ON EQUIVALENT NORMS TO BLOCH NORM IN Cn

Abstract

For f ∈ L2(B, dν), (cid:107) f (cid:107)BM O= (cid:103)|f |2(z) − | ˜f (z)|2. For f continuous on B, (cid:107) f (cid:107)BO= sup{w(f )(z) : z ∈ B} where w(f )(z) = sup{|f (z) − f (w)| In this paper, we will show that if f ∈ BM O, then (cid:107) f (cid:107)BO≤ M (cid:107) f (cid:107)BM O. We will also show that if f ∈ BO, then (cid:107) f (cid:107)BM O≤ M (cid:107) f (cid:107)2 BO. A holomorphic function f : B → C is called a Bloch function (f ∈ B) if (cid:107) f (cid:107)B= supz∈B Qf (z) < ∞. In this paper, we will show that if f ∈ B, then (cid:107) f (cid:107)BO≤(cid:107) f (cid:107)B. We will also show that if f ∈ BM O and f is holomorphic, then (cid:107) f (cid:107)2

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