Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Abstract

Let (hk)k≥0 be the Haar system on [0, 1]. We show that for any vectors ak from a separable Hilbert space H and any ek ∈ [−1, 1], k = 0, 1, 2, . . ., we have the sharp inequality ∣∣∣∣ ∣∣∣∣ n ∑ k=0 ekakhk ∣∣∣∣ ∣∣∣∣ W ([0,1]) ≤ 2 ∣∣∣∣ ∣∣∣∣ n ∑ k=0 akhk ∣∣∣∣ ∣∣∣∣ L∞([0,1]) , n = 0, 1, 2, . . . , where W ([0, 1]) is the weak-L∞ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ||Y ||W (Ω) ≤ 2||X||L∞(Ω), where X, Y stand for H-valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.