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

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1 Abstract and Applied Analysis Volume 2010, Article ID , 15 pages doi: /2010/ Research Article Weighted Differentiation Composition Operators from the Mixed-Norm Space to the nth Weigthed-Type Space on the Unit Disk Stevo Stević Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III, Belgrade, Serbia Correspondence should be addressed to Stevo Stević, Received 26 March 2010; Accepted 9 May 2010 Academic Editor: Narcisa C. Apreutesei Copyright q 2010 Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The boundedness and compactness of the weighted differentiation composition operator from the mixed-norm space to the nth weighted-type space on the unit disk are characterized. 1. Introduction Throughout this paper D will denote the open unit disk in the complex plane C, HD the class of all holomorphic functions on D, andh H D the space of all bounded holomorphic functions on D with the norm f z D fz. The mixed norm space H p,q,γ H p,q,γ D, 0<p,q<, 1 <γ<, consists of all f HD such that f q H p,q,γ 1 0 M q p f, r 1 r γ dr <, 1.1 where 1 2π 1/p M p f, r f re iθ p dθ π 0

2 2 Abstract and Applied Analysis A positive continuous function on D is called weight.letzbe a weight and n N 0. The nth weighted-type space on D, denoted by W n D, consists of all f HD such that b n W D f : zf n z <. z D 1.3 The space was recently introduced by this author in 1 as an extension of several weightedtype spaces which attracted a lot of attention in last few decades. For instance, when n 0, the space becomes the weighted-type space H D see, e.g., 2 4, when n 1, the Blochtype space B D see, e.g., 5 7, andforn 2, the Zygmund-type space Z D. Some information on Zygmund-type spaces on D and some operators on them can be found, for example, in 8 10 and on the unit ball, for example, in 11, 12. The quantity b n W f is a seminorm on the nth weighted-type space Wn D D and a norm on W n D/P n 1, where P n 1 is the set of all polynomials whose degrees are less than or equal to n 1. A natural norm on the nth weighted-type space is introduced as follows: fw n D n 1 f j 0 b n W j0 D f. 1.4 With this norm the nth weighted-type space becomes a Banach space. The little nth weighted-type space, denoted by W n,0d, is a closed subspace of Wn D consisting of those f for which f lim z n z 0. z An analytic self-map ϕ : D D induces the composition operator C ϕ on HD, defined by C ϕ fz fϕz for f HD see, e.g., 8, Let ϕ be an analytic self-map of D, u HD, andm N. Then the weighted differentiation composition operator, denoted by D m ϕ,u, is defined on HD by D m ϕ,ufz uzf m ϕz, f HD. 1.6 Recently there has been some interest in studying some particular cases of operator D m ϕ,u see, e.g., For some other products of linear operators on spaces of holomorphic functions see also recent papers 11, Here we study the boundedness and compactness of the operator D m ϕ,u from H p,q,γ to nth weighted-type spaces, where n N. Throughout this paper, constants are denoted by C; they are positive and may differ from one occurrence to the other. The notation A B means that there is a positive constant C such that B/C A CB.

3 Abstract 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 N 0, n N, p, q > 0, γ > 1, ϕ is an analytic self-map of D and u HD. Then the operator Dϕ,u m : H p,q,γ W n is compact if and only if Dϕ,u m : H p,q,γ W n is bounded and for any bounded sequence f k k N in H p,q,γ which converges to zero uniformly on compact subsets of D, Dϕ,uf m k 0 in W n 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 N 0, 0 <p,q<, 1 <γ< and f H p,q,γ. Then there is a positive constant C independent of f such that fhp,q,γ f n z C 1 z 2. γ1/q1/pn 2.1 Proof. By the monotonicity of the integral means, using the well-known asymptotic formula 1 M q p 0 f, r 1 r γ dr f0 q 1 M q p 0 f n,r 1 r γnq dr, 2.2 and Theorem in 33, we have that 1 f q H p,q,γ M q p f n,r 1 r γnq dr 1 z /2 CM q p C f n, 1 z 1 z 2 γ1nq 2 1 z 2 γ1nqq/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 r β 1 ρr m dr C 1 ρ 1β m, 0 <ρ< A proof of the next lemma can be found in 35, Lemma 2.3.

