Sharp weak-type estimates for maximal operators associated to rare bases

Let B denote a nonempty translation invariant collection of intervals in R-n (which we regard as a rare basis), and define the associated geometric maximal operator M-B byM(B)f(x) = sup (x is an element of R is an element of B) 1 / |R| integral(R) |f|We provide a sufficient condition on B so that the estimate|{x is an element of R-n : M(B)f (x) > alpha}| <= C-n integral(Rn) |f | / alpha (1 + log(+) |f | / alpha) (n-1)is sharp. As a corollary, we obtain sharp weak-type esti-mates for maximal operators associated to several classes of rare bases including unions of one-parameter infinite families of Cordoba, Soria, and Zygmund-type bases.