We establish an estimate for arbitrary geometric maximal operators in the plane: we associate to any family B composed of rectangles and invariant by translations and central dilations a geometric quantity λ_B called its analytic split and satisfying log(λ_B) ≤_p ∥M_B∥_p^p for all 1 < p < ∞, where M_B is the Hardy-Littlewood type maximal operator associated to the family B.

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