4 4 Abstract and Applied Analysis Lemma 2.4. Assume a>0 and a a 1 a n 1 aa 1 a 1a 2 a n 1a n D n a. 2.5 n 2 n 2 n 2 a j a j 1 a j n 1 j0 j0 j0 Then D n a n 1 j1 j!. The following formula nz f ϕ f k ϕz k1 k 1! k n! k 1,...,k n n! n ϕ j kj z, 2.6 j1 j! where the second sum is over all nonnegative integers k 1,k 2,...,k n satisfying k k 1 k 2 k n and k 1 2k 2 nk n n, is attributed to FaàdiBruno36. By using Bell polynomials B n,k x 1,...,x n k1 it can be written as follows: f ϕ nz k0 f k ϕz B n,k ϕ z,ϕ z,...,ϕ n k1 z. 2.7 For n N the last sum can go from k 1sinceB n,0 ϕ z,ϕ z,...,ϕ n1 z 0; however we will keep the summation since for n 0 the only existing term B 0,0 is equal to 1 and we will use it. The Leibnitz formula along with 2.6 yields n l uzg ϕz l un l z g k ϕz B l,k ϕ z,...,ϕ l k1 z. 2.8 l0 k0 Hence we have the next result. Lemma 2.5. Assume that g,u HD and ϕ is an analytic self-map of D. Then n uzg ϕz g k ϕz n l un l zb l,k ϕ z,...,ϕ l k1 z. 2.9 k0 lk 3. The Boundedness and Compactness of D m ϕ,u : H p,q,γ W n This section characterizes the boundedness and compactness of the operator Dϕ,u m : H p,q,γ W n.

5 Abstract and Applied Analysis 5 Theorem 3.1. Suppose that m, n N, 0 <p,q<, 1 <γ<, ϕ is an analytic self-map of the unit disk, u HD, and is a weight. Then the operator Dϕ,u m : H p,q,γ W n is bounded if and only if for each k {0, 1,...,n} z n lk I k : Cn l un l zb l,k ϕ z,...,ϕ l k1 z z D 1 <. ϕz 2 γ1/q1/pmk 3.1 Moreover if D m ϕ,u : H p,q,γ W n is bounded, then the following asymptotic relation holds Dϕ,u m Hp,q,γ W n /P n 1 k0 I k. 3.2 Proof. First assume that Dϕ,u m : H p,q,γ that W n is bounded; then there exists a constant C such D m ϕ,uf W n C f Hp,q,γ 3.3 for all f H p,q,γ. For a fixed w D, t γ 1/q, and constants c 1,...,c n1,set n1 g w z j1 m 1 l0 c j j t 1/p l ĝ w,j z, 3.4 where ĝ w,j z 1 w 2 jt γ1/q 1 wz 1/pjt, j 1,...,n By 33, Theorem , weget M p ĝw,j,r C 1 w 2 jt γ1/q 1 r w jt, j 1,...,n

