Real-variable characterizations of Bergman spaces
aa r X i v : . [ m a t h . F A ] A p r REAL-VARIABLE CHARACTERIZATIONSOF BERGMAN SPACES
ZEQIAN CHEN AND WEI OUYANG
Abstract.
In this paper, we give a survey of results obtained recentlyby the present authors on real-variable characterizations of Bergmanspaces, which are closely related to maximal and area integral functionsin terms of the Bergman metric. In particular, we give a new proofof those results concerning area integral characterizations through us-ing the method of vector-valued Calder´on-Zygmund operators to handleBergman singular integral operators on the complex ball. The proofs in-volve some sharp estimates of the Bergman kernel function and Bergmanmetric. Introduction
There is a mature and powerful real variable Hardy space theory whichhas distilled some of the essential oscillation and cancellation behavior ofholomorphic functions and then found that behavior ubiquitous. A goodintroduction to that is [11]; a more recent and fuller account is in [1, 14, 16]and references therein. However, the real-variable theory of the Bergmanspace is less well developed, even in the case of the unit disc (cf. [12]).Recently, in [7] the present authors established real-variable type maxi-mal and area integral characterizations of Bergman spaces in the unit ballof C n . The characterizations are in terms of maximal functions and areafunctions on Bergman balls involving the radial derivative, the complex gra-dient, and the invariant gradient. Subsequently, in [8] we introduced a familyof holomorphic spaces of tent type in the unit ball of C n and showed thatthose spaces coincide with Bergman spaces. Moreover, the characterizationsextend to cover Besov-Sobolev spaces. A special case of this is a character-ization of H p spaces involving only area functions on Bergman balls.We remark that the first real-variable characterization of the Bergmanspaces was presented by Coifman and Weiss in 1970’s. Recall that ̺ ( z, w ) = (cid:12)(cid:12) | z | − | w | (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) − | z || w | h z, w i (cid:12)(cid:12)(cid:12) , if z, w ∈ B n \{ } , | z | + | w | , otherwise : 46E15; 32A36, 32A50. Key words : Bergman space, Bergman metric, Maximal function, Area function,Bergman integral operator. is a pseudo-metric on B n and ( B n , ̺, dv α ) is a homogeneous space. By theirtheory of harmonic analysis on homogeneous spaces, Coifman and Weiss[11] can use ̺ to obtain a real-variable atomic decomposition for Bergmanspaces. However, since the Bergman metric β underlies the complex geo-metric structure of the unit ball of C n , one would prefer to real-variablecharacterizations of the Bergman spaces in terms of β. Clearly, the resultsobtained in [7, 8] are such a characterization.In this paper, we will give a detailed survey of results obtained in [7, 8].Moreover, we will give a new proof of those results concerning area inte-gral characterizations through using the method of vector-valued Calder´on-Zygmund operator theory to handle Bergman singular integral operators onthe complex ball. This paper is organized as follows. In Section 2, somenotations and a number of auxiliary (and mostly elementary) facts aboutthe Bergman kernel functions are presented. In Section 3, we will discussreal-variable type atomic decomposition of Bergman spaces. In particular,we will present the atomic decomposition of Bergman spaces with respectto Carleson tubes that was obtained in [7]. Section 4 is devoted to presentmaximal and area integral function characterizations of Bergman spaces.Finally, in Section 5, we will give a new proof of those results concerningthe area integral characterizations obtained in [7, 8] using the argument ofCalder´on-Zygmund operator theory through introducing Bergman singularintegral operators on the complex ball.In what follows, C always denotes a constant depending (possibly) on n, q, p, γ or α but not on f, which may be different in different places. Fortwo nonnegative (possibly infinite) quantities X and Y, by X . Y we meanthat there exists a constant C > X ≤ CY.
