Essential norms and weak compactness of integral operators between weighted Bergman spaces
aa r X i v : . [ m a t h . C V ] J un ESSENTIAL NORMS AND WEAK COMPACTNESS OFINTEGRATION OPERATORS BETWEEN WEIGHTEDBERGMAN SPACES
SANTERI MIIHKINEN, PEKKA J. NIEMINEN, AND WEN XUA bstract . We consider Volterra-type integration operators T g betweenBergman spaces induced by weights satisfying a doubling property. Wederive estimates for the operator norms, essential and weak essentialnorms of T g : A p ω → A q ω , 0 < p ≤ q < ∞ . In particular, the opera-tor T g : A ω → A ω is weakly compact if and only if it is compact.
1. I ntroduction
Let D be the unit disk in the complex plane and T be the boundary of D .Let H ( D ) be the algebra of all analytic functions in D . For g ∈ H ( D ), weconsider the generalized Volterra integration operator T g defined by T g ( f )( z ) = Z z f ( ζ ) g ′ ( ζ ) d ζ, z ∈ D for f ∈ H ( D ). The main purpose of the paper is to derive estimates for theoperator norms and essential norms of T g : A p ω → A q ω , 0 < p ≤ q < ∞ , aswell as weak essential norms of T g on A ω , where A p ω is the Bergman spaceinduced by ω in the class b D which consists of radial weights satisfying thedoubling property R r ω ( s ) ds ≤ C R + r ω ( s ) ds with C = C ( ω ) >
0. Essentialnorms of T g between classical weighted Bergman spaces have been esti-mated by R¨atty¨a in [12] for 1 < p ≤ q < ∞ . Later essential norms of T g onHardy spaces, BMOA and the Bloch space have been investigated in [7, 6].Let X and Y be complete metric spaces. For a bounded linear operator T : X → Y , the essential norm (resp. weak essential norm), denoted by k T k e , X → Y ( resp. k T k w , X → Y ), is the distance of T (in the operator norm) fromthe closed ideal of compact operators (resp. weakly compact operators) K : X → Y . Here an operator K : X → Y is weakly compact if K ( B )is compact in the weak topology of Y, where B is the unit ball of X . If Date : July 16, 2018.2000
Mathematics Subject Classification.
Primary 47G10; Secondary30H20,47B38,45P05.
Key words and phrases.
Volterra operator, integral operator, norm, essential norm, weakcompactness, weighted Bergman spaces. either X or Y is reflexive, then every bounded operator T : X → Y isweakly compact. Since A ω is nonreflexive, there are bounded operators on A ω which are not weakly compact. If A ω has so-called Schur property, i.e.weakly convergent sequences in A ω are also norm convergent, then the classof weakly compact operators on A ω coincides with the class of compactoperators on A ω . We do not know if this is the case, therefore we alsoconsider the weak compactness of the operator T g on A ω .There are some previous results on the weak compactness of T g . For ex-ample, it has been shown in [7] that the compactness and weak compactnessof the operator T g are equivalent on Hardy space H and BMOA . In the caseof
BMOA a di ff erent proof of this fact was obtained in an independent workof Blasco et al. [2] using di ff erent techniques.The presence of large class of weights in our setting brings its own dif-ficulties which were not present in the previous works concerning essentialnorms of operator T g . For example, Littlewood-Paley type formula is usu-ally used to get rid of the integral in the definition of T g . However, there isno such formula in general for A p ω , ω ∈ b D unless p =
2, see [11, Chapter4]. In order to circumvent this problem we had to use di ff erent equivalentnorms inherited from the theory of Hardy spaces, see [11, Chapter 4].For each radial weight ω , its associated weight ω ∗ is defined by ω ∗ ( z ) = Z | z | ω ( s ) s log s | z | ds , z ∈ D \ { } . For α ≥ ω ∈ b D , the space C α ( ω ∗ ) consists of g ∈ H ( D ) such that k g k C α ( ω ∗ ) = | g (0) | + k g k ∗ ,α,ω < ∞ , where k g k ∗ ,α,ω = sup I ⊂ T s R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z )( ω ( S ( I ))) α is a seminorm on C α ( ω ∗ ), S ( I ) = { re i θ ∈ D : e i θ ∈ I , − | I | ≤ r < } isthe Carleson square associated with I ⊆ T , | E | is the Lebesgue measure of E ⊆ T and ω ( S ( I )) = R S ( I ) ω ( z ) dA ( z ). We associate each a ∈ D \ { } with theinterval I a = n e i θ : | arg( ae − i θ ) | ≤ −| a | o , and denote S ( a ) = S ( I a ). The space C α ( ω ∗ ) consists of g ∈ H ( D ) such thatlim sup | I |→ R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z )( ω ( S ( I ))) α = . Throughout the paper the notation A . B indicates that there is a constant c independent of said or implied variables or functions such that A ≤ cB . SSENTIAL NORM 3 If A . B and B . A , we write A ≃ B and say that A and B are equivalentquantities.The next result is a generalization of a part of Theorem 4.1 in [11] for theweights in the class b D . Theorem A.
Let < p ≤ q < ∞ , α = p − q ) + , p − q < , ω ∈ b D andg ∈ H ( D ) . Then T g : A p ω → A q ω is bounded if and only if g ∈ C α ( ω ∗ ) . Below are our main results. The first result is a quantitative extension ofTheorem A.
Theorem 1.
