Sharp inequalities for the Haar system and fourier multipliers

A classical result of paley and marcinkiewicz asserts that the haar system h = (hk)k≥ 0 on 0,1 forms an unconditional basis of l p 0,1 provided 1 < p < ∞. that is,if ℘ j denotes the projection onto the subspace generated by (hj)j ∈ j (j is an arbitrary subset of ℕ),then ℘ jl p 0,1 → l p 0,1 ≤ β p for some universal constant β p depending only on p. the purpose of this paper is to study related restricted weak-type bounds for the projections ℘ j. specifically,for any 1 ≤ p < ∞ we identify the best constant c p such that ℘ j χ a l p,∞ 0,1 ≤ c p χ a l p 0,1 for every j ⊆ ℕ and any borel subset a of 0,1. in fact,we prove this result in the more general setting of continuous-time martingales. as an application,a related estimate for a large class of fourier multipliers is established. © 2013 adam oskowski.