Sharp weak type estimates for a family of Córdoba 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 sidelengths of the form $s, t, 2^N st$, where $s, t > 0$ and N lies in a nonempty subset S of the natural numbers. 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

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

but does not satisfy an estimate of the form

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

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

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

Alternatively, if S is an infinite set, then the associated geometric maximal operator $M_\mathcal {B}$ satisfies the weak type estimate

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

but does not satisfy an estimate of the form

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

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

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