Positive Toeplitz operators on the Bergman spaces of the Siegel upper half-space
aa r X i v : . [ m a t h . C V ] A p r POSITIVE TOEPLITZ OPERATORS ON THE BERGMANSPACES OF THE SIEGEL UPPER HALF-SPACE
CONGWEN LIU AND JIAJIA SI
Abstract.
We characterize bounded and compact positive Toeplitz operatorsdefined on the Bergman spaces over the Siegel upper half-space. Introduction
Toeplitz operators on Bergman spaces over the unit disk have been well studied.Especially, positive symbols of bounded and compact Toeplitz operators are com-pletely characterized. See for instance [1] or [16, Chapter 7]. These results werefurther extended to more general settings in [14] and [12]. However, there are onlyfew works on analogous results over unbounded domains. See [2, 3] for a study ofpositive Topelitz operators on harmonic Bergman spaces over the upper half spaceof R n .In this paper we study bounded and compact positive Toeplitz operators onBergman spaces over the Siegel upper half-space.Let C n be the n -dimensional complex Euclidean space. For any two points z = ( z , · · · , z n ) and w = ( w , · · · , w n ) in C n we write z · w := z w + · · · + z n w n , and | z | := √ z · z = p | z | + · · · + | z n | . The unit ball of C n is given by B = { z ∈ C n : | z | < } . The set U = (cid:8) z ∈ C n : Im z n > | z ′ | (cid:9) is the Siegel upper half-space. Here and the throughout the paper, we use thenotatoin z = ( z ′ , z n ) , where z ′ = ( z , · · · , z n − ) ∈ C n − and z n ∈ C . As usual, for p >
0, the space L p ( U ) consists of all Lebesgue measurable functions f on U for which k f k p := (cid:26) Z U | f ( z ) | p dV ( z ) (cid:27) /p Mathematics Subject Classification.
Primary 47B35; Secondary 32A36.
Key words and phrases.
Toeplitz operators; Bergman spaces; Siegel upper half-space; Carlesonmeasure; Berezin transform.This work was supported by the National Natural Science Foundation of China grants11571333. is finite, where V denotes the Lebesgue measure on C n . The Bergman space A p ( U )is the closed subspace of L p ( U ) consisting of holomorphic functions on U . Notethat when 1 ≤ p < ∞ the space A p ( U ) is a Banach space with the norm k · k p .In particular, A ( U ) is a Hilbert space endowed with the usual L inner product.The orthogonal projection from L ( U ) onto A ( U ) can be expressed as an integraloperator: P f ( z ) = Z U K ( z, w ) f ( w ) dV ( w ) , with the Bergman kernel K ( z, w ) = n !4 π n (cid:20) i w n − z n ) − z ′ · w ′ (cid:21) − n − . See, for instance, [6, Theorem 5.1]. This formula enables us to extend the domainof the operator P , which is usually called a Bergman projection, to L p ( U ) for all1 < p < ∞ . Moreover, P is a bounded projection from L p ( U ) onto A p ( U ) for1 < p < ∞ , see [4, Lemma 2.8].Given ϕ ∈ L ∞ ( U ), we define an operator on A p ( U ) by T ϕ f := P ( ϕf ) , f ∈ A p ( U ) .T ϕ is called the Toeplitz operator on A p ( U ) with symbol ϕ . Toeplitz operators canalso be defined for unbounded symbols or even positive Borel measures on U . Let M + be the set of all positive Borel measures µ such that Z U dµ ( z ) | z n + i | α < ∞ for some α >
0. Given µ ∈ M + , the Toeplitz operator T µ with symbol µ is givenby T µ f ( z ) = Z U K ( z, w ) f ( w ) dµ ( w )for f ∈ H ( U ). In general, T µ may not even be defined on all of A p ( U ), 1 < p < ∞ ,but it is always densely defined by the fact that, for each α > n + 1 /p , holomorphicfunctions f on U such that f ( z ) = O ( | z n + i | − α ) form a dense subset of A p ( U ) (seeSection 4). This is inspired by that of Choe et al. [2], in the setting of the upperhalf space of R n .A positive Borel measure µ is called a Carleson measure for the Bergman space A p ( U ) if there exists a positive constant C such that Z U | f ( z ) | p dµ ( z ) ≤ C Z U | f ( z ) | p dV ( z )for all f ∈ A p ( U ). We shall furthermore say that µ is a vanishing Carleson measurefor A p ( U ) if the inclusion map i p : A p ( U ) → L p ( U , µ ) , f f is compact, that is,lim j →∞ Z U | f j ( z ) | p dµ ( z ) = 0whenever { f j } converges to 0 weakly in A p ( U ). OSITIVE TOEPLITZ OPERATORS 3
For a positive Borel measure µ on U , we formally define a function e µ on U by(1.1) e µ ( z ) := Z U | k z ( w ) | dµ ( w ) , z ∈ U , where, for z ∈ U , k z ( w ) := K ( z, w ) / p K ( z, z ) , w ∈ U , and call e µ the Berezin transform of µ . For z ∈ U and r >
0, we define the averagingfunction(1.2) b µ r ( z ) := µ ( D ( z, r )) | D ( z, r ) | , where D ( z, r ) is the Bergman metric ball at z with radius r (see Section 2.3) and | D ( z, r ) | := V ( D ( z, r )) denotes the Lebesgue measure of D ( z, r ).We can now state our main results. Theorem 1.1.