6 6 Abstract and Applied Analysis Applying Lemma 2.3, we have that ĝw,j q H p,q,γ C 0 M q pĝw,j,r 1 r γ dr 1 w 2 qjt γ1 1 r w qjt 1 r γ dr 3.7 C. Therefore g w H p,q,γ, and moreover w D g w Hp,q,γ <. Now we show that for each s {m, m 1,...,m n}, there are constants c 1,c 2,...,c n1, such that g s w w w s 1 w 2 sγ1/q1/p, gt w w 0, t {m,...,m n} \ {s}. 3.8 By differentiating function g w, for each s {m,...,m n}, 3.8 becomes t p 1 m 1 c 1 t p 1 m 2 c 1 c 2 c n1 0, c 2 t p 1 m n 1 c n1 0, j1. s m s m t p 1 m j c 1 t p 1 m n j c n1 1, j1 3.9 n t p 1 m j c 1 j1. n t p 1 m n j j1 c n1 0. Applying Lemma 2.4 with a t 1/p m 1 > 0 and where n n 1, we see that the determinant of system 3.9 is different from zero, as claimed. By g w,k, k {0, 1,...,n}, denote the corresponding family of functions which satisfy 3.8 with s m k. Then, for each fixed k {0, 1,...,n}, inequality 3.3 along with 2.9 and 3.8 implies that for each ϕw / 0 w ϕw km n lk Cn l un l wb l,k ϕ w,...,ϕ l k1 w 1 ϕw 2 γ1/q1/pkm CDϕ,u m gϕw,k W C n w D Dϕ,u m Hp,q,γ W n. 3.10

7 Abstract and Applied Analysis 7 From 3.10 it follows that for each k {0, 1,...,n}, z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z ϕz >1/2 1 C D m ϕz 2 γ1/q1/pkm ϕ,u Hp,q,γ W n Let h k z z k, k m,...,n m Then clearly h k Hp,q,γ 1, for each k N By formula 2.9 applied to the function fz h m z we get nz Dϕ,uh m m n m h m ϕz l un l zb l,0 ϕ z,...,ϕ l1 z l0 m! l un l zb l,0 ϕ z,...,ϕ l1 z, l which along with the boundedness of the operator Dϕ,u m : H p,q,γ that m! z D z l un l zb l,0 ϕ z,...,ϕ z l1 Dϕ,uz m m l0 W n n W and 3.13 implies Dϕ,u m Hp,q,γ W n Now assume that we have proved that for j {0, 1,...,k 1} and a k n z D z l un l zb l,j ϕ z,...,ϕ z l j1 C lj Dϕ,u m Hp,q,γ W n Applying 2.9 to the function fz h mk z, k {0, 1,...,n}, and noticing that h s z 0fors>mk, weget mk D m ϕ,uh mk nz k j0 j0 h mj mk n ϕz l un l zb l,j ϕ z,...,ϕ l j1 z lj k m k k j 1 ϕz k j l un l zb l,j ϕ z,...,ϕ l j1 z. lj 3.17

8 8 Abstract and Applied Analysis From 3.17, the boundedness of the operator Dϕ,u m : H p,q,γ W n, the fact that ϕ 1, the triangle inequality, noticing that m k! is the coefficient at n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z, and finally using hypothesis 3.16 we get z B z l un l zb l,k ϕ z,...,ϕ z l k1 C lk Dϕ,u m Hp,q,γ W n Hence by induction, 3.18 holds for each k {0, 1,...,n}. From 3.18, for each fixed k {0, 1,...,n} z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z ϕz 1/2 1 ϕz 2 γ1/q1/pkm C z B z l un l zb l,k ϕ z,...,ϕ z l k1 C lk Dϕ,u m Hp,q,γ W n Inequalities 3.11 and 3.19 imply I k C k0 Dϕ,u m Hp,q,γ W n Now assume that 3.1 holds. Then for any f H p,q,γ,by2.9 and Lemma 2.2 we have nz z Dϕ,uf m z f mk ϕz n l un l zb l,k ϕ z,...,ϕ l k1 z k0 lk z f mk ϕz l un l zb l,k ϕ z,...,ϕ l k1 z k0 C f Hp,q,γ C f Hp,q,γ k0 lk z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 ϕz 2 γ1/q1/pkm k0 z D z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 ϕz 2 γ1/q1/pkm

