Conditional Carleson measures and related operators on Bergman spaces
aa r X i v : . [ m a t h . F A ] M a y CONDITIONAL CARLESON MEASURES ANDRELATED OPERATORS ON BERGMAN SPACES
A. ALIYAN, Y. ESTAREMI AND A. EBADIANAbstract.
In this paper first we define generalized Carleson mea-sure. Then we consider a special case of it, named conditionalCarleson measure on the Bergman spaces. After that we givea characterization of conditional Carleson measures on Bergmanspaces. Moreover, by using this characterization we find an equiv-alent condition to boundedness of weighted conditional expectationoperators on Bergman spaces. Introduction and Preliminaries
Let C be the complex plane, D = { z ∈ C : | z | < } and T = { z ∈ C : | z | = 1 } . Likewise, we write R for the real line. The normalizedarea measure on D will be denoted by dA . As is known dA ( z ) =1 π dxdy = 1 π rdrdθ , in which z = x + iy = re iθ . For 0 < p < + ∞ and − < α < + ∞ , the (weighted) Bergman space L p,αa = L p,αa ( D ) of D , isthe space of all analytic functions in L p ( D , dA α ), where dA α ( z ) = ( α + 1)(1 − | z | ) α dA ( z ) . Let f ∈ L p ( D , dA α ) and k f k p.α = (cid:20)Z D | f ( z ) | p dA α ( z ) (cid:21) / p . Then for 1 ≤ p < ∞ , the space L p ( D , dA α ) is a Banach space withrespect to the norm k . k p.α . In addition, for 0 < p <
1, the space L p ( D , dA α ) is a complete metric space with respect to the metric de-fined by d ( f, g ) = k f − g k pp.α .Recall that for 1 < p < ∞ and − < α, the projection P α from L α,p into L α,pa is given by P α ( f )( w ) = Z D f ( z )(1 − wz ) α dA α ( z ) , w ∈ D . The projection P α is called the (weighted) Bergman projection on D .Let µ be a finite positive Borel measure on D and 0 < p < ∞ . We say Key words and phrases.
Bergman space,Conditional expectation,generalizedCarleson measure,conditional Carleson measure,Mobius transformation. A. ALIYAN, Y. ESTAREMI AND A. EBADIAN that µ is a Carleson measure on the Bergman space L pa , if there existsa constant C > Z D | f ( z ) | p dµ ( z ) ≤ C Z D | f ( z ) | p dA ( z ) , for all f ∈ L pa . In addition, the functions K α ( ω, z ) = 1(1 − wz ) α z, ω ∈ D are called the (weighted) Bergman kernels of D .Let ( X , M , µ ) be a complete sigma-finite measure space and let A bea sigma-subalgebra of M such that ( X , A , µ | A ) is also sigma-finite.Thecollection of (equivalence classes modulo sets of zero measure) M -measurable complex-valued functions on X will be denoted L ( M ).Moreover, we let L p ( M ) = L p ( X , M , µ ) and L p ( A ) = L p ( X , A , µ | A ),for 1 ≤ p < ∞ . As a consequence of the Radon-Nikodym theoremwe have that for each non-negative function f ∈ L ( M ) there exists aunique non-negative A -measurable function E A ( f ) ∈ L ( A ) such that R ∆ E ( f ) dµ = R ∆ f dµ for all ∆ ∈ A . The function E A ( f ) is called theconditional expectation of f with respect to A .Note that if E A P α = P α E A on L p ( D , M , A ), then L pa ( D ) = L p, a ( D ) isinvariant under the conditional expectation operator E A , i.e., E A ( L pa ( D )) ⊆ L pa ( D ). Suppose that M is the sigma-algebra of the Lebesgue measur-able sets in D , A is a subalgebra of M and E = E A is the relatedconditional expectation operator. For a non-constant analytic self-map ϕ on D and z ∈ D , put c z = { ζ ∈ D : ϕ ( ζ ) = ϕ ( z ) } , where D = { ζ ∈ D : ϕ ′ ( z )( ζ ) = 0 } .The map ϕ has finite multiplicity if we can find some n ∈ N suchthat for each z ∈ D , the level set c z consists of at most n points. By A = A ( ϕ ) we denote the sigma-algebra generated by { ϕ − ( U ) : U ⊂ C is open } . For the Lebesgue measure m on C we get that h = dA ◦ ϕ − dm is almost everywhere finite valued, because the finite measure A ◦ ϕ − isabsolutely continuous with respect to m . Now we define the weightedconditional expectation operator on L p.αα as follow: Definition 1.1.
