A new proof of certain Littlewood-Paley inequalities

The aim of this article is to give a new proof of the L-p-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 f(2) = 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 reminis cent of a well-known identity in the stochastic calculus setting, namely th e Ito formula. Once the above equality is proved, L-p-estimates for g(*)(f) are obviously equivalent to L-p/2-estimates for h(f). We obtain these last estimates (more precisely, H-p/2-estimates for h(f)) by using a slight ext ension of the Coifman-Meyer-Stein theorem relating the so-called rent-space s and the Hardy spaces. We observe that our methods clearly show that the r estriction p > 2n/n+1 is closely related to cancellation and size propertie s of the gradient of the Poisson kernel.