We denote by X ≈ Y when X . Y and Y . X. Any notation and terminology not otherwiseexplained, are as used in [20] for spaces of holomorphic functions in the unitball of C n . Bergman spaces
Let C denote the set of complex numbers. Throughout the paper we fix apositive integer n, and let B n denote the open unit ball in C n . The boundaryof B n will be denoted by S n and is called the unit sphere in C n . Also, wedenote by B n the closed unit ball, i.e., B n = { z ∈ C n : | z | ≤ } = B n ∪ S n . The automorphism group of B n , denoted by Aut( B n ) , consists of all bi-holomorphic mappings of B n . Traditionally, bi-holomorphic mappings arealso called automorphisms.For α ∈ R , the weighted Lebesgue measure dv α on B n is defined by dv α ( z ) = c α (1 − | z | ) α dv ( z )where c α = 1 for α ≤ − c α = Γ( n + α + 1) / [ n !Γ( α + 1)] if α > − dv α is a probability measure on B n . ergman spaces 3 In the case of α = − ( n + 1) we denote the resulting measure by dτ ( z ) = dv ( z )(1 − | z | ) n +1 , and call it the invariant measure on B n , since dτ = dτ ◦ ϕ for any automor-phism ϕ of B n . Recall that for α > − p > L pα ( B n )(or, L pα in short) consists of measurable (complex) functions f on B n with k f k p, α = (cid:18)Z B n | f ( z ) | p dv α ( z ) (cid:19) p < ∞ . The (weighted) Bergman space A pα is then defined as A pα = H ( B n ) ∩ L pα , where H ( B n ) is the space of all holomorphic functions in B n . When α = 0we simply write A p for A p . These are the usual Bergman spaces. Note thatfor 1 ≤ p < ∞ , A pα is a Banach space under the norm k k p, α . If 0 < p < , the space A pα is a quasi-Banach space with p -norm k f k pp,α . Recall that the dual space of A α is the Bloch space B defined as follows(we refer to [20] for details). The Bloch space B of B n is defined to be thespace of holomorphic functions f in B n such that k f k B = sup { | e ∇ f ( z ) | : z ∈ B n } < ∞ . k k B is a semi-norm on B . B becomes a Banach space with the followingnorm k f k = | f (0) | + k f k B . It is known that the Banach dual of A α can be identified with B (withequivalent norms) under the integral pairing h f, g i α = lim r → − Z B n f ( rz ) g ( z ) dv α ( z ) , f ∈ A α , g ∈ B . (e.g., see Theorem 3.17 in [20].)We define the so-called generalized Bergman spaces as follows (e.g., [19]).For 0 < p < ∞ and −∞ < α < ∞ we fix a nonnegative integer k with pk + α > − A pα as the space of all f ∈ H ( B n ) such that (1 −| z | ) k R k f ∈ L p ( B n , dv α ) . One then easily observes that A pα is independent ofthe choice of k and consistent with the traditional definition when α > − . Let N be the smallest nonnegative integer such that pN + α > − k f k p,α = | f (0) | + (cid:18)Z B n (1 − | z | ) pN |R N f ( z ) | p dv α ( z ) (cid:19) p , f ∈ A pα . Equipped with (2.1), A pα becomes a Banach space when p ≥ < p < . Note that the family of the generalized Bergman spaces A pα covers mostof the spaces of holomorphic functions in the unit ball of C n , which has been Z. Chen and W.Ouyang extensively studied before in the literature under different names. For exam-ple, B sp = A pα with α = − ( ps + 1) , where B sp is the classical diagonal Besovspace consisting of holomorphic functions f in B n such that (1 − | z | ) k − s R k f belongs to L p ( B n , dv − ) with k being any positive integer greater than s. Itis clear that A pα = B sp with s = − ( α + 1) /p. Thus the generalized Bergmanspaces A pα are exactly the diagonal Besov spaces. On the other hand, if k is a positive integer, p is positive, and β is real, then there is the Sobolevspace W pk,β consisting of holomorphic functions f in B n such that the par-tial derivatives of f of order up to k all belong to L p ( B n , dv β ) (cf. [2, 3, 6]).It is easy to see that these holomorphic Sobolev spaces are in the scale ofthe generalized Bergman spaces, i.e., W pk,β = A pα with α = − ( pk − β + 1)(e.g., see [19] for an overview). We refer to Arcozzi-Rochberg-Sawyer [4, 5],Tchoundja [17] and Volberg-Wick [18] for some recent results on such Besovspaces and more references.Recall that D ( z, γ ) denotes the Bergman metric ball at zD ( z, γ ) = { w ∈ B n : β ( z, w ) < γ } with γ > , where β is the Bergman metric on B n . It is known that β ( z, w ) = 12 log 1 + | ϕ z ( w ) | − | ϕ z ( w ) | , z, w ∈ B n , whereafter ϕ z is the bijective holomorphic mapping in B n , which satisfies ϕ z (0) = z , ϕ z ( z ) = 0 and ϕ z ◦ ϕ z = id. If B n is equipped with the Bergmanmetric β, then B n is a separable metric space. We shall call B n a separablemetric space instead of ( B n , β ) . For reader’s convenience we collect some elementary facts on the Bergmanmetric and holomorphic functions in the unit ball of C n . Lemma 2.1. (cf. Lemma 1.24 in [20] ) For any real α and positive γ thereexist constant C γ such that C − γ (1 − | z | ) n +1+ α ≤ v α ( D ( z, γ )) ≤ C γ (1 − | z | ) n +1+ α for all z ∈ B n . Lemma 2.2. (cf. Lemma 2.20 in [20] ) For each γ > , − | a | ≈ − | z | ≈ | − h a, z i| for all a in B n with z ∈ D ( a, γ ) . Lemma 2.3. (cf. Lemma 2.27 in [20] ) For each γ > , | − h z, u i| ≈ | − h z, v i| for all z in ¯ B n and u, v in B n with β ( u, v ) < γ. ergman spaces 5 Atomic decomposition
We first recall the following “complex-variable” atomic decomposition forBergman spaces due to Coifman and Rochberg [10] (see also [20], Theorem2.30).