9 Abstract and Applied Analysis 9 We also have that for each s {1,...,n 1} s0 Dϕ,uf m s f mk ϕ0 s C s l us l 0B l,k ϕ 0,...,ϕ l k1 0 D m ϕ,uf k0 C f Hp,q,γ lk s s lk Cs l us l 0B l,k ϕ 0,...,ϕ l k1 0 1, ϕ0 2 γ1/q1/pmk k0 0 u0 f m ϕ0 f Hp,q,γ C u0 1 ϕ0 2. γ1/q1/pm 3.24 Using 3.23, 3.24,and3.1 it follows that the operator Dϕ,u m : H p,q,γ From 3.23 and 3.20 the asymptotic relation 3.2 follows. W n is bounded. Theorem 3.2. Suppose that m, n N, 0 <p,q<, 1 <γ<, ϕ is an analytic self-map of the unit disk, u HD, and is a weight. Then the operator Dϕ,u m : H p,q,γ W n,0 is bounded if and only if Dϕ,u m : H p,q,γ is bounded and for each k {0, 1,...,n} W n lim z z 1 l un l zb l,k ϕ z,...,ϕ z l k lk Proof. The boundedness of Dϕ,u m : H p,q,γ W n,0 clearly implies that Dm ϕ,u : H p,q,γ W n is bounded. Applying 2.9 to the function fz h m z and using the assumption Dϕ,uh m m W n,0 it follows that nz z Dϕ,uh m m m!z l un l zb l,0 ϕ z,...,ϕ z l1 0, 3.26 l0 as z 1, which is 3.25 for k 0. Assume that we have proved the following inequalities: lim z 1 z l un l zb l,j ϕ z,...,ϕ z l j1 0, 3.27 lj for j {0, 1,...,k 1} and a k n. Applying formula 2.9 to the function fz h mk z,k {0, 1,...,n}, we get From 3.17, by using the boundedness of function ϕ, the triangle inequality, noticing that the coefficient at n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z is independent of z, and finally using

10 10 Abstract and Applied Analysis hypothesis 3.27, we easily obtain lim z 1 z l un l zb l,k ϕ z,...,ϕ z l k lk Hence by induction we get that 3.25 holds for each k {0, 1,...,n}. Now assume that Dϕ,u m : H p,q,γ W n is bounded and 3.25 holds for each k {0, 1,...,n}. For each polynomial p we have nz z Dϕ,up m z p k ϕz n l un l zb l,k ϕ z,...,ϕ l k1 z k0 lk p k z l un l zb l,k ϕ z,...,ϕ z l k1 0, lk 3.29 k0 as z 1. From 3.29 we have that, for each polynomial p, Dϕ,up m W n,0. The set of all polynomials is dense in H p,q,γ, so we have that for each f H p,q,γ, there is a sequence of polynomials p k k N such that f p k Hp,q,γ 0ask. Thus the boundedness of D m ϕ,u : H p,q,γ W n implies Dϕ,uf m Dϕ,up m W k n Dϕ,u m Hp,q,γ W n f p Hp,q,γ k 0, as k Hence D m ϕ,uh p,q,γ W n,0, from which the boundedness of Dm ϕ,u : H p,q,γ W n,0 follows, completing the proof of the theorem. Theorem 3.3. Suppose that m, n N, 0 <p,q<, 1 <γ<, ϕ is an analytic self-map of the unit disk, u HD, and is a weight. Then the operator Dϕ,u m : H p,q,γ W n is compact if and only if Dϕ,u m : H p,q,γ is bounded and for each k {0, 1,...,n} W n z n lk lim Cn l un l zb l,k ϕ z,...,ϕ l k1 z ϕz ϕz 2 γ1/q1/pkm 3.31 Proof. First assume that Dϕ,u m : H p,q,γ is bounded and 3.31 holds. By Theorem 3.1 we have that for each k {0, 1,...,n}, 3.1 holds. W n