Suppose that r > , < p < ∞ , < q < ∞ and that µ ∈ M + .Then the following conditions are equivalent: (i) T µ is bounded on A p ( U ) . (ii) e µ is a bounded function on U . (iii) b µ r is a bounded function on U . (iv) µ is a Carleson measure for A q ( U ) . Let ρ ( z ) := Im z n − | z ′ | and b U := { z ∈ C n : ρ ( z ) = 0 } denotes the boundaryof U . Then b U := U ∪ b U ∪ {∞} is the one-point compactification of U . Also, let ∂ b U := b U ∪ {∞} . Thus, z → ∂ b U means ρ ( z ) → ρ ( z ) → ∞ . We denote by C ( U ) the space of complex-value continuous functions f on U such that f ( z ) → z → ∂ b U . Theorem 1.2.
Suppose that r > , < p, q < ∞ and that µ ∈ M + . Then thefollowing conditions are equivalent: (i) T µ is compact on A p ( U ) . (ii) e µ belongs to C ( U ) . (iii) b µ r belongs to C ( U ) . (iv) µ is a vanishing Carleson measure for A q ( U ) . This paper is organized as follows. Section 2 contains the necessary backgroundmaterial and auxiliary results. In Section 3, we characterize Carleson measuresand vanishing Carleson measures for the Bergman spaces over the Siegel upperhalf-space. In Section 4, we show that the Toeplitz operators are well defined on afamily of dense subspaces of the Bergman spaces. The proofs of the theorems 1.1and 1.2 are carried out in Section 5.Throughout the paper, the letter C will denote a positive constant that mayvary at each occurrence but is independent of the essential variables. The letter C with subscripts usually denotes a specific constant.2. Preliminaries
Estimates of the Bergman kernel.
For simplicity, we write ρ ( z, w ) := i w n − z n ) − z ′ · w ′ . CONGWEN LIU AND JIAJIA SI
With this notation, the Bergman kernel of U K ( z, w ) = n !4 π n ρ ( z, w ) n +1 , z, w ∈ U . Note also that ρ ( z ) = ρ ( z, z ). Lemma 2.1.
We have (2.1) | K ( z, w ) | ≤ n − n ! π n min { ρ ( z ) , ρ ( w ) } − n − for any z, w ∈ U .Proof. For each t >
0, we define the nonisotropic dilation δ t by δ t ( u ) = ( tu ′ , t u n ) , u ∈ U . Also, to each fixed z ∈ U , we associate the following (holomorphic) affine self-mapping of U : h z ( u ) := (cid:0) u ′ − z ′ , u n − Re z n − iu ′ · z ′ + i | z ′ | (cid:1) , u ∈ U . All these mappings are holomorphic automorphisms of U . See [11, Chapter XII].Hence the mappings σ z := δ ρ ( z ) − / ◦ h z are holomorphic automorphisms of U .Simple calculations show that σ z ( z ) = i := (0 ′ , i ) and(2.2) ( J C σ z )( u ) = ρ ( z ) − ( n +1) / where ( J C σ z )( u ) stands for the complex jacobian of σ z at u .Thus, by [7, Proposition 1.4.12], we have K ( z, w ) = ( J C σ z )( z ) K ( σ z ( z ) , σ z ( w )) ( J C σ z )( w )(2.3) = K ( i , σ z ( w )) ρ ( z ) − n − . Note that | K ( i , u ) | = n !4 π n n +1 | u n + i | n +1 ≤ n − n ! π n for all u ∈ U . Hence (2.1) follows immediately from (2.3). (cid:3) Lemma 2.2.