Theorem 3.1.
Suppose p > , α > − , and b > n max { , /p } + ( α + 1) /p. Then there exists a sequence { a k } in B n such that A pα consists exactly offunctions of the form (3.1) f ( z ) = ∞ X k =1 c k (1 − | a k | ) ( pb − n − − α ) /p (1 − h z, a k i ) b , z ∈ B n , where { c k } belongs to the sequence space ℓ p and the series converges in thenorm topology of A pα . Moreover, Z B n | f ( z ) | p dv α ( z ) ≈ inf n X k | c k | p o , where the infimum runs over all the above decompositions. By Theorem 3.1 we conclude that for any α > − , A pα as a Banach spaceis isomorphic to ℓ p for every 1 ≤ p < ∞ . Now we turn to the real-variable atomic decomposition of Bergman spaces.To this end, we need some more notations as follows.For any ζ ∈ S n and r > , the set Q r ( ζ ) = { z ∈ B n : d ( z, ζ ) < r } is called a Carleson tube with respect to the nonisotropic metric d. Weusually write Q = Q r ( ζ ) in short.As usual, we define the atoms with respect to the Carleson tube as follows:for 1 < q < ∞ , a ∈ L q ( B n , dv α ) is said to be a (1 , q ) α -atom if there is aCarleson tube Q such that(1) a is supported in Q ;(2) k a k L q ( B n ,dv α ) ≤ v α ( Q ) q − ;(3) R B n a ( z ) dv α ( z ) = 0 . The constant function 1 is also considered to be a (1 , q ) α -atom.Note that for any (1 , q ) α -atom a, k a k ,α = Z Q | a | dv α ≤ v α ( Q ) − /q k a k q,α ≤ . Then, we define A ,qα as the space of all f ∈ A α which admits a decomposition f = X i λ i P α a i and X i | λ i | ≤ C q k f k , α , Z. Chen and W.Ouyang where for each i, a i is an (1 , q ) α -atom and λ i ∈ C so that P i | λ i | < ∞ . Weequip this space with the norm k f k A ,qα = inf n X i | λ i | : f = X i λ i P α a i o where the infimum is taken over all decompositions of f described above.It is easy to see that A ,qα is a Banach space. Theorem 3.2.
Let < q < ∞ and α > − . For every f ∈ A α there exista sequence { a i } of (1 , q ) α -atoms and a sequence { λ i } of complex numberssuch that (3.2) f = X i λ i P α a i and X i | λ i | ≤ C q k f k , α . Moreover, k f k , α ≈ inf X i | λ i | where the infimum is taken over all decompositions of f described above and “ ≈ ” depends only on α and q. Theorem 3.2 is proved in [7] via duality.
Remark 3.1.
One would like to expect that when < p < , A pα also admitsan atomic decomposition in terms of atoms with respect to Carleson tubes.However, the proof of Theorem 3.2 via duality cannot be extended to the case < p < . At the time of this writing, this problem is entirely open.As mentioned in Introduction, by their theory of harmonic analysis onhomogeneous spaces, Coifman and Weiss [11] have obtained a real-variableatomic decomposition in terms of ̺ for Bergman spaces in the case < p ≤ . Real-variable characterizations
Maximal functions.
As is well known, maximal functions play a cru-cial role in the real-variable theory of Hardy spaces (cf. [16]). In [7], theauthors established a maximal-function characterization for the Bergmanspaces. To this end, we define for each γ > f ∈ H ( B n ) :(4.1) (M γ f )( z ) = sup w ∈ D ( z,γ ) | f ( w ) | , ∀ z ∈ B n . The following result is proved in [7].
Theorem 4.1.
Suppose γ > and α > − . Let < p < ∞ . Then for any f ∈ H ( B n ) , f ∈ A pα if and only if M γ f ∈ L p ( B n , dv α ) . Moreover, (4.2) k f k p,α ≈ k M γ f k p,α , where “ ≈ ” depends only on γ, α, p, and n. ergman spaces 7 The norm appearing on the right-hand side of (4.2) can be viewed ananalogue of the so-called nontangential maximal function in Hardy spaces.The proof of Theorem 4.1 is fairly elementary, using some basic facts andestimates on the Bergman balls.
Corollary 4.1.
