On the atomic decomposition for Hardy spaces

Abstract

We give an extremely easy proof of the atomic decomposition for distributions in P(R+ +1 ), 0 < p < 1. Our proof uses only properties of the nontangential maximal function u*. We then adapt our argument to give a "direct" proof of the Chang-Fefferman decomposition for JF(R 2 + XR 2 + ). I. Introduction. Let R w + +1 = {(x, y): x e R", y > 0}. For u(x, y) harmonic on R+ +1 and A > 0 define u*(x) = sup \u(t,y)\.