Let < p < ∞ . Then for each z ∈ U , the Bergman kernel function K z := K ( · , z ) is in A p ( U ) , and (2.4) k K z k p = C n,p ρ ( z ) − ( n +1) /p ′ , where p ′ = p/ ( p − and C n,p is a positive constant depending on n and p .Proof. This is just an application of the following formula from [10, Lemma 5]:(2.5) Z U ρ ( w ) t | ρ ( z, w ) | s dV ( w ) = C n,s,t ρ ( z ) s − t − n − , if t > − s − t > n + 1+ ∞ , otherwise for all z ∈ U , where C n,s,t := 4 π n Γ(1 + t )Γ( s − t − n − ( s/ . We omit the details. (cid:3)
OSITIVE TOEPLITZ OPERATORS 5
Cayley transform and the M¨obius transformations.
Recall that theCayley transform Φ : B → U is given by( z ′ , z n ) (cid:18) z ′ z n , i (cid:18) − z n z n (cid:19)(cid:19) . It is immediate to calculate thatΦ − : ( z ′ , z n ) (cid:18) iz ′ i + z n , i − z n i + z n (cid:19) . We refer to [11, Chapter XII] for the properties of the Cayley transform. For theconvenience of later reference, we record the following lemma from [9].
Lemma 2.3. (i)
The identity (2.6) ρ (Φ( ξ ) , Φ( η )) = 1 − ξ · η (1 + ξ n )(1 + η n ) holds for all ξ, η ∈ B . (ii) The real Jacobian of Φ at ξ ∈ B is (2.7) ( J R Φ) ( ξ ) = 4 | ξ n | n +1) . (iii) The identity (2.8) 1 − Φ − ( z ) · Φ − ( w ) = ρ ( z, w ) ρ ( z, i ) ρ ( i , w ) holds for all z, w ∈ U , where i = (0 ′ , i ) . (iv) The identity (2.9) | Φ − ( z ) | = 1 − ρ ( z ) | ρ ( z, i ) | holds for all z ∈ U . (v) The real Jacobian of Φ − at z ∈ U is (2.10) (cid:0) J R Φ − (cid:1) ( z ) = 14 | ρ ( z, i ) | n +1) . The group of all one-to-one holomorphic mappings of B onto B (the so-calledautomorphisms of B ) will be denoted by Aut( B ). It is generated by the unitarytransformations on C n along with the M¨obius transformations ϕ ξ given by ϕ ξ ( η ) := ξ − P ξ η − (1 − | ξ | ) Q ξ η − η · ξ , where ξ ∈ B , P ξ is the orthogonal projection onto the space spanned by ξ , and Q ξ η = η − P ξ η . See [15, Section 1.2].It is easily shown that the mapping ϕ ξ satisfies ϕ ξ (0) = ξ, ϕ ξ ( ξ ) = 0 , ϕ ξ ( ϕ ξ ( η )) = η. Furthermore, for all η, ω ∈ B , we have(2.11) 1 − ϕ ξ ( η ) · ϕ ξ ( ω ) = (1 − | ξ | )(1 − η · ω )(1 − η · ξ )(1 − ξ · ω ) . CONGWEN LIU AND JIAJIA SI
Bergman metric balls.
Let Ω be a domain in C n and K Ω ( z, w ) be theBergman kernel of Ω. We define g Ω i,j ( z ) := 1 n + 1 ∂ log K Ω ( z, z ) ∂z i ∂ ¯ z j , i, j = 1 , · · · , n and call the complex matrix B Ω ( z ) = (cid:0) g Ω i,j ( z ) (cid:1) ≤ i,j ≤ n the Bergman matrix of Ω. For a C curve γ : [0 , → Ω we define l Ω ( γ ) = Z (cid:16) B Ω ( γ ( t )) γ ′ ( t ) · γ ′ ( t ) (cid:17) / dt. For any two points z and w in Ω, let β Ω ( z, w ) be the infimum of the set consistingof all l Ω ( γ ), where γ is a piecewise smooth curve in Ω from z to w . We will call β Ω the Bergman metric on U . For z ∈ Ω and r > D Ω ( z, r ) denote theBergman metric ball at z with radius r . Thus D Ω ( z, r ) := { w ∈ U : β Ω ( z, w ) < r } . If Ω , Ω are two domains in C n and h is a biholomorphic mapping of Ω ontoΩ , then β Ω ( z, w ) = β Ω ( h ( z ) , h ( w ))for all z, w ∈ Ω . See for instance [7, Proposition 1.4.15]. Hence,(2.12) β U ( z, w ) = β B (Φ − ( z ) , Φ − ( w )) = tanh − (cid:0)(cid:12)(cid:12) ϕ Φ − ( z ) (Φ − ( w )) (cid:12)(cid:12)(cid:1) . It follows that D U ( z, r ) = Φ( D B (Φ − ( z ) , r ))for every z ∈ U and r >
0. Also, a computation shows that(2.13) β U ( z, w ) = tanh − s − ρ ( z ) ρ ( w ) | ρ ( z, w ) | . In the sequel, we simply write β ( z, w ) := β U ( z, w ) and D ( z, r ) := D U ( z, r ) if notcause any confusion. Lemma 2.4.
