\Phi _n}({\mathbb {R}^{n}})} :=\Vert \Phi _n(|f|)\Vert _{L^1({\mathbb {R}^{n}})}<\infty . \end{aligned}$$

In this paper, we introduce a new weak norm space $L_{\Phi _n}^{1,\infty }({\mathbb {R}^{n}})$, which is more larger than $L^{1,\infty }({\mathbb {R}^{n}})$ space, and establish the corresponding limiting weak-type behavior of the strong maximal operator. Similar result has been extended to the multilinear strong maximal operator.