Littlewood–Paley inequalities in uniformly convex and uniformly smooth Banach spaces

It is proved that the inequality δX(ε)⩾cεpδX(ε)⩾cεp, p⩾2p⩾2, where δXδX is the modulus of convexity of X, is sufficient and necessary for the inequality∫D‖∇f(z)‖p(1−|z|)p−1dA(z)⩽C(‖f‖p,Xp−‖f(0)‖p), where f is an X  -valued harmonic function belonging to the Hardy space hp(X)hp(X). The reverse inequality (1