For any r > , there exists a sequence { a k } in U such that (i) U = S ∞ k =1 D ( a k , r ) ; (ii) There is a positive integer N such that each point z in U belongs to at most N of the sets D ( a k , r ) .We are going to call { a k } an r -lattice in the Bergman metric.Proof. The proof is similar to that of [15, Theorem 2.23], so is omitted. (cid:3)
Lemma 2.5.
For any z ∈ U and r > we have (2.14) | D ( z, r ) | = 4 π n n ! tanh n r (1 − tanh r ) n +1 ρ ( z ) n +1 . Proof.
We first show that(2.15) | D ( z, r ) | = ρ ( z ) n +1 | D ( i , r ) | holds for any z ∈ U and r > OSITIVE TOEPLITZ OPERATORS 7
Since the metric β is invariant under the automorphisms, we have β ( w, z ) = β ( σ z ( w ) , σ z ( z )) = β ( σ z ( w ) , i )for all z, w ∈ U , where σ z is as in Subsection 2.1. Hence D ( z, r ) = σ − z ( D ( i , r )). Itfollows that | D ( z, r ) | = Z σ − z ( D ( i ,r )) dV ( w ) = Z D ( i ,r ) (cid:12)(cid:12)(cid:0) J C σ − z (cid:1) ( u ) (cid:12)(cid:12) dV ( u ) . Combining with (2.2), this gives (2.15).It remains to show that | D ( i , r ) | = 4 π n n ! tanh n r (1 − tanh r ) n +1 . Note that D ( i , r ) = Φ( B (0 , R )), where R := tanh( r ) and Φ is the Cayley transform.Thus, | D ( i , r ) | = Z Φ( B (0 ,R )) dV ( w ) ξ =Φ − ( w ) ======== Z B (0 ,R ) | ξ n | n +1) dV ( ξ ) ζ = ξ/R ====== Z B R n | − ( − Re n ) · ζ | n +1) dV ( ζ )= 4 π n n ! R n (1 − R ) n +1 , where the last equality follows a simple calculation, see for instance [8, p.263, (2.12)].The proof is complete. (cid:3) Corollary 2.6.
For any z ∈ U and r > , the averaging function (defined as in (1.2) ) b µ r ( z ) = n !4 π n (1 − tanh r ) n +1 tanh n r µ ( D ( z, r )) ρ ( z ) n +1 . Lemma 2.7.
Given r > , the inequalities (2.16) 1 − tanh( r )1 + tanh( r ) ≤ | ρ ( z, u ) || ρ ( z, v ) | ≤ r )1 − tanh( r ) hold for all z, u, v ∈ U with β ( u, v ) ≤ r .Proof. Let η = Φ − ( z ), ξ = Φ − ( u ) and ζ = Φ − ( v ). We prove only the secondinequality; then the first one follows by symmetry. By (2.12), then we have β B ( ξ, ζ ) = β ( u, v ) ≤ tanh( r ) . Also, ρ ( z, u ) ρ ( z, v ) = (1 − η · ξ )(1 + ¯ ζ n )(1 − η · ζ )(1 + ¯ ξ n ) . CONGWEN LIU AND JIAJIA SI
Let ˜ η = ϕ ξ ( η ) and ˜ ζ = ϕ ξ ( ζ ). Then again by (2.12), | ˜ ζ | ≤ tanh( r ) ∈ (0 , − η · ξ = 1 − ϕ ξ (˜ η ) · ϕ ξ (0) = 1 − | ξ | − ˜ η · ξ and 1 − η · ζ = 1 − ϕ ξ (˜ η ) · ϕ ξ (˜ ζ ) = (1 − | ξ | )(1 − ˜ η · ˜ ζ )(1 − ˜ η · ξ )(1 − ξ · ˜ ζ ) . Thus we get 1 − η · ξ − η · ζ = 1 − ξ · ˜ ζ − ˜ η · ˜ ζ . Likewise, 1 + ¯ ζ n ξ n = 1 − ˜ ̟ · ˜ ζ − ξ · ˜ ζ , where ˜ ̟ := ϕ ξ ( − e n ). Hence, ρ ( z, u ) ρ ( z, v ) − η · ˜ ζ − ˜ ̟ · ˜ ζ (1 − ˜ η · ˜ ζ ) . Since | ˜ ζ | ≤ tanh( r ), we have | − ˜ η · ˜ ζ | ≥ − tanh( r )and | ˜ η · ˜ ζ − ˜ ̟ · ˜ ζ | ≤ | ˜ ζ | ≤ r ) . It follows that (cid:12)(cid:12)(cid:12)(cid:12) ρ ( z, u ) ρ ( z, v ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r )1 − tanh( r ) , which implies the asserted inequality. (cid:3) In what follows we use the notation Q j := D ( i , j )for j = 1 , . . . . Note that Q j ⊂⊂ Q j +1 for all j ∈ N and U = S ∞ j =1 Q j . Lemma 2.8.
