Sharp Weak Type Estimates for a Family of Soria 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 ${\mathcal {B}}$ contains parallelepipeds with side lengths of the form $s, \frac{2^N}{s} , t$, where $s, t > 0$ and N lies in a nonempty subset S of the natural numbers. We show that if S is an infinite set, then the associated geometric maximal operator $M_{\mathcal {B}}$ satisfies the weak type estimate

$$\begin{aligned} \left| \left\{ x \in {\mathbb {R}}^3 : M_{{\mathcal {B}}}f(x) > \alpha \right\} \right| \le C \int \nolimits _{{\mathbb {R}}^3} \frac{|f|}{\alpha } \left( 1 + \log ^+ \frac{|f|}{\alpha }\right) ^{2}, \end{aligned}$$

but does not satisfy an estimate of the form

$$\begin{aligned} \left| \left\{ x \in {\mathbb {R}}^3 : M_{{\mathcal {B}}}f(x) > \alpha \right\} \right| \le C \int \nolimits _{{\mathbb {R}}^3} \phi \left( \frac{|f|}{\alpha }\right) \end{aligned}$$

for any convex increasing function $\phi : \mathbb [0, \infty ) \rightarrow [0, \infty )$ satisfying the condition

$$\begin{aligned} \lim _{x \rightarrow \infty }\frac{\phi (x)}{x (\log (1 + x))^2} = 0\;. \end{aligned}$$