Sharp Inequalities for the Haar System and Fourier Multipliers

Adam Osȩkowski

doi:10.1155/2013/646012

Abstract

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.

Highlights

Our motivation comes from a very natural question about h =(hn)n≥0, the Haar system on [0, 1]
Instead of transforms with values in [0, 1], one can work under the less restrictive assumption of nonsymmetric differential subordination of martingales
By an application of the results of Burkholder and Marcinkiewicz [3], the best constants in the inequalities for the Haar system are the same as those in the corresponding estimates for discrete-time martingales (roughly speaking, any martingale pair (f, g), where g is a transform of f, can be appropriately embedded into a pair consisting of a dyadic martingale and its transform)

Summary

Introduction

(hn)n≥0, the Haar system on [0, 1]. Recall that this collection of functions is given by h0 = [0, 1) , h1. The Haar system is a martingale difference sequence with respect to its natural filtration (on the probability space being Lebesgue’s unit interval) and so is (akhk)k≥0, for given fixed real numbers a0, a1, a2, . Instead of transforms with values in [0, 1], one can work under the less restrictive assumption of nonsymmetric differential subordination of martingales (for the necessary definitions and the precise statement of our results, we refer the reader to Section 2). This setting has the advantage of being more convenient for applications, which constitute the second half of the paper.

A Martingale Inequality

Applications to Fourier Multipliers

Laminates