Given λ ∈ R and j ∈ N , there is a constant C = C ( n, λ, j ) > suchthat (2.17) sup w ∈ Q j ρ ( w ) λ | ρ ( z, w ) | n +1+ λ ≤ C | ρ ( z, i ) | n +1+ λ for all z ∈ U .Proof. Let w ∈ Q j be fixed. We first show that(2.18) 1 − tanh ( j )4 ≤ ρ ( w ) ≤ − tanh ( j ) . Note from (2.13) that ρ ( w ) = | ρ ( w, i ) | (cid:0) − tanh β ( w, i ) (cid:1) . OSITIVE TOEPLITZ OPERATORS 9
Since β ( w, i ) < j and | ρ ( w, i ) | > /
4, the first inequality in (2.18) follows. Again,it follows from (2.13) that ρ ( w ) = (cid:18) ρ ( w ) | ρ ( w, i ) | (cid:19) − tanh β ( w, i ) . The second inequality in (2.18) is then immediate, in view of (2.1).Now, the assertion of the lemma follows from (2.18) and Lemma 2.7. (cid:3)
Growth rate for functions in A p ( U ) .Lemma 2.9. Soppose r > and p > . Then there exists a positive constant C depending on r such that | f ( z ) | p ≤ C ρ ( z ) n +1 Z D ( z,r ) | f ( w ) | p dV ( w ) for all f ∈ H ( U ) and all z ∈ U .Proof. Let f ∈ H ( U ). Then f ◦ Φ ∈ H ( B ). Note that D ( i , r ) = Φ( B (0 , R )) with R = tanh( r ). By the subharmonicity of | f | p , we get | f (Φ(0)) | p ≤ n ! π n R n Z B (0 ,R ) | f (Φ( ξ )) | p dV ( ξ ) w =Φ( ξ ) ======= n !4 π n R n Z D ( i ,r ) | f ( w ) | p | ρ ( w, i ) | n +1) dV ( w ) . Note that f (Φ(0)) = f ( i ) and inf {| ρ ( w, i ) | : w ∈ U} ≥ /
2. Then we have | f ( i ) | p ≤ C Z D ( i ,r ) | f ( w ) | p dV ( w ) , with C := 4 n n ! / ( π n R n ). Replacing f by f ◦ σ − z in the above inequality, we arriveat | f ( z ) | p ≤ C Z D ( i ,r ) | f ( σ − z ( w )) | p dV ( w ) u = σ − z ( w ) ======== C ρ ( z ) n +1 Z D ( z,r ) | f ( u ) | p dV ( u ) . This completes the proof of the Lemma. (cid:3)
Corollary 2.10.
Suppose < p < ∞ . Then | f ( z ) | ≤ (cid:18) n n ! π n (cid:19) /p k f k p ρ ( z ) ( n +1) /p , z ∈ U for all f ∈ A p ( U ) . Weak convergence in A p ( U ) .Lemma 2.11. Assume { f j } is a sequence in A p ( U ) with < p < ∞ . Then f j → weakly in A p ( U ) if and only if { f j } is bounded in A p ( U ) and converges to uniformly on each compact subset of U .Proof. The proof of the sufficiency is not hard. The proof of the necessity is alsoa standard normal family argument. We include a proof for reader’s convenience.Suppose { f j } converges to 0 weakly in A p ( U ). Then, by the uniform boundednessprinciple, { f j } is bounded in A p ( U ). Together with Corollary 2.10, this implies that { f j } is uniformly bounded on each compact subset of U and thus is a normal family.Note that f j → K z ∈ A p ′ ( U )(see [5, Theorem 2.1]) with p ′ = p/ ( p − z ∈ U , K z belongs to A t ( U ) for all 1 < t < ∞ .) Now, by a standard argument, we see that f j → U . The proof is complete. (cid:3) Lemma 2.12.
