arXiv:1805.07573v1 [math.FA] 19 May 2018 ramaueon measure area Let h projection The C h pc falaayi ucin in functions analytic all of space the alsnmauecniinlCreo esr,oistransform measure,Mobius Carleson measure,conditional Carleson − into ndby fined L Let epc otenorm the to respect hnfr1 for Then π 1 p 1 dxdy Let e od n phrases. and words Key ealta o 1 for that Recall ( : D < α < | µ f EAE PRTR NBRMNSPACES BERGMAN ON OPERATORS RELATED L z dA , | ODTOA ALSNMAUE AND MEASURES CARLESON CONDITIONAL ∈ a α,p eafiiepstv oe esr on measure Borel positive finite a be C prtr nBrmnspaces. Bergman equiv- expectatio on give conditional an operators weighted find we of we boundedness Bergman that to characterization conditional condition on this alent After named using measures by it, Carleson Moreover, spaces. of conditional spaces. Bergman of case characterization the special a on a consider measure we Carleson Then sure. Abstract. 1 = = L d α etecmlxplane, complex the be sgvnby given is p sacmlt ercsaewt epc otemti de- metric the to respect with space metric complete a is ) π ( 1 + ( } ,g f, D rdrdθ .AIA,Y SAEIADA EBADIAN A. AND ESTAREMI Y. ALIYAN, A. ≤ ieie ewrite we Likewise, . ∞ P dA , α 1. = ) h wihe)Brmnspace Bergman (weighted) the , < p ( P f α α nrdcinadPreliminaries and Introduction )( nwhich in , nti ae rtw en eeaie alsnmea- Carleson generalized define we first paper this In dA D and ) k scle h wihe)Brmnpoeto on projection Bergman (weighted) the called is k w < p < ∞ f f α ilb eoe by denoted be will = ) − k k ( h space the , z . p.α k g ( = ) p.α ega pc,odtoa expectation,generalized space,Conditional Bergman Z k D ∞ = p.α p nadto,fr0 for addition, In . (1 z . α and Z = − 1)(1 + D f D | w x ( f − R z L 1 z = ( + ) 1 z ) p o h elln.Tenormalized The line. real the for 2+ ( ) L α, < iy { D | | − p z p α dA dA , ( = dA D ∈ z dA | dA , α h projection the 2 re C α α ( ) D ( z saBnc pc with space Banach a is ) α si known is As . iθ z : ) dA α n 0 and ) o 0 For . | w , ,where ), L z 1 / ( | a p,α p < p < z < . ) . = < p < 1 ∈ } < p < L D and a p,α ation. . P ,tespace the 1, ( α ∞ D T from of ) dA esay We . + n = ∞ ( z { D L = ) and z is , α,p D ∈ . 2 A. ALIYAN, Y. ESTAREMI AND A. EBADIAN
p that µ is a Carleson measure on the Bergman space La, if there exists a constant C > 0 such that
|f(z)|pdµ(z) ≤ C |f(z)|pdA(z), D D Z Z p for all f ∈ La. In addition, the functions 1 K (ω, z)= z,ω ∈ D α (1 − wz)2+α are called the (weighted) Bergman kernels of D.