11 Abstract and Applied Analysis 11 Let f i i N be a sequence in H p,q,γ such that i N f i Hp,q,γ L and f i converges to 0 uniformly on compact subsets of D as i. By the assumption, for any ε>0, there is a δ 0, 1, such that for each k {0, 1,...,n} and δ< ϕz < 1 z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 <ε. ϕz 2 γ1/q1/pkm 3.32 We have Dϕ,uf m W i n nz n 1 j0 z Dϕ,uf m i Dϕ,uf m i z D j0 z f mk n z D i ϕz l un l zb l,k ϕ z,...,ϕ l k1 z k0 lk n 1 j f mk j i ϕ0 C j l uj l 0B l,k ϕ 0,...,ϕ l k1 0 j0 k0 ϕz δ n 1 j j0 ϕz >δ f mk i k0 lk z f mk i k0 ϕz l un l zb l,k ϕ z,...,ϕ l k1 z lk j ϕ0 C j l uj l 0B l,k ϕ 0,...,ϕ 0 l k1 J 1 J 2 J 3. lk 3.33 Now we estimate J 1, J 2,andJ 3 : J 1 z ϕz δ f mk i k0 w δ f mk i k0 w δ f mk i k0 ϕz l un l zb l,k ϕ z,...,ϕ l k1 z ϕz δ lk w z l un l zb l,k ϕ z,...,ϕ l k1 z w z D lk z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 ϕz 2 γ1/q1/pmk 3.34 f mk i k0 w δ wi k 0, as i, where in 3.34 we have used the fact that from f i 0 uniformly on compact subsets of D as i it follows that for each s N, f s i 0 uniformly on compact subsets of D as i.

12 12 Abstract and Applied Analysis The fact that n 1 j J 3 j0 k0 f mk i j ϕ0 C j l uj l 0B l,k ϕ 0,...,ϕ 0 l k1 0, 3.35 lk as i, is proved similarly; so we omit it. By Lemma 2.2 and 3.32 we have that J 2 C fi Hp,q,γ k0 ϕz >δ z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 <Cεn1L. ϕz 2 γ1/q1/pkm 3.36 From 3.34, 3.35,and3.36 we obtain lim Dϕ,uf m W i 0. n i 3.37 From this and applying Lemma 2.1 the implication follows. Now assume that Dϕ,u m : H p,q,γ bounded. Let z i i N be a sequence in D such that ϕz i 1asi. If such a sequence does not exist, then the conditions in 3.31 automatically hold. W n is compact; then clearly Dϕ,u m : H p,q,γ W n is Let g w,k, k {0, 1,...,n} be as in Theorem 3.1. Then the sequences g ϕzi,k i N are bounded and g ϕzi,k 0 uniformly on compact subsets of D as i. Since Dϕ,u m : H p,q,γ W n is compact, we have that for each k {0, 1,...,n} lim Dϕ,ug m ϕzi,k i W n On the other hand, from 3.10 we obtain D m ϕ,ug ϕzi,k W n Cz i ϕzi km n lk Cn l un l z i B l,k ϕ z i,...,ϕ l k1 z i 1 ϕzi 2 γ1/q1/pkm, 3.39 which along with ϕz i 1asi and 3.38 implies that z i n lk lim Cn l un l z i B l,k ϕ z i,...,ϕ l k1 z i i 1 ϕzi 2, γ1/q1/pkm 3.40 for each k {0, 1,...,n}, from which 3.31 holds in this case.

