Sharp weak‐type estimate for maximal operators associated to Cartesian families under an arithmetic condition.

Given a set of integers A⊂Z$A \subset \mathbb {Z}$, we consider the smallest family B$\mathcal {B}$ invariant by translations that contains the rectangles R=Ia1×⋯×Ian−1×I−(a1+⋯+an−1)$$\begin{equation*} R = I_{a_1} \times \dots \times I_{a_{n-1}} \times I_{-(a_1+\dots +a_{n-1})} \end{equation*}$$for any a1,⋯,an−1∈An−1$a_1,\dots,a_{n-1} \in A^{n-1}$ and where Ik=[0,2k]$I_k = [0,2^k]$ for k$k$ integer. We prove that if A$A$ contains arbitrary large arithmetic progression, then the maximal operator MB$M_{\mathcal {B}}$ associated to the family B$\mathcal {B}$ satisfies a weak‐type estimate of the form MBf>t≲n∫Rn|f|t1+log+|f|tn−1$$\begin{equation*} {\left|{\left\lbrace M_{\mathcal {B}}f > t \right\rbrace} \right|} \lesssim _n \int _{\mathbb {R}^n} \frac{|f|}{t} {\left(1 + \log ^+ \frac{|f|}{t} \right)}^{n-1} \end{equation*}$$but does not satisfy a weak‐type estimate of the form MBf>t≲n∫Rnϕ|f|t$$\begin{equation*} {\left|{\left\lbrace M_{\mathcal {B}}f > t \right\rbrace} \right|} \lesssim _n \int _{\mathbb {R}^n} \phi {\left(\frac{|f|}{t}\right)} \end{equation*}$$for any nonnegative convex increasing function ϕ$\phi$ such that ϕ(x)=o(x(1+log+x)n−1)$\phi (x) = o(x(1+ \log ^+ x)^{n-1})$ as x$x$ tends to infinity. [ABSTRACT FROM AUTHOR]