Green's formula and the Hardy-Stein identities

Abstract

This is a collection of some known and some new facts on the holomorphic and the harmonic version of the Hardy-Stein identity as well as on their extensions to the real and the complex ball. For example, we prove that if f is holomorphic on the unit disk D, then ??f ??Hp = ?f(0)?p + ?D?f'(z)? p-2 ?f'(z)?2(1-?z?) dA(z), (?) where Hp is the p-Hardy space, which improves a result of Yamashita [Proc. Amer. Math. Soc. 75 (1979), no. 1, 69-72]. An extension of (?) to the unit ball of Cn improves results of Beatrous an Burbea [Kodai Math. J. 8 (1985), 36-51], and of Stoll [J. London Math. Soc. (2) 48 (1993), no. 1, 126-136]. We also prove the analogous result for the harmonic Hardy spaces. The proofs of known results are shorter and more elementary then the existing ones, see Zhu [Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005, Ch. IV]. We correct some constants in that book and in a paper of Jevtic and Pavlovic [Publ. Inst. Math. (Beograd) (N.S.) 64(78) (1998), 36-52].