Suppose γ > and α ∈ R . Let < p < ∞ and k bea nonnegative integer such that pk + α > − . Then for any f ∈ H ( B n ) ,f ∈ A pα if and only if M γ ( R k f ) ∈ L p ( B n , dv α ) , where (4.3) M γ ( R k f )( z ) = sup w ∈ D ( z,γ ) | (1 − | w | ) k R k f ( w ) | , z ∈ B n . Moreover, (4.4) k f k p,α ≈ | f (0) | + k M γ ( R k f ) k p,α , where “ ≈ ” depends only on γ, α, p, k, and n. To prove Corollary 4.1, one merely notices that f ∈ A pα if and only if R k f ∈ L p ( B n , dv α + pk ) and applies Theorem 4.1 to R k f with the help ofLemma 2.2.4.2. Area integral functions.
In order to state the real-variable area in-tegral characterizations of the Bergman spaces, we require some more nota-tion. For any f ∈ H ( B n ) and z = ( z , . . . , z n ) ∈ B n we define R f ( z ) = n X k =1 z k ∂f ( z ) ∂z k and call it the radial derivative of f at z. The complex and invariant gradientsof f at z are respectively defined as ∇ f ( z ) = (cid:16) ∂f ( z ) ∂z , . . . , ∂f ( z ) ∂z n (cid:17) and e ∇ f ( z ) = ∇ ( f ◦ ϕ z )(0) . Now, for fixed γ > < q < ∞ , we define for each f ∈ H ( B n ) and z ∈ B n :(1) The radial area function A ( q ) γ ( R f )( z ) = Z D ( z,γ ) | (1 − | w | ) R f ( w ) | q dτ ( w ) ! q . (2) The complex gradient area function A ( q ) γ ( ∇ f )( z ) = Z D ( z,γ ) | (1 − | w | ) ∇ f ( w ) | q dτ ( w ) ! q . (3) The invariant gradient area function A ( q ) γ ( ˜ ∇ f )( z ) = Z D ( z,γ ) | ˜ ∇ f ( w ) | q dτ ( w ) ! q . The following theorem is proved in [7].
Z. Chen and W.Ouyang
Theorem 4.2.
Suppose γ > , < q < ∞ , and α > − . Let < p < ∞ . Then, for any f ∈ H ( B n ) the following conditions are equivalent: (a) f ∈ A pα . (b) A ( q ) γ ( R f ) is in L p ( B n , dv α ) . (c) A ( q ) γ ( ∇ f ) is in L p ( B n , dv α ) . (d) A ( q ) γ ( ˜ ∇ f ) is in L p ( B n , dv α ) . Moreover, the quantities | f (0) | + k A ( q ) γ ( R f ) k p,α , | f (0) | + k A ( q ) γ ( ∇ f ) k p,α , | f (0) | + k A ( q ) γ ( ˜ ∇ f ) k p,α , are all comparable to k f k p,α , where the comparable constants depend only on γ, q, α, p, and n. In particular, taking the equivalence of (a) and (b), one obtains k f k p,α ≈ | f (0) | + k A ( q ) γ ( R f ) k p,α , which looks tantalizingly simple. However, the authors know no simple proofof this fact even in the case of the usual Bergman space on the unit disc. Corollary 4.2.
Suppose γ > , < q < ∞ , and α ∈ R . Let < p < ∞ and k be a nonnegative integer such that pk + α > − . Then for any f ∈ H ( B n ) ,f ∈ A pα if and only if A ( q ) γ ( R k +1 f ) is in L p ( B n , dv α ) , where (4.5) A ( q ) γ ( R k f )( z ) = Z D ( z,γ ) (cid:12)(cid:12) (1 − | w | ) k R k f ( w ) (cid:12)(cid:12) q dτ ( w ) ! q . Moreover, (4.6) k f k p,α ≈ | f (0) | + k A ( q ) γ ( R k +1 f ) k p,α , where “ ≈ ” depends only on γ, q, α, p, k, and n. To prove Corollary 4.2, one merely notices that f ∈ A pα if and only if R k f ∈ L p ( B n , dv α + pk ) and applies Theorem 4.2 to R k f with the help ofLemma 2.2.4.3. Tent spaces.
The basic functional used below is the one mappingfunctions in B n to functions in B n , given by(4.7) A ( q ) γ ( f )( z ) = Z D ( z,γ ) | f ( w ) | q dτ ( w ) ! q if 1 < q < ∞ , and(4.8) A ( ∞ ) γ ( f )( z ) = sup w ∈ D ( z,γ ) | f ( w ) | , when q = ∞ . Then, the “holomorphic space of tent type” T pq,α in B n is defined as theholomorphic functions f in B n so that A ( q ) γ ( f ) ∈ L pα , when 0 < p ≤ ∞ and α > − , γ > , < q ≤ ∞ . The corresponding classes are then equipped ergman spaces 9 with a norm (or, quasi-norm) k f k T pq,α = k A ( q ) γ ( f ) k p,α . This motivation ariesform the tent spaces in R n , which were introduced and developed by Coif-man, Meyer and Stein in [9].The case of q = ∞ and 0 < p < ∞ was studied in Section 4.1 (see [7] fordetails). Actually, the resulting tent type spaces T p ∞ ,α is Bergman spaces A pα . It is clear that T ∞ q,α with 1 < q < ∞ is imbedded in Bloch space. Onthe other hand, T pq,α are Banach spaces when p ≥ . It is well known that the Hardy-Littlewood maximal function operatorhas played important role in harmonic analysis. To cater our estimates,we use two variants of the non-central Hardy-Littlewood maximal functionoperator acting on the weighted Lebesgue spaces L pα ( B n ), namely,(4.9) M ( q ) γ ( f )( z ) = sup z ∈ D ( w,γ ) v α ( D ( w, γ )) Z D ( w,γ ) | f | q dv α ! q for 0 < q < ∞ . We simply write M γ ( f )( z ) := M (1) γ ( f )( z ) . The following result is proved in [8].