13 Abstract and Applied Analysis The Compactness of the Operator D m ϕ,u : H p,q,γ W n,0 The compactness of Dϕ,u m : H p,q,γ W n,0 is characterized here. The proof of the next lemma is similar to the proof of the corresponding result in 14. Lemma 4.1. Suppose that n N 0 and is a radial weight such that lim z 1 z 0. A closed set K in W n,0 is compact if and only if it is bounded and satisfies lim z 1 f K zf n z Theorem 4.2. Suppose that m, n N, 0 <p,q<, 1 <γ<, ϕ is an analytic self-map of the unit disk, u HD and is a radial weight such that lim z 1 z 0. Then the operator Dϕ,u m : H p,q,γ W n,0 is compact if and only if for each k {0, 1,...,n} z n lk lim Cn l un l zb l,k ϕ z,...,ϕ l k1 z z 1 1 ϕz 2 0. γ1/q1/pkm 4.2 Proof. First assume that Dϕ,u m : H p,q,γ W n,0 is compact. Then it is bounded and since the test functions in 3.12 belong to H p,q,γ D, we have that 3.25 holds. Beside this the operator Dϕ,u m : H p,q,γ W n is compact too, so that 3.31 holds. Hence, if ϕ < 1, from 3.25 for each k {0, 1,...,n} we get z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 ϕz 2 γ1/q1/pkm z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 ϕ 2 γ1/q1/pkm 0, 4.3 as z 1, hence we obtain 4.2 in this case. Now assume ϕ 1. Let ϕz i i N be a sequence such that ϕz i 1asi. Then from 3.31 we have that for every ε>0, there is an r 0, 1 such that for each k {0, 1,...,n} z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 <ε ϕz 2 γ1/q1/pkm 4.4 when r< ϕz < 1, and from 3.25 there exists a σ 0, 1 such that for σ< z < 1 z l un l zb l,k ϕ z,...,ϕ z l k1 <ε 1 r 2 γ1/q1/pkm. 4.5 lk

14 14 Abstract and Applied Analysis Therefore, when σ< z < 1andr< ϕz < 1, we have that z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 <ε. ϕz 2 γ1/q1/pkm 4.6 On the other hand, if ϕz r and σ< z < 1, from 4.5 we obtain z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z 1 ϕz 2 γ1/q1/pkm < z n lk Cn l un l zb l,k ϕ z,...,ϕ l k1 z <ε. 1 r 2 γ1/q1/pkm 4.7 Combining the last two inequalities we obtain 4.2, as desired. Now assume that 4.2 holds. Taking the remum in 3.22 over f in the unit ball of H p,q,γ, then letting z 1 is such obtained inequality and using 4.2 we get lim z 1 f Hp,q,γ 1 nz z Dϕ,uf m Hence by Lemma 4.1 the compactness of the operator D m ϕ,u : H p,q,γ W n,0 follows. References 1 S. Stević, Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane, Abstract and Applied Analysis, vol. 2009, Article ID , 8 pages, K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, Australian Mathematical Society Journal Series A, vol. 54, no. 1, pp , S. Stević, Essential norms of weighted composition operators from the Bergman space to weightedtype spaces on the unit ball, Ars Combinatoria, vol. 91, pp , W. Yang, Weighted composition operators from Bloch-type spaces to weighted-type spaces, Ars Combinatoria, vol. 93, pp , S. Stević, Norms of some operators from Bergman spaces to weighted and Bloch-type space, Utilitas Mathematica, vol. 76, pp , S. Stević, Essential norm of an operator from the weighted Hilbert-Bergman space to the Bloch-type space, Ars Combinatoria, vol. 91, pp , S. Stević, Norm estimates of weighted composition operators between Bloch-type spaces, Ars Combinatoria, vol. 93, pp , B. R. Choe, H. Koo, and W. Smith, Composition operators on small spaces, Integral Equations and Operator Theory, vol. 56, no. 3, pp , S. Stević, Composition operators from the Hardy space to Zygmund-type spaces on the upper halfplane and the unit disk, Journal of Computational Analysis and Applications, vol. 12, no. 2, pp , X. Zhu, Volterra type operators from logarithmic Bloch spaces to Zygmund type spaces, International Journal of Modern Mathematics, vol. 3, no. 3, pp , S. Stević, On an integral operator from the Zygmund space to the Bloch-type space on the unit ball, Glasgow Mathematical Journal, vol. 51, no. 2, pp , 2009.

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