Haar multipliers, paraproducts, and weighted inequalities

Abstract

In this paper we present a brief survey on Haar multipliers, dyadic paraproducts, and recent results on their applications to deduce scalar and vector valued weighted inequalities. We present a new proof of the boundedness of a Haar multiplier in Lp(ℝ). The proof is based on a stopping time argument suggested by P. W. Jones for the case p = 2, that it is adapted to the case 1 < p < ∞ using an new version of Cotlar’s Lemma for Lp. We then prove some weighted inequalities for simple dyadic operators.