Weighted Iterated Radial Composition Operators between Some Spaces of Holomorphic Functions on the Unit Ball

and Applied Analysis 3 where dV z is the Lebesgue volume measure on B. Some facts on mixed-norm spaces in various domains in C can be found, for example, in 6–8 see also the references therein . For 0 < p < ∞ the Hardy space H B H consists of all f ∈ H B such that ∥ ∥f ∥ ∥ Hp : sup 0 0, φ is a holomorphic self-map of B, u ∈ H B , φ is normal and μ is a weight. Then the operator Ru,φ : H p, q, φ → H∞ μ is compact if and only if for every bounded sequence fk k∈N ⊂ H p, q, φ converging to 0 uniformly on compacts of B as k → ∞, one has lim k→∞ ∥ ∥ ∥Ru,φfk ∥ ∥ ∥ H∞ μ 0. 2.1 4 Abstract and Applied Analysis The following lemma is a slight modification of Lemma 2.5 in 8 and is proved similar to Lemma 1 in 35 . Lemma 2.2. Assume μ is a normal weight. Then a closed set K in H∞ μ,0 is compact if and only if it is bounded and lim |z|→ 1 sup f∈K μ z ∣ ∣f z ∣ ∣ 0. 2.2 The following lemma is folklore and in the next form it can be found in 36 . Lemma 2.3. Assume that 0 < p, q < ∞, φ is normal, and m ∈ N. Then for every f ∈ H B the following asymptotic relationship holds: ∫1 0 M p q ( f, r )φ r 1 − r dr ∣ ∣f 0 ∣ ∣p ∫1 0 M p q ( Rf, r ) 1 − r mp φ p r 1 − r dr. 2.3 Lemma 2.4. Assume that m ∈ N, 0 < p, q < ∞, φ is normal and f ∈ H p, q, φ . Then, there is a positive constant C independent of f such that ∣ ∣Rf z ∣ ∣ ≤ C∥∥f∥∥H p,q,φ |z| φ |z| ( 1 − |z| )n/q m . 2.4 Proof. Let g Rm−1f and z ∈ B. By the definition of the radial derivative, the Cauchy-Schwarz inequality and the Chauchy inequality, we have that ∣ ∣Rg z ∣ ∣ ≤ |z|∣∇g z ∣ ≤ C|z| supB z,1−|z| /4 ∣ ∣g w ∣ ∣ 1 − |z| . 2.5 From 2.3 with m → m − 1 we easily obtain the following inequality see, e.g., 8, Lemma 2.1 : ∣ ∣g z ∣ ∣ ≤ C ∥ ∥f ∥ ∥ H p,q,φ φ |z| ( 1 − |z| )n/q m−1 . 2.6 From 2.5 and 2.6 and the asymptotic relations 1 − |w| 1 − |z|, φ |z| φ |w| , for w ∈ B ( z, 1 − |z| 2 ) , 2.7 inequality 2.4 follows. Abstract and Applied Analysis 5 Lemma 2.5. Letand Applied Analysis 5 Lemma 2.5. Let fa,s z 1 1 − 〈z, a〉 s , z ∈ B. 2.8