A classical result of Paley and Marcinkiewicz asserts that the Haar systemh=hkk≥0on0,1forms an unconditional basis ofLp0,1provided1<p<∞. That is, if𝒫Jdenotes the projection onto the subspace generated byhjj∈J(Jis an arbitrary subset ofℕ), then𝒫JLp0,1→Lp0,1≤βpfor some universal constantβpdepending only onp. The purpose of this paper is to study related restricted weak-type bounds for the projections𝒫J. Specifically, for any1≤p<∞we identify the best constantCpsuch that𝒫JχALp,∞0,1≤CpχALp0,1for everyJ⊆ℕand any Borel subsetAof0,1. 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.
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