For < p < ∞ , we have K z k K z k − p → weakly in A p ( U ) as z → ∂ b U .Proof. In view of lemma 2.11, it suffices to prove that K z k K z k − p converges to 0uniformly on every Q j .By Lemmas 2.2 and 2.7, there exists a constant C > w ∈ Q j (cid:12)(cid:12)(cid:12)(cid:12) K z ( w ) k K z k p (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ρ ( z ) ( n +1) /p ′ | ρ ( z, i ) | n +1 for all z ∈ U . Since 2 | ρ ( z, i ) | = | z n + i | ≥ z ∈ U , we havesup w ∈ Q j (cid:12)(cid:12)(cid:12)(cid:12) K z ( w ) k K z k p (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ρ ( z ) ( n +1) /p ′ , which implies that K z k K z k − p → Q j as z → b U . On the other hand,by (2.1) and the fact that 2 | ρ ( z, i ) | ≥ | z | for all z ∈ U ,sup w ∈ Q j (cid:12)(cid:12)(cid:12)(cid:12) K z ( w ) k K z k p (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | z | ( n +1) /p , which implies that K z k K z k − p → Q j as | z | → ∞ . The proof of thelemma is complete. (cid:3) Carleson measures
Theorem 3.1.
Suppose < p < ∞ , r > and µ is a positive Borel measure. Thenthe following conditions are equivalent: (a) µ is a Carleson measure for A p ( U ) . (b) There exists a constant
C > such that Z U ρ ( a ) n +1 | ρ ( z, a ) | n +1) dµ ( z ) ≤ C for all a ∈ U . OSITIVE TOEPLITZ OPERATORS 11 (c)
There exists a constant
C > such that µ ( D ( a, r )) ≤ C ρ ( a ) n +1 for all a ∈ U . (d) There exists a constant
C > such that µ ( D ( a k , r )) ≤ C ρ ( a k ) n +1 for all k ≥ , where { a k } is an r -lattice in the Bergman metric.Proof. It is easy to see that (a) implies (b). In fact, setting f ( z ) = (cid:20) ρ ( a ) n +1 ρ ( z, a ) n +1) (cid:21) /p in (a) immediately yields (b).If (b) is true, then Z D ( a,r ) ρ ( a ) n +1 | ρ ( z, a ) | n +1) dµ ( z ) ≤ C for all a ∈ U . This along with Lemma 2.7 shows that (c) must be true.That (c) implies (d) is trivial.It remains to prove that (d) implies (a). So we assume that there exists a constant C > µ ( D ( a k , r )) ≤ C ρ ( a k ) n +1 for all k ≥
1. If f is holomorphic in U , then Z U | f | p dµ ≤ ∞ X k =1 Z D ( a k ,r ) | f ( z ) | p dµ ( z ) ≤ ∞ X k =1 µ ( D ( a k , r )) sup {| f ( z ) | p : z ∈ D ( a k , r ) } . By Lemmas 2.9 and 2.7, there exists a constant C > {| f ( z ) | p : z ∈ D ( a k , r ) } ≤ C ρ ( a k ) n +1 Z D ( a k , r ) | f ( w ) | p dV ( w )for all k ≥
1. It follows that Z U | f | p dµ ≤ C ∞ X k =1 µ ( D ( a k , r )) ρ ( a k ) n +1 Z D ( a k , r ) | f ( w ) | p dV ( w ) ≤ C C ∞ X k =1 Z D ( a k , r ) | f ( w ) | p dV ( w ) ≤ C C N Z U | f ( w ) | p dV ( w ) , where N is as in Lemma 2.4. This completes the proof of the theorem. (cid:3) It follows from the above theorem that the property of being a Carleson measurefor A p ( U ) is independent of p , so that if µ is a Carleson measure for A p ( U ) for some p , then µ is a Carleson measure for A p ( U ) for all p . Theorem 3.2.