Let < p ≤ q < ∞ , ω ∈ b D, α = p − q ) + and g ∈ C α ( ω ∗ ) . Then there exists η = η ( ω ) > large enough such that the followingquantities are comparable: k T g k A p ω → A q ω ; k g k ∗ ,α,ω = sup I ⊆ T R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) α / ; B = sup a ∈ D Z D ω ( S ( a )) − | a || − ¯ az | ! η ! α | g ′ ( z ) | ω ∗ ( z ) dA ( z ); C = sup z ∈ D | g ′ ( z ) | (1 − | z | ) ω ∗ ( z ) q − p , p < q . Constants of comparison are independent of g.
Theorem 2.
Let < p ≤ q < ∞ , α = p − q ) + , p − q < , ω ∈ b Dand g ∈ C α ( ω ∗ ) . Then there exists η = η ( ω ) > large enough such that thefollowing quantities are comparable: k T g k e , A p ω → A q ω ; A = dist ( g , C α ); B = lim sup | I |→ R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) α / ; C = lim sup | a |→ − Z D ω ( S ( a )) − | a || − ¯ az | ! η ! α | g ′ ( z ) | ω ∗ ( z ) dA ( z ); D = lim sup | z |→ − | g ′ ( z ) | (1 − | z | ) ω ∗ ( z ) q − p , p < q . Theorem 3.
Let ω ∈ b D and g ∈ C ( ω ∗ ) . Then k T g k w , A ω → A ω ≃ dist ( g , C ( ω ∗ )) SANTERI MIIHKINEN, PEKKA J. NIEMINEN, AND WEN XU ≃ lim sup | I |→ R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) / ≃ k T g k e , A ω → A ω . In particular, the operator T g is weakly compact on A ω if and only if it iscompact. The paper is organized as follows. In section 2, we give some preliminaryresults. In section 3, the proofs of norm estimates are presented. In section4, we investigate essential norms between two weighted Bergman spacesand weak compactness on A ω .2. P reliminaries An integrable function ω : D → (0 , ∞ ) is called a weight function orsimply a weight . For 0 < p < ∞ and a weight ω , the weighted Bergmanspace A p ω stands for the space of all functions f ∈ H ( D ) satisfying k f k pA p ω = Z D | f ( z ) | p ω ( z ) dA ( z ) < ∞ , where dA ( z ) = π dxdy is the normalized Lebesgue area measure on D . For ω ( z ) = (1 − | z | ) α , − < α < ∞ , A p ω is the classical weighted Bergman space.If 1 ≤ p < ∞ , then k · k A p ω is a norm which makes A p ω a Banach space. But if0 < p <
1, then it is instead k · k pA p ω which is subadditive and used to inducethe complete translation invariant metric. The operator norm is defined asusual k T g k A p ω → A q ω = sup k f k Ap ω ≤ k T g f k A q ω , although in the case 0 < q < k · k A p ω → A q ω is a quasi-norm, butwe make no distinction between that and the operator norm.A weight ω is radial if ω ( z ) = ω ( | z | ) for all z ∈ D . Let b D be the class ofradial weights such that b ω ( r ) = R r ω ( s ) ds satisfies the doubling property,that is, there exists C = C ( ω ) such that b ω ( r ) ≤ C b ω + r ! , for ∀ ≤ r < . A radial weight ω is called regular if ω is continuous and satisfies b ω ( r ) ω ( r ) ≃ − r , for 0 ≤ r < . The weight ω ∗ is regular if ω ∈ b D . The class of regular weights is denotedby R . Also, a radial weight ω is in the class of rapidly increasing weights I SSENTIAL NORM 5 if it is continuous and satisfieslim r → − b ω ( r )(1 − r ) ω ( r ) = ∞ . See [11] for more information on classes I and R .Recall that non-tangential regions and the tents are defined by Γ ( u ) = ( z ∈ D : | θ − arg z | < − | z | r !) , u = re i θ ∈ D \ { } , T ( z ) = { u ∈ D : z ∈ Γ ( u ) } , z ∈ D . A simple computation shows that ω ( S ( z )) ≃ ω ( T ( z )) ≃ ω ∗ ( z ), as | z | → − ,provided ω ∈ b D . The maximal function related to the measure ω ( · ) dA isdefined by M ω ( ψ )( z ) = sup I : z ∈ S ( I ) ω ( S ( I )) Z S ( I ) | ψ ( ξ ) | ω ( ξ ) dA ( ξ ) , z ∈ D , where ψ ∈ L ω . For more information on A p ω , see [8, 9, 11].Recall that for a given Banach space (or a complete metric space) X ofanalytic functions on D , a positive Borel measure µ on D is called a q-Carleson measure for X if the identity operator I : X → L q ( µ ) is bounded.Pel´aez and R¨atty¨a [11] investigated the q -Carleson measure for A p ω , as wellas the boundedness and compactness of the integral operator T g , where ω ∈I ∪ R . The classes I and R are contained in b D . In fact b D preserves almostall the properties of I ∪ R and so those statements concerning the Carlesonmeasures and the integral operators are also true on A p ω , ω ∈ b D . For thereader’s convenience, we list some results here and skip proofs. The nextlemma is essentially Theorem 2.1 and Corollary 2.2 in [11]. Lemma 1.
Let < p ≤ q < ∞ and ω ∈ b D, and let µ be a positive Borelmeasure on D . Then µ is a q-Carleson measure for A p ω if and only if (1) G , sup I ⊆ T µ ( S ( I ))( ω ( S ( I ))) qp < ∞ . Moreover, if µ is a q-Carleson measure for A p ω , then for all f ∈ A p ω (2) k f k qA q µ . G k f k qA p ω . Furthermore, if α ∈ (0 , ∞ ) such that p α > , then [ M ω (( · ) α )] α : L p ω → L q µ isbounded if and only if µ satisfies (1) and k [ M ω (( · ) α )] α k qL p ω → L q µ ≃ G. Remark 1.
