Sharp Inequalities for the Haar System and Fourier Multipliers

Abstract EN

A classical result of Paley and Marcinkiewicz asserts that the Haar system h=hkk≥0 on 0,1 forms an unconditional basis of Lp0,1 provided 1

That is, if ?J denotes the projection onto the subspace generated by hjj∈J (J is an arbitrary subset of ℕ), then ?JLp0,1→Lp0,1≤βp for some universal constant βp 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 Cp such that ?JχALp,∞0,1≤CpχALp0,1 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.