A new proof of certain littlewood-paley inequalities

The aim of this article is to give a new proof of the Lp-inequalities for the Littlewood-Paley g*-function. Our main tool is a pointwise equality, relating a function f, and the associated functional g*(f), which has the form f2=h(f)+g*2(f), where h(f) is an explicit function. We obtain this equality as a particular case of a more general one, which is reminiscent of a well-known identity in the stochastic calculus setting, namely the Itô formula. Once the above equality is proved, Lp-estimates for g*(f) are obviously equivalent to Lp/2-estimates for h(f). We obtain these last estimates (more precisely, Hp/2-estimates for h(f) by using a slight extension of the Coifman-Meyer-Stein theorem relating the so-called tent-spaces and the Hardy spaces. We observe that our methods clearly show that the restriction p>2n/n+1 is closely related to cancellation and size properties of the gradient of the Poisson kernel.