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We prove the following superexponential distribution inequality: for any integrable g on [0, 1) d with zero average, and any λ > 0 , where S(g) denotes the classical dyadic square function in [0, 1) d . The estimate is sharp when dimension d tends to infinity in the sense that the constant 2 d in the denominator cannot be replaced by C2 d with 0 < C < 1 independent of d when d → ∞. For d = 1 this is a classical result of Chang-Wilson-Wolff ; however, in the case d > 1 they work with specialdoi:10.1090/tran/7795 fatcat:7ckkehzj5vb6dkep7jlzp2pfoy