The operator ψ M ω ( ψ ) is sublinear, but its norm is definedlike in the case of a linear operator. See [11, Theorem 4.2] for the next lemma.
SANTERI MIIHKINEN, PEKKA J. NIEMINEN, AND WEN XU
Lemma 2.
Let < p < ∞ , n ∈ N and f ∈ H ( D ) , and let ω be a radialweight. Then (3) k f k pA p ω = p Z D | f ( z ) | p − | f ′ ( z ) | ω ∗ ( z ) dA ( z ) + ω ( D ) | f (0) | p , and (4) k f k pA p ω ≃ Z D Z Γ ( u ) | f ( n ) ( z ) | (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) zu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) n − dA ( z ) ! p ω ( u ) dA ( u ) + n − X j = | f ( j ) (0) | p , where the constants of comparison depend only on p , n and ω . In particular, (5) k f k A ω = k f ′ k A ω ∗ + ω ( D ) | f (0) | . Recall that the non-tangential maximal function of f in the unit disk isdefined by N ( f )( u ) = sup z ∈ Γ ( u ) | f ( z ) | , u ∈ D \ { } . The following equivalentnorm will be used in our proof also, see [11, Lemma 4.4]. Lemma 3.
Let < p < ∞ and let ω be a radial weight. Then k N ( f ) k A p ω ≃ k f k A p ω , for all f ∈ A p ω . Proposition 4.7 in [11] also holds for weights in the class b D and it statesthat f ∈ C α ( ω ∗ ) , α > M ∞ ( f ′ , r ) . ( ω ∗ ( r )) α − − r , ≤ r < f ∈ C α ( ω ∗ ) if and only if M ∞ ( f ′ , r ) = o ( ω ∗ ( r )) α − − r , r → − . Furthermore, the proof of Proposition 4.7 in [11] implies that
Lemma 4.
Let < α < ∞ , ω ∈ b D and g ∈ C α + ( ω ∗ ) . Then (6) lim sup | z |→ − | g ′ ( z ) | (1 − | z | ) ω ∗ ( z ) − α = lim sup | a |→ − R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) α + . In the next lemma, we classify spaces C α ( ω ∗ ) and C α ( ω ∗ ) according tohow fast the quantity ( ω ∗ ( r )) α − − r grows as r → − . The proof is straightforward and we omit it. SSENTIAL NORM 7
Lemma 5.
Let ω ∈ b D , α > andF α,ω ( r ) = ( ω ∗ ( r )) α − − r , r ∈ ]0 , . Define β = lim inf r → − F α,ω ( r ) . Then(i) If β = , then C α ( ω ∗ ) = C α ( ω ∗ ) = { f ∈ H ( D ) | f is a constant function } ; (ii) If β ∈ ]0 , ∞ [ , then C α ( ω ∗ ) = { f ∈ H ( D ) | f is a constant function } andC α ( ω ∗ ) = { f ∈ H ( D ) | f ′ ∈ H ∞ ( D ) } ; (iii) If β = ∞ , then { f ∈ H ( D ) | f ′ ∈ H ∞ ( D ) } ( C α ( ω ∗ ) ⊂ C α ( ω ∗ ) . A function-theoretic quantity to estimate the distance of a general C α ( ω ∗ )-function from C α ( ω ∗ ) is given by Lemma 6.
Let ω ∈ b D and α ≥ . For g ∈ C α ( ω ∗ ) ,dist ( g , C α ( ω ∗ )) ≃ lim sup | I |→ R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) α . Proof.
The lower estimate is trivial from the definitions of C α ( ω ∗ ) and C α ( ω ∗ ).For the upper estimate we consider three cases. Let β be the numberdefined in Lemma 5. Case 1 - Assume α > β = g , C α ( ω ∗ )) ≃ lim sup | I |→ R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) α . Case 2 - Assume α > β ∈ ]0 , ∞ [.Define G ω, g : ]0 , → R + , G ω, g ( t ) = sup | I | = t R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) α / and G = G ω, id . Now dist( g , C α ( ω ∗ )) = sup t ∈ ]0 , G ω, g ( t ), since C α ( ω ∗ ) = { f ∈ H ( D ) | f is a constant function } by the case (ii) of Lemma 5. It is enough to show thatsup t ∈ ]0 , G ω, g ( t ) . lim sup t → + G ω, g ( t ) , since the direction lim sup t → + G ω, g ( t ) ≤ sup t ∈ ]0 , G ω, g ( t ) SANTERI MIIHKINEN, PEKKA J. NIEMINEN, AND WEN XU is evident.It holds that lim sup t → + G ( t ) ∈ ]0 , ∞ [, since id ∈ C α ( ω ∗ ) \ C α ( ω ∗ ) by thecase (ii) of Lemma 5. Now(7) sup t ∈ ]0 , G ( t ) ≃ lim sup t → + G ( t ) . Since g ′ ∈ H ∞ ( D ), we can assume by rotation invariance that there existthe non-tangential limit g ′ (1) = lim z → z ∈ N g ′ ( z ) s.t. | g ′ (1) | > k g ′ k H ∞ ( D ) , where N ⊂ D is any non-tangential set with vertex at z =
