Sharp Weak Type Estimates for a Family of Zygmund Bases

Abstract

Let $\mathcal{B}$ be a collection of rectangular parallelepipeds in ${\mathbb{R}}^3$ whose sides are parallel to the coordinate axes and such that $\mathcal{B}$ consists of parallelepipeds with side lengths of the form $s,2^{j}s,t$, where $s,t>0$ and $j$ lies in a nonempty subset $S$ of the integers. In this paper, we prove the following: If $S$ is a finite set, then the associated geometric maximal operator $M_{\mathcal{B}}$ satisfies the weak type estimate of the form $$\left|\left\{x\in {\mathbb{R}}^3:M_{\mathcal{B}}f(x)>\alpha \right\}\right|\le C{\int }_{{\mathbb{R}}^3}\frac{|f|}{\alpha }\left(1+{\log }^{+}\frac{|f|}{\alpha }\right)$$ but does not satisfy an estimate of the form $$\left|\left\{x\in {\mathbb{R}}^3:M_{\mathcal{B}}f(x)>\alpha \right\}\right|\le C{\int }_{{\mathbb{R}}^3}\varphi \left(\frac{|f|}{\alpha }\right)$$ for any convex increasing function $\varphi :[0,\infty )\to [0,\infty )$ satisfying the condition $$\underset{x\to \infty }{\lim }\frac{\varphi (x)}{x(\log (1+x))}=0.$$ On the other hand, if $S$ is an infinite set, then the associated geometric maximal operator $M_{\mathcal{B}}$ satisfies the weak type estimate $$\left|\left\{x\in {\mathbb{R}}^3:M_{\mathcal{B}}f(x)>\alpha \right\}\right|\le C{\int }_{{\mathbb{R}}^3}\frac{|f|}{\alpha }{\left(1+{\log }^{+}\frac{|f|}{\alpha }\right)}^2$$ but does not satisfy an estimate of the form $$\left|\left\{x\in {\mathbb{R}}^3:M_{\mathcal{B}}f(x)>\alpha \right\}\right|\le C{\int }_{{\mathbb{R}}^3}\varphi \left(\frac{|f|}{\alpha }\right)$$ for any convex increasing function $\varphi :[0,\infty )\to [0,\infty )$ satisfying the condition $$\underset{x\to \infty }{\lim }\frac{\varphi (x)}{x(\log (1+x){)}^2}=0.$$