Extended Cesàro Operators from Logarithmic-Type Spaces to Bloch-Type Spaces

and Applied Analysis 3 2. Main Results and Proofs In this section, we give our main results and their proofs. Before stating these results, we need some auxiliary results, which are incorporated in the lemmas which follows. Lemma 2.1. Assume that g ∈ H Bn and μ : Bn → 0,∞ are normal. Then Tg : H∞ log → Bμ is compact if and only if Tg : H∞ log → Bμ is bounded and for any bounded sequence fk k∈N in H∞ log which converges to zero uniformly on compact subsets of Bn as k → ∞, one has ‖Tgfk‖Bμ → 0 as k → ∞. The proof of Lemma 2.1 follows by standard arguments see, e.g., Lemmas 3 in 20, 21, 29 . Hence, we omit the details. Lemma 2.2. Assume that μ : Bn → 0,∞ is normal. A closed set K in Bμ,0 is compact if and only if it is bounded and satisfies lim |z|→ 1 sup f∈K μ z ∣Rf z ∣∣ 0. 2.1 This lemma can be found in 5 , and its proof is similar to the proof of Lemma 1 in 40 . Hence, it will be omitted. The following result was proved in 8 . Lemma 2.3. There exist two functions f1, f2 ∈ H∞ log B1 such that ∣f1 z ∣∣ ∣f2 z ∣∣ ≥ ln 1 1 − |z| , z ∈ B1. 2.2 Now we are in a position to state and prove our main results. Theorem 2.4. Assume that g ∈ H Bn and μ : Bn → 0,∞ is normal. Then Tg : H∞ log → Bμ is bounded if and only if M sup z∈Bn μ z ∣Rg z ∣∣ ln e 1 − |z| < ∞. 2.3 Moreover, if Tg : H∞ log → Bμ is bounded then the following asymptotic relation holds ∥Tg ∥∥ H∞ log →Bμ sup z∈Bn μ z ∣Rg z ∣∣ ln e 1 − |z| < ∞. 2.4 Proof. Assume that 2.3 holds. Then, for any f ∈ H∞ log, we have μ z ∣R ( Tgf ) z ∣∣ μ z ∣Rg z ∣∣f z ∣∣ ≤ μ z ∣Rg z ∣∣ ( ln e 1 − |z| ) ∥f ∥∥ H∞ log . 2.5 4 Abstract and Applied Analysis In addition, it is easy to see that Tgf 0 0. Therefore we have ∥Tgf ∥∥ Bμ sup z∈Bn μ z ∣R ( Tgf ) z ∣∣ ≤ M∥f∥H∞ log 2.6 as desired. Conversely, assume that Tg : H∞ log → Bμ is bounded. For a ∈ Bn, set fa z ln e 1 − 〈z, a〉 . 2.7 It is easy to see that fa ∈ H∞ log and supa∈Bn‖fa‖H∞ log < ∞. For any b ∈ Bn, we have ∞ > ∥Tgfb ∥∥ Bμ sup z∈Bn μ z ∣R ( Tgfb ) z ∣∣ sup z∈Bn μ z ∣Rg z ∣∣fb z ∣∣ ≥ μ b ∣Rg b ∣ln e 1 − |b| , 2.8 from which 2.3 follows, moreover sup z∈Bn μ z ∣Rg z ∣ln e 1 − |z| ≤ C∥Tg ∥∥ H∞ log →Bμ . 2.9 From 2.6 and 2.9 , we see that 2.4 holds. The proof is completed. Theorem 2.5. Assume that g ∈ H Bn and μ : Bn → 0,∞ is normal. Then Tg : H∞ log → Bμ is compact if and only if lim |z|→ 1 μ z ∣Rg z ∣∣ ln e 1 − |z| 0. 2.10 Proof. Suppose that Tg : H∞ log → Bμ is compact. Let zk k∈N be a sequence in Bn such that limk⇀∞|zk| 1. Set fk z ( ln e 1 − 〈z, zk〉 )2( ln e 1 − |zk| )−1 , k ∈ N. 2.11 Abstract and Applied Analysis 5and Applied Analysis 5 It is easy to see that supk∈N‖fk‖H∞ log < ∞. Moreover fk → 0 uniformly on compact subsets of Bn as k → ∞. By Lemma 2.1, lim k→∞ ∥Tgfk ∥∥ Bμ 0. 2.12 In addition, ∥Tgfk ∥∥ Bμ sup z∈Bn μ z ∣Rg z fk z ∣∣ ≥ μ zk ∣Rg zk ∣ln e 1 − |zk| , 2.13 which together with 2.12 implies that lim k→∞ μ zk ∣Rg zk ∣ln e 1 − |zk| 0. 2.14 From the above inequality we see that 2.10 holds. Conversely, assume that 2.10 holds. From Theorem 2.4 we see that Tg : H∞ log → Bμ is bounded. In order to prove that Tg : H∞ log → Bμ is compact, according to Lemma 2.1, it suffices to show that if fk k∈N is a bounded sequence in H ∞ log converging to 0 uniformly on compact subsets of Bn, then lim k→∞ ∥Tgfk ∥∥ Bμ 0. 2.15 Let fk k∈N be a bounded sequence in H ∞ log such that fk → 0 uniformly on compact subsets of Bn as k → ∞. By 2.10 we have that for any ε > 0, there is a constant δ ∈ 0, 1 , such that μ z ∣Rg z ∣ln e 1 − |zk| < ε 2.16 whenever δ < |z| < 1. Let K {z ∈ Bn : |z| ≤ δ}. From 2.10 we see that g ∈ Bμ. Equality 2.16 along with the fact that g ∈ Bμ implies ∥Tgfk ∥∥ Bμ sup z∈Bn μ z ∣R ( Tgfk ) z ∣∣ sup z∈Bn μ z ∣Rg z fk z ∣∣ ≤ ( sup {z∈Bn:|z|≤δ} sup {z∈Bn:δ≤|z|<1} ) μ z ∣Rg z ∣∣fk z ∣∣ ≤ ∥g∥Bμsup z∈K ∣fk z ∣∣ sup {z∈Bn:δ≤|z|<1} μ z ∣g z ∣∣ ( ln e 1 − |z| ) ∥fk ∥∥ H∞ log ≤ ∥g∥Bμsup z∈K ∣fk z ∣∣ Cε. 2.17 6 Abstract and Applied Analysis Observe that K is a compact subset of Bn, so that lim k→∞ sup z∈K ∣fk z ∣∣ 0. 2.18