The limiting weak type behaviors and The lower bound for a new weak $L\log L$ type norm of strong maximal operators

It is well known that the weak ($1,1$) bounds doesn’t hold for the strong maximal operators, but it still enjoys certain weak $L\log L$ type norm inequality. Let ${\mathrm{\Phi }}_n(t)=t(1+({\log }^{+}t{)}^{n-1})$ and the space $L_{{\mathrm{\Phi }}_n}({\mathbb{R}}^n)$ be the set of all measurable functions on ${\mathbb{R}}^n$ such that $|f|{L{\Phi_n}({\mathbb R^{n}})} :=|\Phi_n(|f|)|_{L^1({\mathbb R^{n}})}<\infty$. In this paper, we introduce a new weak norm space $L_{{\mathrm{\Phi }}_n}^{1,\mathrm{\infty }}({\mathbb{R}}^n)$, which is more larger than $L^{1,\mathrm{\infty }}({\mathbb{R}}^n)$ space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that $\max \{2^n((n-1)!{)}^{-1},1\}$ is a lower bound for the best constant of the $L_{{\mathrm{\Phi }}_n}\to L_{{\mathrm{\Phi }}_n}^{1,\mathrm{\infty }}$ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.