On the Bounds of Weak $(1,1)$ Norm of Hardy-Littlewood Maximal Operator with $Llog L({mathbb S^{n-1}})$ Kernels

Let $Omegain L^1{({mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_Omega$ be the Hardy-Littlewood maximal operator associated with $Omega$ defined by $M_Omega(f)(x) = sup_{r>0}frac1{r^n}int_{|x-y|<r}|Omega(x-y)f(y)|dy.$ It was shown by Christ and Rubio de Francia that $|M_Omega(f)|_{L^{1,infty}({mathbb R^n})} le C(|Omega|_{Llog L({mathbb S^{n-1}})}+1)|f|_{L^1({mathbb R^n})}$ provided $Omegain Llog L {({mathbb S^{n-1}})}$. In this paper, we show that, if $Omegain Llog L({mathbb S^{n-1}})$, then for all $fin L^1({mathbb R^n})$, $M_Omega$ enjoys the limiting weak-type behaviors that $$lim_{lambdato 0^+}lambda|{xin{mathbb R^n}:M_Omega(f)(x)>lambda}| = n^{-1}|Omega|_{L^1({mathbb S^{n-1}})}|f|_{L^1({mathbb R^n})}.$$ This removes the smoothness restrictions on the kernel $Omega$, such as Dini-type conditions, in previous results. To prove our result, we present a new upper bound of $|M_Omega|_{L^1to L^{1,infty}}$, which essentially improves the upper bound $C(|Omega|_{Llog L({mathbb S^{n-1}})}+1)$ given by Christ and Rubio de Francia. As a consequence, the upper and lower bounds of $|M_Omega|_{L^1to L^{1,infty}}$ are obtained for $Omegain Llog L {({mathbb S^{n-1}})}$.