Hardy-Sobolev spaces and algebras of holomorphic functions on the unit ball in ℂ n

Abstract

For 0 < p ≤ ∞ let H p (𝔹 n ) denote the usual Hardy space of holomorphic functions on the unit ball 𝔹 n in ℂ n , n ≥ 2. If f is a holomorphic function on 𝔹 n , the radial derivative ℛ f is defined by and for m = 2, 3, … , the mth order radial derivative ℛ m is defined by ℛ m f = ℛ(ℛ m − 1 f). The Hardy-Sobolev space of order m is defined as the set of holomorphic functions f on 𝔹 n for which ℛ m f ∈ H p (𝔹 n ). The main result is as follows: Theorem: Let m ∈ {1, 2, 3, …}. If a. m ≥ n and , or b. 1 ≤ m < n (n ≥ 2) and , then is an algebra. Furthermore, ∥ ∥ p,m,λ defined for p ≥ 1 by where is a sequence of positive numbers satisfying is a Banach algebra norm on whenever p, m with p ≥ 1 satisfy one of (a) or (b) above. In the article we also prove that if m > n and , then is a p-Banach algebra under a suitable p-norm on . By examples it is shown that the results are best possible.