Suppose < p < ∞ , r > and µ is a positive Borel measure. Thenthe following conditions are equivalent: (a) µ is a vanishing Carleson measure for A p ( U ) . (b) The measure µ satisfies lim a → ∂ b U Z U ρ ( a ) n +1 | ρ ( z, a ) | n +1) dµ ( z ) = 0(c) The measure µ has the property that lim a → ∂ b U µ ( D ( a, r )) ρ ( a ) n +1 = 0 . (d) For { a k } an r -lattice in the Bergman metric, we have lim k →∞ µ ( D ( a k , r )) ρ ( a k ) n +1 = 0 . Proof.
If (a) is true, it means that the inclusion map i p is compact. By Lemma2.11 and Lemma 2.12, we see that k a converges to 0 uniformly on each compactsubset of U as a → ∂ b U , and so does g a := k /pa . It is obvious that k g a k p = 1. Thenby Lemma 2.11 again, g a converges to 0 weakly in A p ( U ) as a → ∂ b U . Therefore, Z U ρ ( a ) n +1 | ρ ( z, a ) | n +1) dµ ( z ) = C Z U | g a ( z ) | p dµ ( z ) → a → ∂ b U . So (b) follows.If (b) holds, then lim a → ∂ b U Z D ( a,r ) ρ ( a ) n +1 | ρ ( z, a ) | n +1) dµ ( z ) = 0 . By Lemma 2.7, | ρ ( z, a ) | and ρ ( z ) are comparable when z ∈ D ( a, r ). So (c) follows.Note that a k → ∂ b U as k → ∞ if { a k } is an r -lattice in the Bergman metric.That (c) implies (d) is immediate.It remains to prove that (d) implies (a). Assume that (d) is true and { f j } is asequence in A p ( U ) that converges to 0 weakly. We only need to prove that(3.1) lim j →∞ Z U | f j ( z ) | p dµ ( z ) = 0 . By Lemma 2.11, { f j } has the property that sup j k f j k p ≤ M for some positive con-stant M and converges to 0 uniformly on each compact subset of U . By assumption,given ε > N such that µ ( D ( a k , r )) ρ ( a k ) n +1 < ε for all k ≥ N . OSITIVE TOEPLITZ OPERATORS 13
By the last part of the proof of Theorem 3.1, there is a constant
C > ∞ X k = N Z D ( a k ,r ) | f j ( w ) | p dµ ( w ) ≤ C ∞ X k = N µ ( D ( a k , r )) ρ ( a k ) n +1 Z D ( a k , r ) | f j ( w ) | p dV ( w ) ≤ εCN Z U | f j ( w ) | p dV ( w ) ≤ εCN M p for all j , where C , N , and M are all independent of ε . Sincelim j →∞ N − X k =1 Z D ( a k ,r ) | f j ( z ) | p dµ ( z ) = 0by uniform convergence. Therefore,lim sup j →∞ Z U | f j ( z ) | p dµ ( z ) ≤ lim sup j →∞ N − X k =1 Z D ( a k ,r ) | f j ( z ) | p dµ ( z ) + ∞ X k = N Z D ( a k ,r ) | f j ( w ) | p dV ( w ) ≤ εCN M p . Since ε is arbitrary, (3.1) follows. The proof of the theorem is complete. (cid:3) It follows from the above theorem that the property of being a vanishing Carlesonmeasure for A p ( U ) depends neither on p nor on r .4. Dense subspaces of A p ( U )Given α real, we denote by S α the vector space of functions f holomorphic in U satisfying sup z ∈U | z n + i | α | f ( z ) | < ∞ . Theorem 4.1. If ≤ p < ∞ and α > n + 1 /p , then S α is a dense subspace of A p ( U ) .Proof. It is immediate from (2.5) that S α is contained in A p ( U ) whenever α >n + 1 /p .We now prove the density of S α in A p ( U ). Let f be arbitrary in A p ( U ). Put f j = f · χ Q j for j = 1 , , . . . , where Q j := D ( i , j ) and χ Q j is the characteristicfunction of Q j . Clearly, k f j − f k p → j → ∞ .Given λ > −
1, let P λ be the integral operator given by P λ g ( z ) = c λ Z U ρ ( w ) λ ρ ( z, w ) n +1+ λ g ( w ) dV ( w ) , z ∈ U , where c λ = Γ( n + 1 + λ ) / (4 π n Γ(1 + λ )). It was shown in [5, Theorem 3.1] that P λ is a bounded projection from L p ( U ) onto A p ( U ), provided that λ > /p − Take λ = α − n −
1. By H¨older’s inequality, we obtain | P α − n − f j ( z ) | ≤ c α − n − k f k p | Q j | /p ′ sup w ∈ Q j ρ ( w ) α − n − | ρ ( z, w ) | α for all z ∈ U , where p ′ = p/ ( p −
1) and | Q j | stands for the Lebesgue measure of Q j .Combining this inequality with Lemma 2.8, we obtain | P α − n − f j ( z ) | ≤ C k f k p | ρ ( z, i ) | α for all z ∈ U , where C > n , α , j and p . Thus, P α − n − f j ∈ S α .Since f ∈ A p ( U ) and P α − n − is a bounded projection from L p ( U ) onto A p ( U ), k P α − n − f j − f k p = k P α − n − ( f j − f ) k p ≤ k P α − n − k k f j − f k p → j → ∞ . This implies that S α is dense in A p ( U ). (cid:3) Corollary 4.2.
