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

Abstract

It is proved that the inequality delta(X) (epsilon) >= c epsilon(p) p >= 2, where delta(X) is the modulus of convexity of X, is sufficient and necessary for the inequality integral parallel to del f(z) parallel to(p) (1-vertical bar z vertical bar)(p-1) dA(z) LT = C(parallel to f parallel to(p)(p.X) - parallel to f(0) parallel to(p)), where f is an X-valued harmonic function belonging to the Hardy space h(P)(X). The reverse inequality (1 LT p LT = 2) holds if and only if rho X (tau) LT = C tau(p) P, where rho X is the modulus of smoothness of X.