Theorem 4.3.
Suppose γ > , < q < ∞ , and α > − . Let < p < ∞ . Then for any f ∈ H ( B n ) , the following conditions are equivalent: (1) f ∈ A pα . (2) A ( q ) γ ( f ) is in L p ( B n , dv α ) . (3) M ( q ) γ ( f ) is in L p ( B n , dv α ) . Moreover, k f k A pα ≈ k f k T pq,α ≈ k M ( q ) γ ( f ) k p,α , where the comparable constants depend only on γ, q, α, p, and n. Note that the Bergman metric β is non-doubling on B n and so ( B n , β, dv α )is a non-homogeneous space. The proof of the above theorem does involvesome techniques of non-homogeneous harmonic analysis developed in [15]. Corollary 4.3.
Suppose γ > , < q < ∞ , and α ∈ R . Let < p < ∞ and k be a nonnegative integer such that pk + α > − . Then for any f ∈ H ( B n ) ,f ∈ A pα if and only if A ( q ) γ ( R k f ) is in L p ( B n , dv α ) if and only if M ( q ) γ ( R k f ) is in L p ( B n , dv α ) , where (4.10) A ( q ) γ ( R k f )( z ) = Z D ( z,γ ) (cid:12)(cid:12) (1 − | w | ) k R k f ( w ) (cid:12)(cid:12) q dτ ( w ) ! q and (4.11) M ( q ) γ ( R k f )( z ) = sup z ∈ D ( w,γ ) Z D ( w,γ ) (cid:12)(cid:12) (1 − | u | ) k R k f ( u ) (cid:12)(cid:12) q dv α ( u ) v α ( D ( w, γ )) ! q Moreover, (4.12) k f k p,α ≈ | f (0) | + k A ( q ) γ ( R k f ) k p,α ≈ | f (0) | + k M ( q ) γ ( R k f ) k p,α , where “ ≈ ” depends only on γ, q, α, p, k, and n. To prove Corollary 4.3, one merely notices that f ∈ A pα if and only if R k f ∈ L p ( B n , dv α + pk ) and applies Theorem 4.3 to R k f with the help ofLemma 2.2. When α > − , we can take k = 1 and then recover Theorem4.2.As mentioned in Section 2, the family of the generalized Bergman spaces A pα covers most of the spaces of holomorphic functions in the unit ball of C n , such as the classical diagonal Besov space B sp and the Sobolev space W pk,β . In particular, H ps = A pα with α = − s − , where H ps is the Hardy-Sobolevspace defined as the set (cid:26) f ∈ H ( B n ) : k f k p H ps = sup
Vector-valued kernels and Calder´on-Zygmund operators onhomogeneous spaces.
