Two presentations of a weak type inequality for geometric maximal operators

Let φ: [ 0, ∞) → [ 0, ∞) be a Young's function satisfying the Δ₂ -condition and let M_B be the geometric maximal operator associated to a homothecy invariant basis B acting on measurable functions on Rⁿ. Let Q be the unit cube in Rⁿ and let L^φ(Q) be the Orlicz space associated to φ with the norm given by {equation presented} We show that M_B satisfies the weak type estimate {equation presented} for all measurable functions f on Rⁿ and α > 0 if and only if M_B satisfies the weak type estimate {equation presented} for all measurable functions f supported on Q and α > 0. As a consequence of this equivalence, we prove that if φ satisfies the above conditions and B is a homothecy invariant basis differentiating integrals of all measurable functions f on R_n such that {equation presented}, then the associated maximal operator M_B satisfies both of the above weak type estimates.