An equivalent norm on BMO spaces

Abstract

Let p0. A Borel function f, locally integrable in the unit ball B, is said to be a BMOp(B) function if fBMOp=supB(ar)B1V(B(ar))B(ar)f(x)−fB(ar)pdV(x)1p+ where the supremum is taken over all balls B(ar) in B, and fB(ar) is the mean value of f over B(ar) . Let (B) denote the set of harmonic functions in open unit ball B, far(x) denotes f(a+rx) for arbitrary function f. The main result of this paper is to prove the following theorem: Let u(B), p1. Then a) upBMOp=supaB0r1−ap(p−1)2n(n−2)Buar(x)−uar(0)p−2uar(x)2(2x2−n+(n−2)x2−n)dVN(x) for n3, and b) upBMOp=supaB0r1−ap(p−1)Buar(x)−uar(0)p−2uar(x)2ln1x−1+xdVN(x)for n=2.