Recall that a quasimetric on a set X is a map ρ from X × X to [0 , ∞ ) such that(1) ρ ( x, y ) = 0 if and only if x = y ;(2) ρ ( x, y ) = ρ ( y, x );(3) there exists a positive constant C ≥ ρ ( x, y ) ≤ C [ ρ ( x, z ) + ρ ( z, y )] , ∀ x, y, z ∈ X, (the quasi-triangular inequality).For any x ∈ X and r > , the set B ( x, r ) = { y ∈ X : ρ ( x, y ) < r } is calleda ρ -ball of center x and radius r. A space of homogeneous type is a topological space X endowed with aquasimetrc ρ and a Borel measure µ such that ergman spaces 11 (a) for each x ∈ X, the balls B ( x, r ) form a basis of open neighborhoods of x and, also, µ ( B ( x, r )) > r > C > x ∈ X and r > , one has µ ( B ( x, r )) ≤ Cµ ( B ( x, r )) . ( X, ρ, µ ) is called a space of homogeneous type or simply a homogeneousspace. We will usually abusively call X a homogeneous space instead of( X, ρ, µ ) . We refer to [11, 16] for details on harmonic analysis on homoge-neous spaces.Let E be a Banach space. Let L p ( µ, E ) be the usual Bochner-Lebesguespace for 1 ≤ p ≤ ∞ , and let L , ∞ ( µ, E ) be defined by L , ∞ ( µ, E ) := (cid:8) f : X E | f is strongly measurable such that k f k L , ∞ ( µ,E ) < ∞ (cid:9) , where k f k L , ∞ ( µ,E ) := sup t> tµ (cid:0) { x ∈ X : k f ( x ) k E > t } (cid:1) . Note that k f k L , ∞ is not actually a norm in the sense that it does not satisfy the triangleinequality. However, we still have k cf k L , ∞ ( µ,E ) = | c |k f k L , ∞ ( µ,E ) and k f + g k L , ∞ ( µ,E ) ≤ k f k L , ∞ ( µ,E ) + k g k L , ∞ ( µ,E ) )for every c ∈ C and f, g ∈ L , ∞ ( µ, E ) . If E = C we simply write L p ( µ, E ) = L p ( µ ) and L , ∞ ( µ, E ) = L , ∞ ( µ ) . Fix m > △ = { ( x, x ) : x ∈ X } . Avector-valued m -dimensional Calder´on-Zygmund kernel with respect to ρ isa continuous mapping K : X × X \△ 7→ E for which we have(a) there exists a constant C > k K ( x, y ) k E ≤ C ρ ( x, y ) , ∀ x, y ∈ X × X \△ ;(b) there exist constants 0 < ǫ ≤ C , C > k K ( x, y ) − K ( x ′ , y ) k E + k K ( y, x ) − K ( y, x ′ ) k E ≤ C ρ ( x, x ′ ) ǫ ρ ( x, y ) m + ǫ whenever x, x ′ , y ∈ X and ρ ( x, x ′ ) ≤ C ρ ( x, y ) . Given a vector-valued m -dimensional Calder´on-Zygmund kernel K, wecan define (at least formally) a Calder´on-Zygmund singular integral operatorassociated with this kernel by T f ( x ) = Z X K ( x, y ) f ( y ) dµ ( x ) . Proposition 5.1.
Let E be a Banach space. If a Calder´on-Zygmund singu-lar integral operator T is bounded from L q ( µ ) into L q ( µ, E ) for some fixed ≤ q < ∞ , then T can be extended to an operator on L p ( µ ) for every ≤ p < ∞ such that (a) T is L p -bounded for every < p < ∞ , i.e., k T f k L p ( µ,E ) ≤ C p k f k L p ( µ ) ;(b) T is of weak type (1 , , i.e., k T f k L , ∞ ( µ,E ) ≤ C k f k L ( µ ) for all f ∈ L ( µ ); (c) T is bounded from L ∞ ( µ ) into BMO(
X, ρ, µ ; E );(d) T is bounded from H ( X, ρ, µ ) into L ( µ ) . H ( X, ρ, µ ) and BMO(
X, ρ, µ ; E ) can be defined in a natural way, see[11] for the details. This result must be known for experts in the field ofvector-valued harmonic analysis, and the proof can be obtained by merelymodifying the proof of Theorem V.3.4 in [13].5.2. Bergman singular integral operators.
We now turn our attentionto the special case of the unit ball B n . Recall that ̺ ( z, w ) = (cid:12)(cid:12) | z | − | w | (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) − | z || w | h z, w i (cid:12)(cid:12)(cid:12) , if z, w ∈ B n \{ } , | z | + | w | , otherwise . It is know that ̺ is a pseudo-metric on B n and ( B n , ̺, v α ) is a homogeneousspace for α > − E be a Banach space. Suppose α > − . We are interested in vector-valued Bergman type integral operators on the unit ball B n in C n . Moreprecisely, we are interested in Bergman type integral operators whose kernelswith values in E satisfy the following estimates(5.1) k K ( z, w ) k E ≤ C̺ ( z, w ) n +1+ α , ∀ ( z, w ) ∈ B n × B n \ { ( ζ, ζ ) : ζ ∈ B n } , and(5.2) k K ( z, w ) − K ( z, ζ ) k E + k K ( w, z ) − K ( ζ, z ) k E ≤ C̺ ( w, ζ ) β ̺ ( z, ζ ) n +1+ α + β , for z, w, ζ ∈ B n so that ̺ ( z, ζ ) ≥ δ̺ ( w, ζ ) , with some (fixed) α > − , δ > , and 0 < β ≤ . That is, K is ( n + 1 + α )-dimensional Calder´on-Zygmundkernel K with values in E on the homogeneous space ( B n , ̺, v α ) . Once the kernel has been defined, then a α -time Bergman singular integraloperator T is defined as a Calder´on-Zygmund singular integral operator witha vector-valued kernel K by(5.3) T f ( z ) = Z B n K ( z, w ) f ( w ) dv α ( w ) , z, w ∈ B n . If T is bounded from L pα into L p ( B n , v α ; E ) for any 1 < p < ∞ , we call it a α -time Bergman integral operator (BIO). We denote by BIO α ( E ) all suchoperators. If E = C we write BIO α ( C ) = BIO α . The examples that we keep in mind are the Bergman projection operator P α from L α onto A α , which can be expressed as P α f ( z ) = Z B n K α ( z, w ) f ( w ) dv α ( w ) , ∀ f ∈ L ( B n , dv α ) , where(5.4) K α ( z, w ) = 1 (cid:0) − h z, w i (cid:1) n +1+ α , z, w ∈ B n with α > − . ergman spaces 13 Indeed, we have
Proposition 5.2. (Proposition 2.13 in [17] ) (i) there exists a constant C > such that | K α ( z, w ) | ≤ C ̺ ( z, w ) n +1+ α , ∀ z, w ∈ B n . (ii) There are two constants C , C > such that for all z, w, ζ ∈ B n satisfying ̺ ( z, ζ ) > C ̺ ( w, ζ ) one has | K α ( z, w ) − K α ( z, ζ ) | ≤ C ̺ ( w, ζ ) ̺ ( z, ζ ) n +1+ α + . It is well known that P α extends to a bounded operator on L pα for 1
Theorem 5.1.