1. Also, there exist r ∈ [0 , S = S ( r ) and a non-tangential set T ⊂ S with vertex at z = | g ′ ( z ) | ≥ k g ′ k H ∞ ( D ) for all z ∈ T and ω ∗ ( T ) ≃ ω ∗ ( S ) . Let S = S ( I ) be any Carleson window s.t. | I | ≤ − r . Choose a Carlesonwindow S ′ = S ′ ( I ′ ) ⊂ S with | I ′ | = | I | and a non-tangential set T ′ ⊂ S ′ ∩ T with vertex at z = ω ∗ ( T ′ ) ≃ ω ∗ ( S ′ ). Now we can estimatesup t ≤ − r G ω, g ( t ) ≥ R S ′ | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ′ ) α ≥ R T ′ | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ′ ) α & k g ′ k H ∞ ( D ) ω ∗ ( T ′ ) ω ( S ′ ) α ! ≃ k g ′ k H ∞ ( D ) ω ∗ ( S ′ ) ω ( S ′ ) α ! = k g ′ k H ∞ ( D ) ω ∗ ( S ) ω ( S ) α ! . Hence sup t ≤ − r G ω, g ( t ) & k g ′ k H ∞ ( D ) sup t ≤ − r G ( t )and letting r → − we get(8) lim sup t → + G ω, g ( t ) & k g ′ k H ∞ ( D ) lim sup t → + G ( t ) . Now by (7) and (8) we getsup t ∈ ]0 , G ω, g ( t ) ≤ k g ′ k H ∞ ( D ) sup t ∈ ]0 , G ( t ) ≃ k g ′ k H ∞ ( D ) lim sup t → + G ( t ) . lim sup t → + G ω, g ( t ) . Thus we have established the upper estimate in the case β ∈ ]0 , ∞ [. Case 3 - Assume α = β = ∞ .Now it holds that { f ∈ H ( D ) | f ′ ∈ H ∞ ( D ) } ⊂ C α ( ω ∗ ) . Set g r ( z ) = g ( rz ) for 0 < r <
1. Then g r ∈ C α ( ω ∗ ). Fix 0 < δ <
1. Nowdist( g , C α ( ω ∗ )) ≤ lim sup r → − k g − g r k C α ( ω ∗ ) ≤ lim sup r → − sup | I |≥ δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) − rg ′ ( rz ) | ω ∗ ( z ) dA ( z ) SSENTIAL NORM 9 + sup | I | <δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) − rg ′ ( rz ) | ω ∗ ( z ) dA ( z ) ! = lim sup r → − sup | I |≥ δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) − rg ′ ( rz ) | ω ∗ ( z ) dA ( z ) ! + lim sup r → − sup | I | <δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) − rg ′ ( rz ) | ω ∗ ( z ) dA ( z ) ! , wheresup | I |≥ δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) − rg ′ ( rz ) | ω ∗ ( z ) dA ( z ) . k g ′ − ( g r ) ′ k A ω ∗ → , r → − . Thus we havedist( g , C α ( ω ∗ )) . lim sup r → − sup | I | <δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) − rg ′ ( rz ) | ω ∗ ( z ) dA ( z ) ! . sup r > − δ sup | I | <δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) + sup | I | <δ ω ( S ( I )) α Z S ( I ) r | g ′ ( rz ) | ω ∗ ( z ) dA ( z ) ! = sup | I | <δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! + sup r > − δ sup | I | <δ ω ( S ( I )) α Z S ( I ) r | g ′ ( rz ) | ω ∗ ( z ) dA ( z ) ! . (9)Given an interval I ⊂ T , let e i θ ∈ I be the center point of I and define aCarleson window S ′ ( I ) = { re i θ ∈ D : | θ − θ | < | I | , − | I | ≤ r < } . Now rS ( I ) ⊂ S ′ ( I ) for all r ∈ ]1 − δ, , when δ is small enough. Also, itholds that ω ( S ′ ( I )) ω ( S ( I )) . I ⊂ T by the doubling property. Thus by the change of variables u = rz , we get 1 ω ( S ( I )) α Z S ( I ) r | g ′ ( rz ) | ω ∗ ( z ) dA ( z ) = ω ( S ( I )) α Z rS ( I ) | g ′ ( u ) | ω ∗ ( u / r ) dA ( u ) ≤ ω ( S ′ ( I )) ω ( S ( I )) ! α ω ( S ′ ( I )) α Z S ′ ( I ) | g ′ ( u ) | ω ∗ ( u ) dA ( u ) . ω ( S ′ ( I )) α Z S ′ ( I ) | g ′ ( u ) | ω ∗ ( u ) dA ( u )for all r ∈ ]1 − δ,
1[ and consequentlysup r > − δ sup | I | <δ ω ( S ( I )) α Z S ( I ) r | g ′ ( rz ) | ω ∗ ( z ) dA ( z ) ! . sup | I | <δ ω ( S ′ ( I )) α Z S ′ ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! . Now the estimate (9) becomesdist( g , C α ( ω ∗ )) . sup | I | <δ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! + sup | I | <δ ω ( S ′ ( I )) α Z S ′ ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! . (10)Letting δ → + in (10), we getdist( g , C α ( ω ∗ )) . lim sup | I |→ ω ( S ( I )) α Z S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! . The proof is complete. (cid:3)
3. N orm E stimate Define(11) f a , p ( z ) = (1 − | a | ) γ + p (1 − ¯ az ) γ + p ω ( S ( a )) p , where γ = β ( ω ) > a ∈ D k f a , p k A p ω .
