SHARP WEAK TYPE ESTIMATES FOR A FAMILY OF ZYGMUND BASES

Let B be the collection of rectangular parallelepipeds in R3 whose sides are parallel to the coordinate axes and such that B consists of parallelepipeds with side lengths of the form s, 2j s, t, where s, t > 0 and j lies in a nonempty subset S of the integers. In this paper, we prove the following: If S is a finite set, then the associated geometric maximal operator MB satisfies the weak type estimate l{ x ∈ R3 : MBf(x) > α l ≤ C l R3 |f| α ( 1 + log+ |f| α ) but does not satisfy an estimate of the form l { x ∈ R3 : MBf(x) > α l ≤ C ( R3 φ l |f| α ) for any convex increasing function φ : [0,∞) → [0,∞) satisfying the condition lim x→∞ φ(x) x(log(1 + x)) = 0. On the other hand, if S is an infinite set, then the associated geometric maximal operator MB satisfies the weak type estimate l ( x ∈ R3 : MBf(x) > α l ≤ C l R3 |f| α l 1 + log+ |f| α ) 2 but does not satisfy an estimate of the form l ( x ∈ R3 : MBf(x) > α l ≤ C l R3 φ ( |f| α ) for any convex increasing function φ : [0,∞) → [0,∞) satisfying the condition lim x→∞ φ(x) x(log(1 + x))2 = 0.