Let u be an analytic function on D , 0 < p < ∞ and α, β > −
1. If A is a sigma-subalgebra of the sigma-algebra of theLebesgue measurable sets, then the weighted conditional expectationoperator T = M u E A on L p.αα is defined as T ( f )( z ) = u ( z ) E A ( f )( z ), foreach f ∈ L p.αα and z ∈ D such that E A ( f ) is an analytic function.Joseph Ball in [2] studied conditional expectation on H p and he findthat H p is invariant under E A ( f ) for an inner function f . After thatAleksandrov in [1] proved that for a complete sigma-subalgebra A of M , the sigma-algebra of Lebesgue measurable subsets of T . Then A = A ( f ) for some inner function f if and only if P E A = E A P , inwhich P is the Riesz projection, i.e. the orthogonal projection from L onto the Hardy space H . If we compose the discussions presentedin [8], [9] and [13], then we will get a formula for conditional expecta-tion operator corresponding to A ( ϕ ), in which ϕ is an arbitrary non-constant analytic self-map on D .In [5], the authors investigated some basic properties of some condi-tional expectation-type Toeplitz operators on Bergman spaces. In thispaper we consider the weighted conditional expectation operators onthe Bergman spaces and we will give some results based on Carlesonmeasures. Here we recall some basic Lemmas that we need them in thesequel. Lemma 1.2. [6] If < p < ∞ and α > − . Then | f ( z ) | ≤ k f k p,α (1 − | z | ) (2+ α ) /p for all f ∈ L p,αa and z ∈ D . Lemma 1.3. [14]
Let < p < ∞ and α > − . Then for each z ∈ D we have sup {| f ( z ) | : f ∈ L p,αa , k f k L p,αa ≤ } = 1(1 − | z | ) (2+ α ) /p . Let a ∈ D and ϕ a ( z ) = a − z − az , z ∈ D . The analytic transformation ϕ a is called the involutive Mobius trans-formation of D , which interchanges the origin and z . It is clear for all z ∈ D , ϕ a ( ϕ a ( z )) = z or ϕ a − = ϕ a and ϕ ′ a ( z ) = − − | a | (1 − ¯ az ) . The metric defined on D by ρ ( a, z ) = | ϕ a ( z ) | = (cid:12)(cid:12) a − z − az (cid:12)(cid:12) , a, z ∈ D , is called pseudo-hyperbolic distance. Maybe one of important proper-ties of the pseudo-hyperbolic distance is that it is Mobius invariant,that is ρ ( w, z ) = ρ ( ϕ a ( w ) , ϕ a ( z )) . In addition the metric given by β ( a, z ) = 12 log ρ ( a, z )1 − ρ ( a, z ) , a, z ∈ D , A. ALIYAN, Y. ESTAREMI AND A. EBADIAN is called Bergman metric on D ( or some where is called the hyperbolicmetric or the Poincar´e metric on D ). Hence we have β (0 , z ) = 12 log | z | − | z | , z ∈ D . The Bergman metric is also Mobius invariant. This means that β ( w, z ) = β ( ϕ a ( w ) , ϕ a ( z )) . for all ϕ ∈ Aut ( D ) and all z, w ∈ D . For any a ∈ D and r >