1, and f a ( z ) → D as | a | → Lemma 7.
Let < p ≤ q < ∞ , s = (cid:16) p − q (cid:17) + , p − q < , ω ∈ b D andg ∈ C s ( ω ∗ ) . Then lim sup | a |→ k T g ( f a , p ) k A q ω ≥ lim sup | a |→ R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) s (12) Proof.
We split the analysis into two cases.
Case 1 - Assume p = q . For this, we divide the proof of the claim (12)into three sub-cases. SSENTIAL NORM 11
Sub-case 1 : p >
2. We may assume that | a | > /
2. For z ∈ S ( a ), it iseasy to see that f a , p ( z ) ≃ ω ( S ( a )) − p , and so(13) Z S ( a ) | f a , p | p | g ′ ( z ) | ω ∗ ( z ) dA ( z ) & ω ( S ( a )) Z S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) . Furthermore, by applying Fubini’s theorem, H¨older’s inequality, Lemma 3and (4), we obtain Z S ( a ) | f a , p | p | g ′ ( z ) | ω ∗ ( z ) dA ( z ) . Z D | f a , p | p | g ′ ( z ) | Z T ( z ) ω ( u ) dA ( u ) dA ( z ) = Z D Z Γ ( u ) | f a , p ( z ) | p | g ′ ( z ) | dA ( z ) ω ( u ) dA ( u ) ≤ Z D N ( f a , p )( u ) p − Z Γ ( u ) | f a , p ( z ) | | g ′ ( z ) | dA ( z ) w ( u ) dA ( u ) ≤ Z D N ( f a , p )( u ) p ω ( u ) dA ( u ) ! p − p Z D Z Γ ( u ) | f a , p ( z ) | | g ′ ( z ) | dA ( z ) ! p ω ( u ) dA ( u ) p . k f a , p k p − A p ω k T g ( f a , p ) k A p ω . k T g ( f a , p ) k A p ω . This last estimate, along with (13) gives k T g f a , p k A p ω & ω ( S ( a )) Z S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) . Sub-case 2 : p =
2. The desired estimate follows from (5) immediately.
Sub-case 3 : 0 < p <
2. Let 1 < α, β < ∞ be such that β/α = p / < | a | > / α ′ and β ′ be the conjugate indexes of α and β respectively.It follows from Fubini’s theorem , H¨older’s inequality, and (4) that1 ω ( S ( a )) p Z S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ≃ Z S ( a ) | g ′ ( z ) | | f a , p ( z ) | ω ∗ ( z ) dA ( z ) ≃ Z D Z S ( a ) ∩ Γ ( u ) | g ′ ( z ) | | f a , p ( z ) | dA ( z ) ! α + α ′ ω ( u ) dA ( u ) ≤ Z D Z Γ ( u ) | g ′ ( z ) | | f a , p ( z ) | dA ( z ) ! βα ω ( u ) dA ( u ) β (14) · Z D Z Γ ( u ) ∩ S ( a ) | g ′ ( z ) | | f a , p ( z ) | dA ( z ) ! β ′ α ′ ω ( u ) dA ( u ) β ′ ≃ k T g ( f a , p ) k p β A p ω ω ( S ( a )) − p α ′ k S g ( χ S ( a ) ) k α ′ L β ′ α ′ ω for | a | > / , where S g ( ψ )( u ) = Z Γ ( u ) | ψ ( z ) | | g ′ ( z ) | dA ( z ) , u ∈ D \ { } , for any bounded function ψ on D . From 1 < β < α , we obtain β ′ α ′ > (cid:16) β ′ α ′ (cid:17) ′ = β ( α − α − β >
1. Thereby(15) k S g ( χ S ( a ) ) k L β ′ α ′ ω = sup k f k L β ( α − α − βω ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D f ( u ) S g ( χ S ( a ) )( u ) ω ( u ) dA ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Combining Fubini’s theorem, H¨older’s inequality, and Lemma 1, we con-clude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D f ( u ) S g ( χ S ( a ) )( u ) ω ( u ) dA ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z D | f ( u ) | Z Γ ( u ) ∩ S ( a ) | g ′ ( z ) | dA ( z ) ω ( u ) dA ( u ) = Z S ( a ) | g ′ ( z ) | Z T ( z ) | f ( u ) | ω ( u ) dA ( u ) dA ( z ) . Z S ( a ) M ω ( | f | )( z ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ≤ Z S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! α ′ β ′ Z S ( a ) M ω ( | f | )( z ) (cid:18) β ′ α ′ (cid:19) ′ | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! − α ′ β ′ ≤ Z S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! α ′ β ′ sup b ∈ D µ a ( S ( b )) ω ( S ( b )) ! − α ′ β ′ k f k L (cid:18) β ′ α ′ (cid:19) ′ ω , where d µ a ( z ) = χ S ( a ) ( z ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ). The last estimate, along with (14)and (15) gives R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) p . k T g ( f a , p ) k p β A p ω · (cid:16)R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) (cid:17) β ′ ω ( S ( a )) p · α ′ sup b ∈ D µ a ( S ( b )) ω ( S ( b )) ! (1 − α ′ β ′ ) · α ′ , SSENTIAL NORM 13 so that R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) β . k T g ( f a , p ) k p β A p ω sup b ∈ D µ a ( S ( b )) ω ( S ( b )) ! β (1 − βα ) , from which we obtain R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) . k T g ( f a , p ) k pA p ω sup b ∈ D µ a ( S ( b )) ω ( S ( b )) ! − p = k T g ( f a , p ) k pA p ω sup b : S ( b ) ⊆ S ( a ) µ a ( S ( b )) ω ( S ( b )) ! − p . (16)It is easy to see thatlim sup | a |→ sup b : S ( b ) ⊆ S ( a ) µ a ( S ( b )) ω ( S ( b )) = lim sup | a |→ sup b : S ( b ) ⊆ S ( a ) R S ( b ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( b )) = lim sup | a |→ R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) . The last equality and (16) yield lim sup | a |→ R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) p = lim sup | a |→ R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) (cid:18) sup b : S ( b ) ⊆ S ( a ) R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) (cid:19) − p . lim sup | a |→ k T g ( f a , p ) k pA p ω . (17)Thereforelim sup | a |→ k T g ( f a , p ) k A p ω & lim sup | a |→ R S ( a ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( a )) . Case 2 - Assume p < q .For all h ∈ A q ω , we have k h k qA q ω ≥ Z D \ D (0 , r ) | h ( z ) | p ω ( z ) dA ( z ) & M pp ( r , h ) Z r ω ( s ) ds , r ≥ , where M p ( r , h ) = (cid:18) π R π | h ( re i θ ) | p d θ (cid:19) / p . Then M qq ( r , T g f a , p ) . k T g f a , p k qA q ω R r ω ( s ) ds , r ≥ . By Cauchy’s integral formula, we get two well-known estimates as M ∞ ( r , f ′ ) . M ∞ ( ρ, f ) / (1 − r ) and M ∞ ( r , f ) . M q ( ρ, f )(1 − r ) − / q , ρ = (1 + r ) /
2. Then | g ′ ( a ) | ≃ ω ∗ ( a ) p | ( T g f a , p ) ′ ( a ) | . ω ∗ ( a ) p M ∞ (cid:16) + | a | , T g f a , p (cid:17) − | a | . ω ∗ ( a ) p M q (cid:16) + | a | , T g f a , p (cid:17) (1 − | a | ) + q . ω ∗ ( a ) p k T g f a , p k A q ω (1 − | a | ) + q (cid:18)R + | a | ω ( s ) ds (cid:19) q ≃ ω ∗ ( a ) p − q k T g f a , p k A q ω − | a | . The last inequality is due to ω ∗ ( a ) ≃ (1 −| a | ) R | a | ω ( s ) ds . (1 −| a | ) R + | a | ω ( s ) ds ,for | a | > . Thus k T g f a , p k A q ω & | g ′ ( a ) | ω ∗ ( a ) q − p (1 − | a | ) , | a | > (cid:3) Proof of Theorem 1.
Clearly, k g k ∗ ,α,ω ≃ B and k g k ∗ ,α,ω ≃ C follow by theproof of Lemma 5.3 and Proposition 4.7 in [11], respectively. The proof ofLemma 7 also deduces that sup a ∈ D k T g f a , p k A q ω & k g k ∗ ,α,ω . So k T g k A p ω → A q ω & k g k ∗ ,α,ω . It remains to prove k T g k A p ω → A q ω . k g k ∗ ,α,ω .Notice(18) k T g k qA p ω → A q ω = sup k f k Ap ω ≤ k T g ( f ) k qA q ω . Two cases have to be analyzed.
Case 1 -Assume q ≥
2. Applying (3) and H¨older’s inequality, for q > k T g f k qA q ω ≃ Z D | T g f ( z ) | q − | f ( z ) | | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ≤ Z D | T g f ( z ) | q − p + pqp | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! p ( q − q − p + pq · Z D | f ( z ) | q − p + pqq | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ! q q − p + pq , whence(19) k T g f k qA q ω . U p ( q − q − p + pq V q q − p + pq , SSENTIAL NORM 15 where(20) U = R D | T g f ( z ) | q − p + pqp | g ′ ( z ) | ω ∗ ( z ) dA ( z ); V = R D | f ( z ) | q − p + pqq | g ′ ( z ) | ω ∗ ( z ) dA ( z )Noticing that (19) is also true for q =
2. We have to control U and V fromabove. To do so, set d µ ( z ) = | g ′ ( z ) | ω ∗ ( z ) dA ( z ). The assumption g ∈ C α ( ω ∗ )and Theorem A yield the boundedness of T g : A p ω → A q ω . Moreover, Lemma1 and the fact that α = (cid:16) p − q (cid:17) + = (cid:16) q − p + pqp (cid:17) / q = (cid:16) q − p + pqq (cid:17) / p ensure d µ is a q − p + pqp -Carleson measure for A q ω and also a q − p + pqq -Carleson mea-sure for A p ω . Consequently, the inequality (2) is applied to deduce that U . sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! k T g f k q − p + pqp A q ω , (21)and(22) V . sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! k f k q − p + pqq A p ω . A combination of (19), (21) and (22) gives when n → ∞ that k T g f k qA q ω . sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! p ( q − q − p + pq k T g f k q − A q ω · sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! q q − p + pq k f k A p ω = sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! k T g f k q − A q ω k f k A p ω . It follows that k T g k A p ω → A q ω = sup k f k Ap ω ≤ k T g f k A q ω . sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! . Case 2 -Assume 0 < p ≤ q <
2. From the equation (4), H¨older’s inequal-ity, Fubini’s theorem and Lemma 3 it follows that k T g f k qA q ω ≃ Z D Z Γ ( u ) | f ( z ) | | g ′ ( z ) | dA ( z ) ! q ω ( u ) dA ( u ) ≤ Z D N ( f )( u ) p (2 − q )2 Z Γ ( u ) | f ( z ) | − pq + p | g ′ ( z ) | dA ( z ) ! q ω ( u ) dA ( u ) ≤ Z D N ( f )( u ) p ω ( u ) dA ( u ) ! − q · Z D Z Γ ( u ) | f ( z ) | − pq + p | g ′ ( z ) | dA ( z ) ω ( u ) dA ( u ) ! q = k N ( f ) k p (2 − q )2 A p ω Z D | f ( z ) | − pq + p | g ′ ( z ) | ω ( T ( z )) dA ( z ) ! q ≃ k f k p (2 − q )2 A p ω V q , where V is defined in (20). Now (22) ensures that k T g f k qA q ω . k f k p (2 − q )2 A p ω sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! k f k q − p + pqq A p ω ! q = sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! q k f k qA p ω . Consequently we get thatsup k f k Ap ω ≤ k T g f k A q ω . sup I ⊆ T µ ( S ( I )) ω ( S ( I )) α ! , and then k T g k A p ω → A q ω . k g k ∗ ,α,ω . The proof is complete. (cid:3)
