Sharp weak type estimates for a family of Cardoba bases

Let B be a collection of rectangular parallelepipeds in R-3 whose sides are parallel to the coordinate axes and such that 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-B satisfies the weak type estimate vertical bar{x is an element of R-3 : M(B)f(x) > alpha}vertical bar <= C integral(R3)vertical bar f vertical bar alpha(1 + log(+)vertical bar f vertical bar/alpha) but does not satisfy an estimate of the form vertical bar{x is an element of R-3 : M(B)f(x) > alpha}vertical bar <= C integral(R3) phi vertical bar f vertical bar/alpha) for any convex increasing function phi : [0, infinity) -> [0, infinity) satisfying the condition lim(x ->infinity) phi(x)/x(log(1 + x)) = 0. Alternatively, if S is an infinite set, then the associated geometric maximal operator M-B satisfies the weak type estimate vertical bar{x is an element of R-3 : M(B)f(x) > alpha}vertical bar <= C integral(R3) vertical bar f vertical bar/alpha (1 + log+ vertical bar f vertical bar/alpha)(2) but does not satisfy an estimate of the form vertical bar{x is an element of R-3 : M(B)f(x) > alpha}vertical bar <= C integral(R3) phi(vertical bar f vertical bar/alpha) for any convex increasing function phi : [0, infinity) -> [0, infinity) satisfying the condition lim(x ->infinity) phi(x)/x(log(1 + x))(2) = 0.