0, let D ( a, r ) = { z ∈ D : β ( a, z ) < r } be the Bergman metric disk with ”center” a and ”radius” r . It followsfrom the expression for β ( a, z ) that D ( a, r ) is a Euclidean disk withEuclidean center and radius C = 1 − s − s | a | a, R = 1 − | a | − s | a | s, where s = tanh r .We have D ( a, r ) = ϕ a ( D (0 , r )). As is known for each z ∈ D , the Riesz representation theorem implies that there exists aunique function K z in L a ( D ) such that for all f ∈ L a ( D ) f ( z ) = Z D f ( w ) K z ( w ) dA ( w ) . Let K ( z, a ) be the function on D × D defined by K ( z, a ) = K z ( a ). K ( z, a ) is called the Bergman kernel of D or the reproducing kernel of L a ( D ) because of the formula f ( z ) = Z D f ( w ) K ( z, w ) dA ( w )reproduces each f ∈ L a ( D ). And it is known that K ( z, w ) = − za ) .Moreover the functions K α ( ω, z ) = 1(1 − wz ) α z, ω ∈ D are called (weighted) Bergman kernels of D . If we set k a ( z ) = K ( z, a ) p K ( a, a ) = 1 − | a | (1 − ¯ az ) , then k a ( z ) ∈ L a ( D ) and they are called normalized reproducing kernelsof L ( D ). Easily we get that the derivative of ϕ a at z is equal to k a ( z ).This implies that | ϕ ′ a ( z ) | = | k a ( z ) | for all a, z ∈ D . Remark . Let 0 < p < ∞ , α > −
1, and a ∈ D . The normalizedBergman kernel function for L p,αa is defined by( k a ( z )) α +2 p = f p,αa ( z ) = (cid:0) − | a | (cid:1) (2+ α ) /p (1 − az ) α ) /p . Direct computations shows that k f p,αa k pL p,αa = (1 − | a | ) α (cid:13)(cid:13) (1 − ¯ az ) − − α (cid:13)(cid:13) L α, a . Therefore for all α > p > || f p,αa || pL p,αa = 1. Let f ∈ L p,αa and a ∈ D . If we set F ( z ) = f ( ϕ a ( z )) f p,αa ( z ) z ∈ D , then k F k L p,αa = k f k L p,αa .Here we recall some results of [10] that are basic and useful for inthe sequel. Lemma 1.5. [10]
For each r > there exists a positive constant C r such that C − r ≤ − | a | − | z | ≤ C r and C − r ≤ − | a | | − ¯ az | ≤ C r , for all a and z in D with β ( a, z ) < r . Moreover, if r is bounded above,then we may choose C r to be independent of r . By some elementary calculations we have(1 − | a | ) | − az | ≈ − | z | ) ≈ − | a | ) ≈ | D ( a, r ) | , for β ( a, z ) ≤ R and | D ( z, r ) | A ≈ (1 − | z | ) ≈ (1 − | ω | ) ≈ | D ( ω, s ) | A for β ( z, ω ) ≤ R .Note that for all z, w ∈ D , we have11 − | w | ≤ | − zw | ≤ − | z | = 1 + | z | − | z | ≤ − | z | . The notation A ≈ B means that there is a positive constant C inde-pendent of A and B such that C − B ≤ A ≤ CB .Now by using the results of [10] we obtain that there is a cover ofdisjoint balls in A for D . A. ALIYAN, Y. ESTAREMI AND A. EBADIAN
Lemma 1.6.
There is a positive integer N such that for any r ≤ ,there exists a sequence { a n } in D such that D ( a n , r ) ∈ A and satisfyingthe following conditions:(1) D = S + ∞ n =1 D ( a n , r ) (2) D ( a n , r ) ∩ D ( a m , r ) = ∅ if n = m ; (3) Any point in D belongs to at most N of the sets D ( a n , r ) . Here we recall a theorem concerning of the form of the functions inthe range of conditional expectation operators.
