Norm Equivalence and Composition Operators between Bloch/lipschitz Spaces of the Unit Ball

For p > 0, let B(Bn) and Lp(Bn) respectively denote the p-Bloch and holomorphic p-Lipschitz spaces of the open unit ball Bn in C . It is known that B(Bn) and L1−p(Bn) are equal as sets when p ∈ (0, 1). We prove that these spaces are additionally norm-equivalent, thus extending known results for n = 1 and the polydisk. As an application, we generalize work by Madigan on the disk by investigating boundedness of the composition operator Cφ from Lp(Bn) to Lq(Bn). 1. Background and Terminology Let n ∈ N, and suppose that D is a domain in C. Denote the linear space of complex-valued, holomorphic functions on D by H(D). If X is a linear subspace of H(D) and φ : D → D is holomorphic, then one can define the linear operator Cφ : X → H(D) by Cφ(f) = f ◦ φ for all f ∈ X . Cφ is called the composition operator induced by φ. The problem of relating properties of symbols φ and operators such as Cφ that are induced by these symbols is of fundamental importance in concrete operator theory. However, efforts to obtain characterizations of self-maps that induce bounded composition operators on many function spaces have not yielded completely satisfactory results in the several-variable case, leaving a wealth of basic, open problems. In this paper, we try to make progress toward the goal of characterizing the holomorphic self-maps of the open unit ball Bn in C n that induce bounded composition operators between holomorphic pLipschitz spaces Lp(Bn) for 0 < p < 1 by translating the problem to (1 − p)-Bloch spaces B(Bn) via an auxiliary Hardy/Littlewood-type norm-equivalence result of potential independent interest. The function-theoretic characterization of analytic self-maps of B1 that induce bounded composition operators on Lp(B1) for 0 < p < 1 is due to K. Madigan [Mad], and the case of the open unit polydisk ∆ was handled in a joint paper by the present authors with Z. Zhou [CSZ], in which a full characterization of the holomorphic self-maps 1 2 DANA CLAHANE AND STEVO STEVIĆ φ of ∆ that induce bounded composition operators between Lp(∆ ) and Lq(∆ ), and, more generally, between Bloch spaces B(∆) and B(∆), is obtained for p, q ∈ (0, 1), along with analogous characterizations of compact composition operators between these spaces. Although our main results concerning composition operators, Theorem 2.1 and Corollary 2.2, are not full characterizations, they do generalize Madigan’s result for the disk to Bn; on the other hand, we obtain a complete Hardy-Littlewood norm-equivalence result for p-Bloch and (1− p)-Lipschitz spaces of Bn for all n ∈ N. This norm-equivalence result should lead to an eventual extension to Bn of the characterizations of bounded composition operators established on B1 in [Mad] and on ∆ in [CSZ]. Most of our several complex variables notation is adopted from [Ru]. If z = (z1, ..., zn) and w = (w1, ..., wn) are points in C , then we define a complex inner product by 〈z, ω〉 = ∑n k=1 zkw̄k and put |z| := √ 〈z, z〉. We call Bn := {z ∈ C n : |z| < 1} the (open) unit ball of C. Let p ∈ (0,∞). The p-Bloch space B(Bn) consists of the set of all f ∈ H(Bn) with the property that there is an M ≥ 0 such that b(f, z, p) := (1− |z|)|∇f(z)| ≤M for all z ∈ Bn B(Bn) is a Banach space with norm ||f ||Bp given by ||f ||Bp = |f(0)|+ sup z∈Bn b(f, z, p). The little p-Bloch space B 0(Bn) is defined as the closed subspace of B(Bn) consisting of the functions that satisfy lim z→∂Bn (1− |z|)|∇f(z)| = 0. For p ∈ (0, 1), Lp(Bn) denotes the holomorphic p-Lipschitz space, which is the set of all f ∈ H(Bn) such that for some C > 0, (1) |f(z)− f(w)| ≤ C|z − w| for every z, w ∈ Bn. These functions extend continuously to Bn (cf. [CSZ, Lemma 4.4]). Therefore, if A(Bn) is the ball of algebra [Ru, Ch. 6], then Lp(Bn) = Lipp(Bn) ∩ A(Bn), where Lipp(Bn) is the set of all f : Bn → C satisfying Equation (1) for some C > 0 and all z ∈ Bn. Lp(Bn) is endowed with a complete norm || · ||Lp that is given by (2) ||f ||Lp = |f(0)|+ sup z 6=w:z,w∈Bn { |f(z)− f(w)| |z − w|p }