Let E be a Banach space and α > − . Suppose T is aCalder´on-Zygmund singular integral operator associated with a kernel satis-fying (5.1) and (5.2) . If T is bounded on L q ( v α ) for some fixed < q < ∞ , then T is bounded from L p ( B n , v α ) into L p ( B n , v α ; E ) for every < p < ∞ , and is of weak type (1 , . Area functions as vector-valued Bergman integral operators.
Given γ > < q < ∞ . Let E = L q ( B n , χ D (0 ,γ ) dτ ) . We consider theoperator(5.5) [ T tent f ( z )]( w ) = Z B n f ( u ) dv α ( u )(1 − h ϕ z ( w ) , u i ) n +1+ α , ∀ z, w ∈ B n , with the kernel K tent ( z, u )( w ) = 1(1 − h ϕ z ( w ) , u i ) n +1+ α . By the reproduce kernel formula (e.g., Theorem 2.2 in [20]) we have[ T tent f ( z )]( w ) = f ( ϕ z ( w )) , ∀ f ∈ H ( B n ) , and hence k T tent f ( z ) k E = A ( q ) γ ( f )( z ) , for any f ∈ H ( B n ) . Theorem 5.2.
Let γ > , α > − , < q < ∞ , and < p < ∞ . Then T tent ∈ BIO α ( E ) . Consequently, k A ( q ) γ ( f ) k L pα . k f k L pα , ∀ f ∈ A pα ( B n ) . Proof.
Let f ∈ L qα ( B n ) . Then k T tent f k qL q ( v α ,E ) = Z B n Z B n (cid:12)(cid:12)(cid:12)(cid:12)Z B n f ( u ) dv α ( u )(1 − h ϕ z ( w ) , u i ) n +1+ α (cid:12)(cid:12)(cid:12)(cid:12) q χ D (0 ,γ ) ( w ) dτ ( w ) dv α ( z )= Z B n Z B n | P α f ( w ) | q χ D ( z,γ ) ( w ) dτ ( w ) dv α ( z ) ≈ k P α f k qL qα . k f k qL qα by the L q -boundedness of P α for 1 < q < ∞ . This concludes that T tent isbounded from L qα into L q ( v α , E ) . By Theorem 5.1, it remains to show that K tent satisfies the conditions(5.1) and (5.2). It is easy to check the condition (5.1). Indeed, by Lemmas2.1 and 2.3 and Proposition 5.2 (i) we have k K tent ( z, u ) k E = (cid:16) Z B n | − h ϕ z ( w ) , u i| n +1+ α ) χ D (0 ,γ ) ( w ) dτ ( w ) (cid:17) = (cid:16) Z D ( z,γ ) | − h w, u i| n +1+ α ) dτ ( w ) (cid:17) . | − h z, u i| n +1+ α ≤ C̺ ( z, u ) n +1+ α . This concludes that K tent satisfies (5.1).To check the condition (5.2), we need the following variant of Proposition5.2 (ii). Lemma 5.1.