Theorem 1.7. [13]
Suppose that A = A ( ϕ ) for some ϕ ∈ A ( D ) with finite multiplicity. Suppose that none of the ζ j ( w ) belongs to { z : ϕ ′ ( z ) = 0 } and that w / ∈ f ( T ) . Then for every f in L pa ( D ) and ζ in ϕ − ( w ) , E A ( f )( ζ ) = P ζ j ǫc z f ( ζ j ) | ϕ ′ ( ξ j ) | P ζ j ǫc z | ϕ ′ ( ξ j ) | . Also, the function ω defined as ω ( ζ ) = | ϕ ′ ( ζ j ) | P ζ j ǫc z | ϕ ′ ( ζ j ) | is constant on each level set. In particular if E A P α = P α E A , then ω is constant on D . Every non-constant analytic self-map ϕ on D has finite multiplicity,since if there is w ∈ ϕ ( D ) such that ϕ − ( { w } ) is infinite, then ϕ ( z ) = w for all z ∈ D . Therefore the Theorem 1.7 holds for all non-constantanalytic self-map ϕ on D . Now we recall some assertions that we willuse them in the sequel. Lemma 1.8. [12]
There is a constant
C > such that | f ( z ) | p ≤ C (1 − | z | ) α +2 Z D ( z,r ) | f ( ω ) | p dA α ( ω ) , for all f analytic, z ∈ D , p > , and r ≤ . We especially note thatthe constant C above is independent of r and p . The restriction r < above can be replaced by r < R for any positive number R. Lemma 1.9. [12]
For any r > and a ∈ D , we have the followingequalities: | D ( a, r ) | = (1 − | a | ) s (1 − | a | s ) , inf z ∈ D ( a,r ) | k a ( z ) | = (1 − s | a | ) (1 − | a | ) , sup z ∈ D ( a,r ) | k a ( z ) | = (1 + s | a | ) (1 − | a | ) , where s = tanh r ∈ (0 , ./ for r ∈ (0 , and | D ( a, r ) | is the (nor-malized) area of D(a,r). Lemma 1.10.
Let < p ≤ q < ∞ and z ∈ D . Then for each f ∈ L p,αa ( D ) we have (cid:18)Z D | f ( z ) | q dµ ( z ) (cid:19) ≤ ( k f k L p,αa ) q ( µ ( D )) p − qp . Proof. (cid:18)Z | f ( z ) | q dµ ( z ) (cid:19) ≤ (cid:18)Z ( | f ( z ) | q ) pq dµ ( z ) (cid:19) qp (cid:18)Z dµ ( z ) (cid:19) p − qp = (cid:18)Z ( | f ( z ) | q ) pq dµ ( z ) (cid:19) qp ( µ ( D )) p − qp = ( k f k L p,αa ) q ( µ ( D )) p − qp . (cid:3) Hence by this lemma we conclude that R D | k a ( z ) | q dA α ( z ) ≤ µ ( D ) p − qp .2. Main Results
In this section first we define Generalized Carleson measures.
Definition 2.1.
Let µ finite positive Borel measure on D and p >
0. We say that µ is a ( L α,pa ( D ) , p ) -generalized Carleson measure on L α,pa ( D ) if there exists a closed subspace M ⊆ L α,pa ( D ) such that Z D | f ( z ) | p dµ ≤ C k f k pL α,pa for all f ∈ M . Moreover, if we set M = L α,pa ( A ) = E A ( L α,pa ( D )) forsome sigma-subalgebra A , then we say that µ is a ( L α,pa , p ) -conditionalCarleson measure on L α,pa ( D ) if there exists C > Z D | E A ( f )( z ) | p dµ ≤ C k f k pL p,αa for all f ∈ L α,pa ( D ).Now we find an upper bound for the evaluation function f → E A ( f )( z ). Theorem 2.2.
Let f ∈ L p,αa ( D ) such that E A ( f ) ∈ L p,αa ( A ) , as statedin Theorem 1.7, < p < ∞ and α > − . If there exists r > suchthat c z ⊂ D ( z, r ) , then we have | E A ( f )( z ) | ≤ E A | f | ( z ) ≤ C r sup a ∈ D | k a ( z ) | k f k L p,αa for all z ∈ D . A. ALIYAN, Y. ESTAREMI AND A. EBADIAN
Proof.