Let (X , M,µ) be a complete sigma-finite measure space and let A be a sigma-subalgebra of M such that (X , A,µ |A) is also sigma-finite.The collection of (equivalence classes modulo sets of zero measure) M- measurable complex-valued functions on X will be denoted L0(M). p p p p Moreover, we let L (M) = L (X , M,µ) and L (A) = L (X , A,µ|A), for 1 ≤ p < ∞. As a consequence of the Radon-Nikodym theorem we have that for each non-negative function f ∈ L0(M) there exists a 0 unique non-negative A-measurable function EA(f) ∈ L (A) such that E(f)dµ = fdµ for all ∆ ∈ A . The function EA(f) is called the ∆ ∆ Rconditional expectationR of f with respect to A. p D p D p,0 D Note that if EAPα = PαEA on L ( , M, A), then La( )= La ( ) is p D invariant under the conditional expectation operator EA, i.e., EA(La( )) ⊆ p D La( ). Suppose that M is the sigma-algebra of the Lebesgue measur- able sets in D, A is a subalgebra of M and E = EA is the related conditional expectation operator. For a non-constant analytic self- map ϕ on D and z ∈ D, put cz = {ζ ∈ D0 : ϕ(ζ) = ϕ(z)}, where ′ D0 = {ζ ∈ D : ϕ (z)(ζ) =06 }. The map ϕ has finite multiplicity if we can find some n ∈ N such that for each z ∈ D, the level set cz consists of at most n points. By A = A(ϕ) we denote the sigma-algebra generated by {ϕ−1(U): U ⊂ − C C dA◦ϕ 1 is open} . For the Lebesgue measure m on we get that h = dm is almost everywhere finite valued, because the finite measure A◦ϕ−1 is absolutely continuous with respect to m. Now we define the weighted p.α conditional expectation operator on Lα as follow: Definition 1.1. Let u be an analytic function on D, 0
A = A(f) for some inner function f if and only if P EA = EAP , in which P is the Riesz projection, i.e. the orthogonal projection from L2 onto the Hardy space H2. If we compose the discussions presented in [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 this paper we consider the weighted conditional expectation operators on the Bergman spaces and we will give some results based on Carleson measures. Here we recall some basic Lemmas that we need them in the sequel. Lemma 1.2. [6] If 0
p,α 1 sup{|f(z)| : f ∈ La , kfk p,α ≤ 1} = . La (1 −|z|2)(2+α)/p Let a ∈ D and a − z ϕ (z)= , z ∈ D. a 1 − az
The analytic transformation ϕa is called the involutive Mobius trans- formation of D, which interchanges the origin and z. It is clear for all −1 z ∈ D,ϕa(ϕa(z)) = z or ϕa = ϕa and 1 −|a|2 ϕ′ (z)= − . a (1 − az¯ )2
The metric defined on D by a − z ρ(a, z)= |ϕ (z)| = , a, z ∈ D, a 1 − az 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(ϕa(w),ϕa(z)). In addition the metric given by 1 1+ ρ(a, z) β(a, z)= log , a, z ∈ D, 2 1 − ρ(a, z) 4 A. ALIYAN, Y. ESTAREMI AND A. EBADIAN is called Bergman metric on D( or some where is called the hyperbolic metric or the Poincar´emetric on D). Hence we have 1 1+ |z| β(0, z)= log , z ∈ D. 2 1 −|z| The Bergman metric is also Mobius invariant. This means that