There exist two constants C , C > such that for all z, u, ζ ∈ B n satisfying ̺ ( z, ζ ) > C ̺ ( u, ζ ) one has | K α ( w, u ) − K α ( w, ζ ) | ≤ C ̺ ( u, ζ ) ̺ ( z, ζ ) n +1+ α + , for all w ∈ D ( z, γ ) . The proof can be obtained by slightly modifying the proof of Proposition2.13 (2) in [17] with the help of Lemmas 2.2 and 2.3. We omit the details. ergman spaces 15
Now we turn out to proceed our proof. Suppose z, u, ζ ∈ B n . Note that k K tent ( z,u ) − K tent ( z, ζ ) k E = (cid:16) Z B n (cid:12)(cid:12)(cid:12) − h ϕ z ( w ) , u i ) n +1+ α − − h ϕ z ( w ) , ζ i ) n +1+ α (cid:12)(cid:12)(cid:12) χ D (0 ,γ ) ( w ) dτ ( w ) (cid:17) = (cid:16) Z χ D ( z,γ ) (cid:12)(cid:12)(cid:12) − h w, u i ) n +1+ α − − h w, ζ i ) n +1+ α (cid:12)(cid:12)(cid:12) dτ ( w ) (cid:17) = (cid:16) Z χ D ( z,γ ) (cid:12)(cid:12)(cid:12) K α ( w, u ) − K α ( w, ζ ) (cid:12)(cid:12)(cid:12) dτ ( w ) (cid:17) . Then, by Lemma 5.1 there exist two constants C , C > z, u, ζ ∈ B n , k K tent ( z, u ) − K tent ( z, ζ ) k E ≤ C ̺ ( u, ζ ) ̺ ( z, ζ ) n +1+ α + whenever ̺ ( z, ζ ) > C ̺ ( u, ζ ) . On the other hand, since k K tent ( u,z ) − K tent ( ζ, z ) k E = (cid:16) Z B n (cid:12)(cid:12)(cid:12) − h ϕ u ( w ) , z i ) n +1+ α − − h ϕ ζ ( w ) , z i ) n +1+ α (cid:12)(cid:12)(cid:12) χ D (0 ,γ ) ( w ) dτ ( w ) (cid:17) = (cid:16) Z B n (cid:12)(cid:12)(cid:12) K α ( ϕ u ( w ) , z ) − K α ( ϕ ζ ( w ) , z ) (cid:12)(cid:12)(cid:12) χ D (0 ,γ ) ( w ) dτ ( w ) (cid:17) and ̺ ( ϕ u ( w ) , ϕ ζ ( w )) . | − h ϕ u ( w ) , ϕ ζ ( w ) i| ≈ | − h u, ζ i| , ∀ w ∈ D (0 , γ ) , by Lemma 2.3 and the inequality ̺ ( z, w ) . | − h z, w i| (e.g., Eq.(6) in [17]), then by slightly modifying the proof of Proposition2.13 (2) in [17] we can prove that k K tent ( u, z ) − K tent ( ζ, z ) k E . ̺ ( u, ζ ) ̺ ( z, ζ ) n +1+ α + . The details are left to readers. This completes the proof. (cid:3)
Evidently, we can define:(i) The radial area integral operator(5.6) [ T radial f ( z )]( w ) = Z B n K radial ( z, u )( w ) f ( u ) dv α ( u ) , ∀ z, w ∈ B n , with the Bergman kernel K radial ( z, u )( w ) = ( n + 1 + α ) (1 − | ϕ z ( w ) | ) h ϕ z ( w ) , u i (1 − h ϕ z ( w ) , u i ) n +2+ α , ∀ z, u, w ∈ B n . It is easy to check that[ T radial f ( z )]( w ) = (1 − | ϕ z ( w ) | ) R f ( ϕ z ( w ))for any f ∈ H ( B n ) . (ii) The complex gradient area integral operator(5.7) [ T grad f ( z )]( w ) = Z B n K grad ( z, u )( w ) f ( u ) dv α ( u ) , ∀ z, w ∈ B n , with the Bergman kernel K grad ( z, u )( w ) = ( n + 1 + α )(1 − | ϕ z ( w ) | )¯ u (1 − h ϕ z ( w ) , u i ) n +2+ α , ∀ z, u, w ∈ B n . It is easy to check that[ T grad f ( z )]( w ) = (1 − | ϕ z ( w ) | ) ∇ f ( ϕ z ( w ))for any f ∈ H ( B n ) . (iii) The invariant gradient area integral operator(5.8) [ T invgrad f ( z )]( w ) = Z B n K invgrad ( z, u )( w ) f ( u ) dv α ( u ) , ∀ z, w ∈ B n , with the Bergman kernel K invgrad ( z, u )( w ) = ( n + 1 + α ) (1 − | ϕ z ( ω ) | ) n +1+ α ϕ ϕ z ( ω ) ( u ) | − h ϕ z ( ω ) , u i| n +1+ α ) , ∀ z, u, w ∈ B n . It is easy to check that[ T invgrad f ( z )]( w ) = ˜ ∇ f ( ϕ z ( w ))for any f ∈ H ( B n ) . Similarly, we have
Theorem 5.3.
Let γ > , < q < ∞ , and α > − . Then T radial , T grad , and T invgrad are all in BIO α ( E ) . Consequently, A ( q ) γ ( R f ) , A ( q ) γ ( ∇ f ) , and A ( q ) γ ( ˜ ∇ f ) are all bounded on A pα for every < p < ∞ . The proof is the same as that of Theorem 5.2 and the details are omitted.
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