It is known that for the conditional expectation E A , we have | E A ( f ) | ≤ E A ( | f | ). Moreover, there exists some ζ n ∈ c z such that foreach ζ j ∈ c z we have | f ( ζ j ) | ≤ | f ( ζ n ) | . Hence we get that E A | f | ( z ) = ( X ζ j ∈ c z ω ( ζ j )( | f | ( ζ j ))) ≤ ( X ζ j ∈ c z ω ( ζ j ))( | f ( ζ n ) | ) . Also by the Lemma1.2 we have( X ζ j ∈ c z ω ( ζ j ))( | f ( ζ n ) | ) = | f ( ζ n ) |≤ k f k L p,αa (1 − | ζ n | ) (2+ α ) /p . Moreover, by Lemmas 1.5 and Lemma1.3 we conclude that k f k L p,αa (1 − | ζ n | ) (2+ α ) /p ≤ C r k f k L p,αa (1 − | z | ) (2+ α ) /p = C sup b ∈ D | k b ( z ) | k f k L p,αa . for β ( z, ζ n ) ≤ r . This completes the proof. (cid:3) Here we get the next corollary.
Corollary 2.3.
Under the assumptions of Theorem 2.2 we have | E A ( k a )( z ) | ≤ E A ( | k a ( z ) | ) ≤ C r sup a ∈ D | k a ( z ) | z ∈ D for all a ∈ D . Let 0 < p < ∞ and α > −
1. Then the function Ψ αa ( µ )( z ) defined asΨ αa ( µ )( z ) = Z D − | a | | − az | ! (2+ α ) dµ ( z ) , is well defined. In the sequel we provide some equivalence conditionsto conditional- Carleson measure on the Bergman spaces. Theorem 2.4.
Let u be an analytic function and µ be a finite Borelmeasure on D , < p ≤ q < ∞ and α, β > − . In addition, let A = A ( ϕ ) , in which ϕ is a non-constant analytic self map on D . If ∞ P k =1 1(1 − tanh ( r ) | a k | ) < ∞ for the sequence { a k } in the Lemma 1.6 andsome < r < . Then the followings are equivalent: (1) There exists C > such that for every f ∈ L p,αa ( D ) we have Z D | E A ( f )( z ) | p dµ ( z ) ≤ C k f k pL p,αa . (2) There exists C such that µ ( D ( a, r )) ≤ C − | a | (1 − tanhr | a | ) ! ( α +2) , for all a ∈ D such that D ( a, r ) ∈ A . (3) There exists C such that Ψ αa ( µ )( z ) ≤ C , for all a ∈ D such that f p,αa is A -measurable.Proof. (3) −→ (1) Let (3) holds. we have Z D | E ( f )( z ) | p dµ ( z ) ≤ Z D ( | f ( z ) | ) p dµ ( z ) ≤ Z D (cid:18) − | z | (cid:19) ( α +2) dµ ( z ) k f k pL p,αa ≈ Z D ( (1 − | a | ) | − ¯ az | ) ( α +2) dµ ( z ) k f k pL p,αa = Ψ αa ( µ )( z ) k f k pL p,αa ≤ C k f k pL p,αa . (1) −→ (3) Let 0 < p < ∞ and α, β > −
1. Since || f p,αa || pL p,αa = 1, thenwe have Ψ αa ( µ )( z ) = Z D − | a | | − ¯ az | ! ( α +2) dµ ( z )= Z D | f p,αa ( z ) | p dµ ( z )= Z D | E ( f p,αa ( z )) | p dµ ( z ) ≤ C . (2) −→ (3)Let 0 < p < ∞ and α > −
