On the Littlewood–Paley gg-function and Calderón’s area theorem

This paper contains some known and some new properties of the Littlewood–Paley gg-function. By using the Fefferman–Stein vector maximal theorem and the area theorem of Calderón, we prove the equivalence of the following four conditions, for all p>0p>0: (a)f∈Hpf∈Hp;(b)∫02π(∫01(1−r)|f′(reiθ)|2dr)p/2dθ<∞;(c)∫02π(∫01(1−r)supρ