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

It is well known that the weak ($1,1$) bounds doesnt hold for the strong maximal operators, but it still enjoys certain weak $Llog L$ 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 $|f|_{L_{Phi_n}({mathbb R^{n}})} :=|Phi_n(|f|)|_{L^1({mathbb R^{n}})}<infty$. In this paper, we introduce a new weak norm space $L_{Phi_n}^{1,infty}({mathbb R^{n}})$, which is more larger than $L^{1,infty}({mathbb R^{n}})$ 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 $L_{Phi_n}to L_{Phi_n}^{1,infty}$ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.