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

Abstract: Let $\Omega\in L^1{({\mathbb S^{n-1}})}$, 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 $|M_\Omega(f)|{L^{1,\infty}({\mathbb R^n})} \le C(|\Omega|{L\log L({\mathbb S^{n-1}})}+1)|f|_{L^1({\mathbb R^n})}$ provided $\Omega\in L\log L {({\mathbb S^{n-1}})}$. In this paper, we show that, if $\Omega\in L\log L({\mathbb S^{n-1}})$, then for all $f\in L^1({\mathbb R^n})$, $M_\Omega$ enjoys the limiting weak-type behaviors that $$\lim_{\lambda\to 0^+}\lambda|{x\in{\mathbb R^n}:M_\Omega(f)(x)>\lambda}| = n^{-1}|\Omega|_{L^1({\mathbb S^{n-1}})}|f|_{L^1({\mathbb R^n})}.$$ 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 $|M_\Omega|{L^1\to L^{1,\infty}}$, which essentially improves the upper bound $C(|\Omega|{L\log L({\mathbb S^{n-1}})}+1)$ given by Christ and Rubio de Francia. As a consequence, the upper and lower bounds of $|M_\Omega|_{L^1\to L^{1,\infty}}$ are obtained for $\Omega\in L\log L {({\mathbb S^{n-1}})}$.