β(w, z)= β(ϕa(w),ϕa(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) f(z)= f(w)Kz(w)dA(w). D Z Let K(z, a) be the function on D × D defined by K(z, a) = Kz(a). K(z, a) is called the Bergman kernel of D or the reproducing kernel of 2 D La( ) because of the formula f(z)= f(w)K(z,w)dA(w) D Z reproduces each f ∈ L2(D). And it is known that K(z,w) = 1 . a (1−za)2 Moreover the functions 1 K (ω, z)= z,ω ∈ D α (1 − wz)2+α are called (weighted) Bergman kernels of D. If we set K(z, a) 1 −|a|2 ka(z)= = , K(a, a) (1 − az¯ )2 2 D then ka(z) ∈ La( ) and theyp are called normalized reproducing kernels of L2(D). Easily we get that the derivative of ϕa at z is equal to ka(z). This implies that ′ |ϕ a(z)| = |ka(z)| for all a, z ∈ D. 5 Remark 1.4. Let 0 2 (2+α)/p α+2 1 −|a| p p,α (ka(z)) = fa (z)= 2(2+α)/p . (1 − az) Direct computations shows that p,α p 2 2+α −2−α 2 kf k p,α = (1 −|a| ) (1 − az¯ ) α,2 . a La La p,α p p,α Therefore for all α> 1 and p> 0 we have ||f || p,α = 1. Let f ∈ L a La a and a ∈ D. If we set p,α D F (z)= f(ϕa(z))fa (z) z ∈ , then kF k p,α = kfk p,α . La La Here we recall some results of [10] that are basic and useful for in the sequel. Lemma 1.5. [10] For each r > 0 there exists a positive constant Cr such that 2 −1 1 −|a| C ≤ ≤ Cr r 1 −|z|2 and 1 −|a|2 C−1 ≤ ≤ C , r |1 − az¯ | r for all a and z in D with β(a, z) Now by using the results of [10] we obtain that there is a cover of disjoint balls in A for D. 6 A. ALIYAN, Y. ESTAREMI AND A. EBADIAN Lemma 1.6. There is a positive integer N such that for any r ≤ 1, there exists a sequence {an} in D such that D(an,r) ∈A and satisfying the following conditions: D +∞ (1) = n=1 D(an,r) r r (2) D(an, ) ∩ D(am, )= ∅ if n =6 m; S 4 4 (3)Any point in D belongs to at most N of the sets D(an, 2r). Here we recall a theorem concerning of the form of the functions in the 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 ζj(w) belongs to ′ T p D {z : ϕ (z)=0} and that w∈ / f( ). Then for every f in La( ) and ζ in ϕ−1(w), f(ζj ) ′ 2 ζj ǫcz |ϕ (ξj )| EA(f)(ζ)= 1 . ′ 2 Pζj ǫcz |ϕ (ξj )| Also, the function ω defined as P 1 ′ 2 |ϕ (ζj )| ω(ζ)= 1 ′ 2 ζj ǫcz |ϕ (ζj )| is constant on each level set. InP particular if EAPα = PαEA , 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−1({w}) is infinite, then ϕ(z)= w for all z ∈ D. Therefore the Theorem 1.7 holds for all non-constant analytic self-map ϕ on D. Now we recall some assertions that we will use them in the sequel. Lemma 1.8. [12] There is a constant C > 0 such that p C p |f(z)| ≤ |f(ω)| dAα(ω), (1 −|z|2)α+2 ZD(z,r) for all f analytic, z ∈ D,p > 0, and r ≤ 1. We especially note that the constant C above is independent of r and p. The restriction r < 1 above can be replaced by r < R for any positive number R. Lemma 1.9. [12] For any r > 0 and a ∈ D, we have the following equalities: 2 (1 −|a|2) s2 |D(a, r)| = 2 , (1 −|a|2s2) 4 2 (1 − s |a|) inf |ka(z)| = 2 , z∈D(a,r) (1 −|a|2) 7 4 2 (1 + s |a|) sup |ka(z)| = , 2 2 z∈D(a,r) (1 −|a| ) where s = tanh r ∈ (0, ./762) for r ∈ (0, 1) and |D(a, r)| is the (nor- malized) area of D(a,r). Lemma 1.10. Let 0 < p ≤ q < ∞ and z ∈ D. Then for each p,α D f ∈ La ( ) we have p−q q q |f(z)| dµ(z) ≤ (kfk p,α ) (µ(D)) p . La ZD Proof. q p−q p p q q p |f(z)| dµ(z) ≤ (|f(z)| ) q dµ(z) dµ(z) Z Z q Z p − q p p q = (|f(z)| ) q dµ(z) (µ(D)) p Z p−q q p,α p = (kfkLa ) (µ(D)) . − q p q Hence by this lemma we conclude