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 $M_{\Omega }(f)(x)={\sup }_{r>0}\frac{1}{r^n}{\int }_{|x-y|<r}|\Omega (x-y)f(y)|dy.$ It was shown by Christ and Rubio de Francia that $\Vert M_{\Omega }(f){\Vert }_{L^{1,\infty }({\mathbb{R}}^n)}\le C(\Vert \Omega {\Vert }_{L\log L({\mathbb{S}}^{n-1})}+1)\Vert f{\Vert }_{L^1({\mathbb{R}}^n)}$ 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 $$\underset{\lambda \to 0^{+}}{\lim }\lambda |\{x\in {\mathbb{R}}^n:M_{\Omega }(f)(x)>\lambda \}|=n^{-1}\Vert \Omega {\Vert }_{L^1({\mathbb{S}}^{n-1})}\Vert f{\Vert }_{L^1({\mathbb{R}}^n)}.$$ 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 $\Vert M_{\Omega }{\Vert }_{L^1\to L^{1,\infty }}$, which essentially improves the upper bound $C(\Vert \Omega {\Vert }_{L\log L({\mathbb{S}}^{n-1})}+1)$ given by Christ and Rubio de Francia. As a consequence, the upper and lower bounds of $\Vert M_{\Omega }{\Vert }_{L^1\to L^{1,\infty }}$ are obtained for $\Omega \in L\log L({\mathbb{S}}^{n-1})$.

18 pages