Weighted Differentiation Composition Operators from the Mixed-Norm Space to the nth Weigthed-Type Space on the Unit Disk

and Applied Analysis 3 2. Auxiliary Results Here we quote some auxiliary results which will be used in the proofs of the main results. The first lemma can be proved in a standard way see, e.g., in 13, Proposition 3.11 or in 15, Lemma 3 . Lemma 2.1. Assume that m ∈ N0, n ∈ N, p, q > 0, γ > −1, φ is an analytic self-map of D and u ∈ H D . Then the operator D φ,u : Hp,q,γ → W n μ is compact if and only if D φ,u : Hp,q,γ → W n μ is bounded and for any bounded sequence fk k∈N in Hp,q,γ which converges to zero uniformly on compact subsets of D, D φ,ufk → 0 inW n μ as k → ∞. The next lemma is known, but we give a proof of it for the benefit of the reader. Lemma 2.2. Assume that n ∈ N0, 0 < p, q < ∞, −1 < γ < ∞ and f ∈ Hp,q,γ . Then there is a positive constant C independent of f such that ∣ ∣ ∣f n z ∣ ∣ ∣ ≤ C ∥ ∥f ∥ ∥ Hp,q,γ ( 1 − |z| ) γ 1 /q 1/p n . 2.1 Proof. By the monotonicity of the integral means, using the well-known asymptotic formula ∫1 0 M q p ( f, r ) 1 − r dr ∣∣f 0 ∣q ∫1 0 M q p ( f n , r ) 1 − r γ dr, 2.2 and Theorem 7.2.5 in 33 , we have that ∥ ∥f ∥ ∥ q Hp,q,γ ≥ ∫1 1 |z| /2 M q p ( f n , r ) 1 − r γ dr ≥ CM p ( f n , 1 |z| 2 )( 1 − |z| )γ 1 nq ≥ C ( 1 − |z| )γ 1 nq q/p∣ ∣ ∣f n z ∣ ∣ ∣ q , 2.3 from which the result follows. The following lemma can be found in 34 . Lemma 2.3. For β > −1 and m > 1 β one has