that |ka(z)| dAα(z) ≤ µ(D) p . D R 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 > α,p D 0. We say that µ is a (La ( ),p) -generalized Carleson measure on α,p D α,p D La ( ) if there exists a closed subspace M ⊆ La ( ) such that p p |f(z)| dµ ≤ C kfk α,p La ZD α,p α,p D for all f ∈ M. Moreover, if we set M = La (A) = EA(La ( )) for α,p some sigma-subalgebra A, then we say that µ is a (La ,p) -conditional α,p D Carleson measure on La ( ) if there exists C > 0 such that p p |EA(f)(z)| dµ ≤ C kfkLp,α D a Z α,p D for all f ∈ La ( ). Now we find an upper bound for the evaluation function f → EA(f)(z). p,α D p,α Theorem 2.2. Let f ∈ La ( ) such that EA(f) ∈ La (A), as stated in Theorem 1.7, 0 |E (f)(z)| ≤ E |f| (z) ≤ C sup |k (z)|kfk p,α A A r a La a∈D0 for all z ∈ D0. 8 A. ALIYAN, Y. ESTAREMI AND A. EBADIAN Proof. It is known that for the conditional expectation EA, we have |EA(f)| ≤ EA(|f|). Moreover, there exists some ζn ∈ cz such that for each ζj ∈ cz we have |f(ζj)|≤|f(ζn)|. Hence we get that EA|f| (z) = ( ω(ζj)(|f|(ζj))) ζXj ∈cz ≤ ( ω(ζj))(|f(ζn)|). ζXj ∈cz Also by the Lemma1.2 we have ( ω(ζj))(|f(ζn)|) = |f(ζn)| ζXj ∈cz kfk p,α ≤ La . 2 (2+α)/p (1 −|ζn| ) Moreover, by Lemmas 1.5 and Lemma1.3 we conclude that kfk p,α Cr kfk p,α La ≤ La 2 (2+α)/p 2 (2+α)/p (1 −|ζn| ) (1 −|z| ) p,α = C sup |kb(z)|kfkLa . b∈D0 for β(z,ζn) ≤ r. This completes the proof. Here we get the next corollary. Corollary 2.3. Under the assumptions of Theorem 2.2 we have |EA(ka)(z)| ≤ EA(|ka(z)|) ≤ Cr sup |ka(z)| z ∈ D0 a∈D0 for all a ∈ D0. α Let 0 2 (2+α) α 1 −|a| Ψa (µ)(z)= 2 dµ(z), D |1 − az| Z ! is well defined. In the sequel we provide some equivalence conditions to conditional- Carleson measure on the Bergman spaces. Theorem 2.4. Let u be an analytic function and µ be a finite Borel measure on D, 0 < p ≤ q < ∞ and α,β > −1. In addition, let A = A(ϕ), in which ϕ is a non-constant analytic self map on D. If ∞ 1 < ∞ for the sequence {a } in the Lemma 1.6 and (1−tanh(r)|a |)2 k k=1 k someP 0 p p |EA(f)(z)| dµ(z) ≤ C1 kfkLp,α . D a Z 9 (2) There exists C2 such that (α+2) 1 −|a|2 µ(D(a, r)) ≤ C2 2 , (1 − tanhr |a|) ! for all a ∈ D such that D(a, r) ∈A. (3) There exists C3 such that α Ψa (µ)(z) ≤ C3, p,α for all a ∈ D such that fa is A-measurable. Proof. (3) −→ (1) Let (3) holds. we have |E(f)(z)|pdµ(z) ≤ (|f(z)|)pdµ(z) D0 Z DZ0 (α+2) 1 p ≤ 2 dµ(z) kfkLp,α D 1 −|z| a Z 0 2 (α+2) (1 −|a| ) p ≈ ( 2 ) dµ(z) kfkLp,α D |1 − az¯ | a Z 0 α p = Ψ (µ)(z) kfk p,α a La p ≤ C kfk p,α . 3 La p,α p (1) −→ (3) Let 0 10 A. ALIYAN, Y. ESTAREMI AND A. EBADIAN such that n (α+2) µ(En) ≤ (2 (1 −|a|)) ≤ (2n(1 −|a|2))(α+2). Moreover we have 1 −|a|2 1 ≤ |1 − az¯ |2 1 −|a| 2 ≤ a ∈ E , 1 −|a|2 1 1 −|a|2 |1 − az¯ |2 1 ≤ 22n(1 −|a|) 2 ≤ a ∈ E \ E . 22n(1 −|a|2) n n−1 Since for each r> 0 we have 0 < tanhr < 1, then we conclude that (2+α) (2+α) 1 −|a|2 1 −|a|2 dµ(ω) ≤ dµ(ω) |1 − aω|2 |1 − aω|2 ZD0 ! ZE1 ! (2+α) ∞ 1 −|a|2 + 2 dµ(ω) n=2 En\En−1 |1 − aω| ! X Z ∞ 2 ≤ µ(En) 2n 2 (2+α) n=1 (2 (1 −|a| )) X ∞ 1 ≤ C 2 2n(α+2) n=1 X ≤ C3. 