Sharp Weak Type Estimates for Maximal Operators associated to Rare Bases

Let $\mathcal{B}$ denote a nonempty translation invariant collection of intervals in ${\mathbb{R}}^n$ (which we regard as a rare basis), and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$M_{\mathcal{B}}f(x)=\underset{x\in R\in \mathcal{B}}{\sup }\frac{1}{\mathrm{\mid }R\mathrm{\mid }}{\int }_R\mathrm{\mid }f\mathrm{\mid }\mathrm{.}$$ We provide a sufficient condition on $\mathcal{B}$ so that the estimate $$\mathrm{\mid }\{x\in {\mathbb{R}}^n:M_{\mathcal{B}}f(x)>\alpha \}\mathrm{\mid }\le C_n{\int }_{{\mathbb{R}}^n}\frac{\mathrm{\mid }f\mathrm{\mid }}{\alpha }{(1+{\log }^{+}\frac{\mathrm{\mid }f\mathrm{\mid }}{\alpha })}^{n-1}$$ is sharp. As a corollary we obtain sharp weak type estimates for maximal operators associated to several classes of rare bases including C'ordoba, Soria and Zygmund bases.