4. E ssential N orm and W eak C ompactness Proof of Theorem 2.
Lemma 5.3 in [11] and Lemma 4 show that B ≃ C and B ≃ D . A ≃ B follows by Lemma 6.To prove k T g k e , A p ω → A q ω . A , observe that for any h ∈ C α ( ω ∗ ), by Theorem1, k T g k e , A p ω → A q ω . k T g − T h k A p ω → A q ω = k T g − h k A p ω → A q ω . k g − h k C α ( ω ∗ ) , whence, taking infimum over h , we obtain k T g k e , A p ω → A q ω . A .Next, we turn to establishing(23) k T g k e , A p ω → A q ω & B = lim sup | I |→ R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) α / . Given a subarc I of T , consider f a , p defined in (11), where a = (1 − | I | ) ζ and ζ is the center point of I . Then S ( a ) = S ( I ). For the moment fix a compactoperator K : A p ω → A q ω . Thenlim | a |→ k K f a , p k A q ω = , SSENTIAL NORM 17 and so we find that k T g − K k & lim sup | a |→ k ( T g − K ) f a , p k A q ω & lim sup | a |→ ( k T g f a , p k A q ω − k K f a , p k A q ω ) & lim sup | a |→ k T g f a , p k A q ω . Upon taking the infimum of both sides of this inequality over all compactoperators K : A p ω → A q ω , it follows from Lemma 7 that(24) k T g k e , A p ω → A q ω & lim sup | a |→ k T g f a , p k A q ω & B . The proof is complete. (cid:3)
Next, we consider the weak essential norm of T g on A ω . Recall thatthe notion of weak compactness of an operator is non-trivial only on non-reflexive spaces. The non-reflexivity of A ω can be shown e.g. by construct-ing an isomorphic copy of the sequence space ℓ inside A ω . For this oneuses suitable normalized functions so that the closed subspace spanned bythese functions is isomorphic to ℓ . One may use e.g. functions f r k ,γ ( z ) = g r k ,γ ( z ) k g r k ,γ k A ω , z ∈ D , where r k ∈ (0 , , r k → ffi ciently fast, g r k ,γ ( z ) = (cid:16) − r k − r k z (cid:17) γ and γ >
0. Thefunctions f r k ,γ have the properties(i) R D \ D (1 ,ε ) | f r k ,γ | ω dA → , as k → ∞ for all ε > R D ∩ D (1 ,δ ) | f r k ,γ | ω dA → , as δ → k = , , . . . .The condition (i) follows from the doubling property of b ω and the choicefor the parameter γ to be large enough. The condition (ii) is evident. Theseproperties and the fact that r k → ffi ciently fast ensure that the map U : ℓ → A ω , U (( α k ) ∞ k = ) = ∞ X k = α k f r k ,γ is an isomorphism onto its image.In order to deal with the weak essential norm of T g on A ω we utilize theclassical Dunford-Pettis criterion (see e.g. [1, Theorem 5.2.9]), which statesthat a bounded set S ⊂ L µ , (where the measure µ is a probability measure)is relatively compact in the weak topology of L µ if and only if it is equi-integrable, i.e., lim µ ( A ) → sup f ∈ S Z A | f | d µ = . The application of this criterion in our setting is based on the next lemma.
Lemma 8.
Let ω ∈ b D. Suppose g ∈ C ( ω ∗ ) . For all non-zero a ∈ D , letJ ( a ) = { re i θ : | θ − arg a | < (1 − | a | ) / , − | a | < r < } andf a ( z ) = f a , ( z ) = (1 − | a | ) γ + (1 − az ) γ + ω ( S ( a )) , where γ is large enough so that lim | a |→ (1 − | a | ) γ − R | a | ω ( s ) ds = . Then lim | a |→ Z D \ J ( a ) | T g f a ( z ) | ω ( z ) dA ( z ) = . Proof.
We may assume that g (0) = < a < ≤ r < | θ | ≤ π , we have | − are i θ | ≥ c | θ | , where c > z ∈ D \ J ( a ) and a > , we have | f a ( z ) | . (1 − a ) γ + | θ | γ + ω ( S ( a )) ≤ (1 − a ) ( γ + ω ( S ( a )) ≃ (1 − a ) ( γ + (1 − a ) R | a | ω ( s ) ds = (1 − a ) γ − R | a | ω ( s ) ds . Proposition 5.1 in [11] and g ∈ C ( ω ∗ ) imply that g ∈ A ω . Therefore, byusing equation (4) twice we obtain Z D \ J ( a ) | T g f a ( z ) | ω ( z ) dA ( z ) ≃ Z D Z Γ ( u ) \ J ( a ) | f a ( z ) | | g ′ ( z ) | dA ( z ) ! ω ( u ) dA ( u ) . (1 − a ) γ − R | a | ω ( s ) ds k g k A ω , which tends to 0 as | a | →
1. The proof is complete. (cid:3)
Proof of Theorem 3.