1. First we assume that a = 0, hence µ ( D ) ≤ C . If | a | ≤ ( a ∈ D ), then by [7] we obtain that Z D − | a | | − aω | ! (2+ α ) dµ ( ω ) ≤ µ ( D ) ≤ C . If | a | > ( a ∈ D ), then we define E n as E n = { z ∈ D : (cid:12)(cid:12)(cid:12)(cid:12) z − a | a | (cid:12)(cid:12)(cid:12)(cid:12) < n (1 − | a | ) } n = 1 , , , ... A. ALIYAN, Y. ESTAREMI AND A. EBADIAN such that µ ( E n ) ≤ (2 n (1 − | a | )) ( α +2) ≤ (2 n (1 − | a | )) ( α +2) . Moreover we have1 − | a | | − ¯ az | ≤ − | a |≤ − | a | a ∈ E , − | a | | − ¯ az | ≤ n (1 − | a | ) ≤ n (1 − | a | ) a ∈ E n \ E n − . Since for each r > < tanhr <
1, then we conclude that Z D − | a | | − aω | ! (2+ α ) dµ ( ω ) ≤ Z E − | a | | − aω | ! (2+ α ) dµ ( ω )+ ∞ X n =2 Z E n \ E n − − | a | | − aω | ! (2+ α ) dµ ( ω ) ≤ ∞ X n =1 n (1 − | a | )) (2+ α ) µ ( E n ) ≤ C ∞ X n =1 n ( α +2) ≤ C . (3) −→ (2)Suppose (3) is true then with the Lemma 1.3 and 1.9, we have (cid:16) (1 − tanh ( r ) | a | ) −| a | (cid:17) ( α +2) µ ( D ( a, r )) ≤ Z D ( a,r ) (1 − tanh ( r ) | a | ) − | a | ! ( α +2) dµ ( z ) ≤ Z D ( a,r ) (1 + tanh ( r ) | a | ) − | a | ! ( α +2) dµ ( z )= Z D ( a,r ) sup z ∈ D ( a,r ) | f p,αa ( z ) | p dµ ( z ) ≤ Z D sup z ∈ D | f p,αa ( z ) | p dµ ( z )= Z D (cid:18) − | z | (cid:19) (2+ α ) dµ ( z ) ≈ Z D − | a | | − ¯ az | ! (2+ α ) dµ ( z )= Ψ αa ( µ )( z ) ≤ C . For all a ∈ D .(2) −→ (1)By some properties of conditional expectation, we have Z D | E ( f )( z ) | p dµ ≤ Z D | f ( z ) | p dµ ≤ ∞ X k =1 µ ( D ( a k , r )) sup z ∈ D ( a k ,r ) | f ( z ) | p . Here by the Lemma 1.8 ,[[6] page 60] we havesup {| f ( z ) | p : z ∈ D ( a k , r ) } ≤ C (1 − | a k | ) α +2 Z D ( a k, r ) | f ( w ) | p dA α ( w ) . A. ALIYAN, Y. ESTAREMI AND A. EBADIAN using Holders inequality and let q be the conjugate index of q ′ ,then ∞ X k =1 µ ( D ( a k , r )) sup z ∈ D ( a k ,r ) | f ( z ) | p ≤ ∞ X k =1 µ ( D ( a k , r ))(1 − | a k | ) α +2 Z D ( a k ,r ) | f ( w ) | p dA α ( w ) ≤ ∞ X k =1 (cid:18) µ ( D ( a k , r ))(1 − | a k | ) α +2 (cid:19) q q ∞ X k =1 (cid:18)Z D ( a k ,r ) | f ( w ) | p dA α ( w ) (cid:19) q ′ q ′ ≤ ∞ X k =1 (cid:18) − tanh ( r ) | a k | ) (cid:19) α +2 q q × n X k =1 (cid:18)Z D ( a k , r ) | f ( w ) | p dA α ( w ) (cid:19) q ′ q ′ ≤ N k f k pL p,αa . Then Z D | E ( f )( z ) | p dµ ≤ C k f k pL p,αa . (cid:3) Notice
In Theorem 2.4 the notation A ≈ B means that there existsa positive constant C independent of A and B such that C − B ≤ A ≤ CB . Moreover, the best constants C , C and C are in factcomparable, i,e., there exists a positive constant M such that1 M C ≤ C ≤ MC , M C ≤ C ≤ MC . Theorem 2.5.