11 (3) −→(2) Suppose (3) is true then with the Lemma 1.3 and 1.9, we have 2 (α+2) (1−tanh(r)|a|) µ(D(a, r)) 1−|a|2 (α+2) (1 − tanh(r) |a|)2 ≤ dµ(z) 1 −|a|2 ZD(a,r) ! (α+2) (1 + tanh(r) |a|)2 ≤ dµ(z) 1 −|a|2 ZD(a,r) ! p,α p = sup |fa (z)| dµ(z) ZD(a,r) z∈D(a,r) p,α p ≤ sup |fa (z)| dµ(z) ZD0 z∈D0 1 (2+α) = 2 dµ(z) D 1 −|z| Z 0 (2+α) 1 −|a|2 ≈ 2 dµ(z) D |1 − az¯ | Z 0 ! α = Ψa (µ)(z) ≤ C2. For all a ∈ D0. (2) −→ (1) By some properties of conditional expectation, we have |E(f)(z)|pdµ ≤ |f(z)|pdµ D0 D0 Z Z∞ p ≤ µ(D(ak,r)) sup |f(z)| . z∈D(ak,r) Xk=1 Here by the Lemma 1.8 ,[[6] page 60] we have p C p sup{|f(z)| : z ∈ D(ak,r)} ≤ |f(w)| dAα(w). 2 α+2 (1 −|ak| ) ZD(ak,r) 12 A. ALIYAN, Y. ESTAREMI AND A. EBADIAN using Holders inequality and let q be the conjugate index of q′,then ∞ p µ(D(ak,r)) sup |f(z)| k=1 z∈D(ak,r) X∞ µ(D(ak,r)) p ≤ |f(w)| dAα(w) (1 −|a |2)α+2 k=1 k ZD(ak,r) X q q′ ∞ 1 ∞ 1 q q′ µ(D(ak,r)) p ≤ |f(w)| dAα(w) 2 α+2 (1 −|ak| ) D(ak,r) Xk=1 Xk=1 Z q ∞ α+2 1 q ≤ 2 (1 − tanh(r) |ak|) Xk=1 q′ n 1 q′ p × |f(w)| dAα(w) D(ak,2r) Xk=1 Z p ≤ N kfk p,α . La Then p p |E(f)(z)| dµ ≤ C1 kfkLp,α . D a Z 0 Notice In Theorem 2.4 the notation A ≈ B means that there exists a positive constant C independent of A and B such that C−1B ≤ A ≤ CB. Moreover, the best constants C1, C2 and C3 are in fact comparable, i,e., there exists a positive constant M such that 1 C ≤ C ≤ MC , M 1 2 1 1 C ≤ C ≤ MC . M 1 3 1 Theorem 2.5. Under the assumptions of the Theorem 2.4 we obtain that the weighted Conditional expectation operator MuE is bounded p,α p,β from La into La if and only if there exists some C such that α β Ψa (µu)(z) ≤ C ′ for a, z ∈ D0 = {z ∈ D: ϕ (z) =06 }. Proof. Suppose that MuE is bounded. So we can find C > 0 such that p p kMuE(f)k p,β ≤ C kfk p,α , La La 13 and p p p kMuE(f)k p,β = |E(f)(z)| |u(z)| dAβ(z) La D Z p β = |E(f)(z)| dµu(z) D Z p ≤ C kf(z)k p,α , La (2.1) β p β p,α In which µu = D |u(z)| dAβ(z). This means that dµu is an (La )- expectation Carleson measure. By theorem2.4, this is equivalent to R 2 (2+α) 1 −|a| p α β 2 |u(z)| dAβ(z) ≤ Ψa (µu)(z) ≤ C. D |1 − az| Z 0 ! This completes the proof. 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, p,α q,β then the Multiplication operator Mu is bounded from La into La if and only if there exists some C such thet (2+α)q 2 p 1 −|a| q |u(z)| dAβ(z) ≤ C. |1 − az¯ |2 ZD ! References [1] A. B. Aleksandrov, Measurable partitions of the circumf erence, induced by in- ner functions, Translated from Zepiski Nauchnykh Seminarov Leningradskogo otdeleniya 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 in Bergman spaces and their derivatives. American Journal of Mathematics, 1985. 107(1): p. 85-111. 2 [5] Jabbarzadeh,M.R.and M.Moradi,C-E type Toeplitz operators on La(D) . 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[12] Zhu, K., Operator theory in function spaces. 2007: American Mathematical Soc. [13] Jabbarzadeh, M. and M. Hassanloo, Conditional expectation operators on the Bergman spaces. Journal of Mathematical Analysis and Applications, 2012. 385(1): p. 322-325. [14] Ueki, S.-i., Order bounded weighted composition operators mapping into the Bergman 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] Department of mathematics, Payame Noor university , P. O. Box: 19395-3697, Tehran, Iran,