Since compact operators are also weakly compact, wehave k T g k w , A ω → A ω ≤ k T g k e , A ω → A ω . lim sup | I |→ R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) / . SSENTIAL NORM 19
To verify the lower estimate for k T g k w , A ω → A ω , suppose that W : A ω → A ω isany weakly compact operator. Assume that J ( a ) and f a satisfy the condi-tions from Lemma 8. As k f a k A ω ≃
1, we have k T g − W k & k ( T g − W ) f a k A ω = Z D | T g f a ( z ) − W f a ( z ) | ω ( z ) dA ( z ) ≥ Z J ( a ) | T g f a ( z ) − W f a ( z ) | ω ( z ) dA ( z )(25) ≥ Z J ( a ) | T g f a ( z ) | ω ( z ) dA ( z ) − Z J ( a ) | W f a ( z ) | ω ( z ) dA ( z ) . Now by Lemma 8 lim | a |→ Z D \ J ( a ) | T g f a ( z ) | ω ( z ) dA ( z ) = , and therefore(26) lim sup | a |→ Z J ( a ) | T g f a ( z ) | ω ( z ) dA ( z ) = lim sup | a |→ k T g f a k A ω . Since the set { W f a : a ∈ D } is relatively weakly compact in A ω and henceequi-integrable in L ω , it holds that(27) lim | a |→ Z J ( a ) | W f a ( z ) | ω ( z ) dA ( z ) = | a |→ in (25) we have k T g − W k ≥ lim sup | a |→ k T g f a k A ω by (26) and (27). Hence by taking infimum over all weakly compact opera-tors W : A ω → A ω we get k T g k w , A ω → A ω ≥ lim sup | a |→ k T g f a k A ω . Finally, it follows from Lemma 7 thatlim sup | a |→ k T g f a k A ω & lim sup | I |→ R S ( I ) | g ′ ( z ) | ω ∗ ( z ) dA ( z ) ω ( S ( I )) / . The last inequality, along with Lemma 6 completes the proof. (cid:3) R eferences [1] F. Albiac and N. Kalton, Topics in Banach Space Theory, Springer-Verlag, NewYork, 2006.[2] O. Blasco, S. D´ıaz-Madrigal, J. Mart´ınez, M- Papadimitrakis and A. Siskakis, Semi-groups of composition operators and integral operators in spaces of analytic func-tions, Ann. Acad. Sci. Fenn. Math. (2013), no. 1, 67-89.[3] P.L. Duren, Theory of H p Spaces, Dover Publications, New York, 2000.[4] F. Hirsch and G. Lacombe, Elements of Functional Analysis, Graduate Texts inMathematics, 192. Springer-Verlag, New York, 1999.[5] B. Liu, E. Saksman, H-O. Tylli, Small composition operators on analytic vector-valued function spaces, Pacific J. Math. (1998), no. 2, 295-309.[6] J. Liu, Z. Lou and C. Xiong, Essential norms of integral operators on spaces ofanalytic functions, Nonlinear Anal. (2012), 5145–5156.[7] J. Laitila, S. Miihkinen, P.J. Nieminen, Essential norms and weak compactness ofintegration operators, Arch. Math. (2011), 39–48.[8] J.A. Pel´aez, J. R¨atty¨a, Embedding theorems for Bergman spaces via harmonic anal-ysis, Math. Ann. DOI 10.1007 / s00208-014-1108-5.[9] J.A. Pel´aez, J. R¨atty¨a, Generalized Hilbert operators on weighted Bergman spaces,Adv. Math. (2013), 227–267.[10] J.A. Pel´aez, J. R¨atty¨a, Trace class criteria for Toeplitz and composition operators onsmall Bergman spaces, arXiv:1501.00131 [math.FA][11] J.A. Pel´aez, J. R¨atty¨a, Weighted Bergman spaces induced by rapidly increasingweights, Mem. Amer. Math. Soc. (2014), no. 1066.[12] J. R¨atty¨a, Integration operator acting on Hardy and weighted Bergman spaces, Bull.Austral. Math. Soc. (2007), 431–446.[13] J.H. Shapiro, C. Sundberg, Compact composition operators on L , Proc. Amer.Math. Soc. (1990), no. 2, 443-449.[14] A. Siskakis, Weighted integrals of analytic functions, Acta Sci. Math. (Szeged) (2000), no. 3-4, 651–664.[15] J. Xiao, The Q p Carleson measure problem, Adv. Math. (2008), 2075–2088.[16] K. Zhu, Operator Theory in Function Spaces. Second edition. Mathematical Surveysand Monographs, 138. American Mathematical Society, Providence, RI, 2007.S anteri M iihkinen : D epartment of M athematics and S tatistics , U niversity of H elsinki ,B ox
68, 00014 H elsinki , F inland
E-mail address : [email protected] P ekka J. N ieminen : D epartment of M athematics and S tatistics , U niversity of H elsinki ,B ox
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E-mail address : [email protected] W en X u : D epartment of P hysics and M athematics , U niversity of E astern F inland , P.O. B ox oensuu , F inland E-mail address ::