Under the assumptions of the Theorem 2.4 we obtainthat the weighted Conditional expectation operator M u E is boundedfrom L p,αa into L p,βa if and only if there exists some C such that Ψ αa ( µ βu )( z ) ≤ C for a, z ∈ D = { z ∈ D : ϕ ′ ( z ) = 0 } .Proof. Suppose that M u E is bounded. So we can find C > k M u E ( f ) k pL p,βa ≤ C k f k pL p,αa , and k M u E ( f ) k pL p,βa = Z D | E ( f )( z ) | p | u ( z ) | p dA β ( z )= Z D | E ( f )( z ) | p dµ βu ( z ) ≤ C k f ( z ) k pL p,αa , (2.1)In which µ βu = R D | u ( z ) | p dA β ( z ). This means that dµ βu is an ( L p,αa )-expectation Carleson measure. By theorem2.4, this is equivalent to Z D − | a | | − az | ! (2+ α ) | u ( z ) | p dA β ( z ) ≤ Ψ αa ( µ βu )( z ) ≤ C. This completes the proof. (cid:3)
Example 2.6.
Let u be an analytic function on D , 0 < p ≤ q < ∞ and α, β > −
1. If the conditional expectation operator E is identity,then the Multiplication operator M u is bounded from L p,αa into L q,βa ifand only if there exists some C such thet Z D − | a | | − ¯ az | ! (2+ α ) qp | u ( z ) | q dA β ( z ) ≤ C. References [1] A. B. Aleksandrov, Measurable partitions of the circumf erence, induced by in-ner functions, Translated from Zepiski Nauchnykh Seminarov Leningradskogootdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol 149,(1986) 103-106.[2] J. A. Ball, Hardy space expectation operators and reducing subspace, Proc.Amer. Math. Soc. 47 (1975) 351-357.[3] Hastings, W.W., A Carleson measure theorem for Bergman spaces. Proceed-ings of the American Mathematical Society, 1975. 52(1): p. 237-241.[4] Luecking, D.H., Forward and reverse Carleson inequalities for functions inBergman spaces and their derivatives. American Journal of Mathematics, 1985.107(1): p. 85-111.[5] Jabbarzadeh,M.R.and M.Moradi,C-E type Toeplitz operators on L a ( D ) . Op-erators and Matrices, 2017. 11: p. 875-884.[6] Vukoti, D., A sharp estimate for Proceedings of the American MathematicalSociety, 1993. 117(3): p. 753-756.[7] Aulaskari, R., D.A. Stegenga, and J. Xiao, Some subclasses of BMOA andtheir characterization in terms of Carleson measures. Rocky Mountain Journalof Mathematics, 1996. 26: p. 485-506.[8] Hornor, W.E. and J.E. Jamison, Properties of lsometry-Inducing Maps of theUnit Disc. Complex Variables and Elliptic Equations, 1999. 3: p. 69-84.[9] Carswell, B.J. and M.I. Stessin, Conditional expectation and the Bergmanprojection. Journal of Mathematical Analysis and Applications, 2008. 341(1):p. 270-275. A. ALIYAN, Y. ESTAREMI AND A. EBADIAN [10] Zhu, K., Spaces of holomorphic functions in the unit ball. Vol. 226. 2005:Springer Science .[11] Cuckovi, Z. and R. Zhao, Weighted composition operators between differentweighted Bergman spaces and different Hardy spaces. Illinois Journal of math-ematics, 2007. 51(2): p. 479-498.[12] Zhu, K., Operator theory in function spaces. 2007: American MathematicalSoc.[13] Jabbarzadeh, M. and M. Hassanloo, Conditional expectation operators on theBergman spaces. Journal of Mathematical Analysis and Applications, 2012.385(1): p. 322-325.[14] Ueki, S.-i., Order bounded weighted composition operators mapping into theBergman space. Complex Analysis and Operator Theory, 2012. 6(3): p. 549-560.
A. Aliyan
E-mail address : [email protected] Y. estaremi
E-mail address : [email protected] A. Ebadian
E-mail address : [email protected]@gmail.com