The Toeplitz operator T µ with symbol µ ∈ M + is densely definedon A p ( U ) for every < p < ∞ .Proof. It suffices to show that Z U | K ( z, w ) f ( w ) | dµ ( w ) < ∞ holds for every f ∈ S α with α >
0, and for each fixed z ∈ U . Indeed, it follows by(2.1) that there exists a constant C > n and α such that Z U | K ( z, w ) f ( w ) | dµ ( w ) ≤ C ρ ( z ) − n − Z U dµ ( w ) | w n + i | α < ∞ , as desired. (cid:3) Proofs of the theorems
Just like the cases of the unit disk or bounded symmetric domains, the key stepis the justification of the equality(5.1) h T µ f, g i = Z U f gdµ, where h· , ·i denotes the duality pairing between A p ( U ) and A p ′ ( U ). This wouldenable us to make a connection between Carleson measures and positive Toeplitzoperators. Lemma 5.1.
Let < α < n + 1 . Then there exists a constant C > such that (5.2) Z U dV ( u ) | ρ ( z, u ) | n +1 | ρ ( u, i ) | α ≤ C | ρ ( z, i ) | α (cid:18) | ρ ( z, i ) | ρ ( z ) (cid:19) for any z ∈ U . OSITIVE TOEPLITZ OPERATORS 15
Proof.
Given z ∈ U , let η := Φ − ( z ). Making the change of variables u = Φ( ξ ) inthe integral and using Lemma 2.3, we obtain Z U dV ( u ) | ρ ( z, u ) | n +1 | ρ ( u, i ) | α = 4 | η n | n +1 Z B dV ( ξ ) | − η · ξ | n +1 | ξ n | n +1 − α . By [13, Theorem 3.1], the last integral is dominated by a constant multiple of1 | η n | n +1 − α log e | − η · ϕ η ( − e n ) | = 1 | η n | n +1 − α (cid:18) | η n | − | η | (cid:19) , where e n := (0 ′ , C > Z U dV ( u ) | ρ ( z, u ) | n +1 | ρ ( u, i ) | α ≤ C | η n | α (cid:18) − | η | (cid:19) = C | ρ ( z, i ) | α (cid:18) | ρ ( z, i ) | ρ ( z ) (cid:19) as desired. (cid:3) Lemma 5.2.
Suppose that < p < ∞ , n + 1 /p < α < n + 1 and that µ ∈ M + isa Carleson measure for A ( U ) . Then T µ maps S α into A p ( U ) .Proof. The proof of Corollary 4.2 shows that T µ is well defined on S α . Let f ∈ S α .We proceed to show that T µ f in A p ( U ). Since µ is a Carleson measure for A ( U ),there exists a constant C > Z U | K ( z, w ) f ( w ) | dµ ( w ) ≤ C Z U | K ( z, w ) f ( w ) | dV ( w ) ≤ C Z U dV ( u ) | ρ ( z, u ) | n +1 | ρ ( u, i ) | α . This, together with (5.2), implies(5.3) | T µ f ( z ) | ≤ C | ρ ( z, i ) | α (cid:18) | ρ ( z, i ) | ρ ( z ) (cid:19) for all z ∈ U . Since log x < x ǫ holds for any x > ǫ >
0, we have(5.4) log | ρ ( z, i ) | ρ ( z ) < | ρ ( z, i ) | ǫ ρ ( z ) ǫ for all z ∈ U . Choose ǫ small enough so that 0 < ǫ < min { /p, α − ( n + 1) /p } .Then we have Z U | T µ f ( z ) | p dV ( z ) ≤ C Z U dV ( z ) | ρ ( z, i ) | pα + Z U ρ ( z ) − pǫ | ρ ( z, i ) | p ( α − ǫ ) dV ( z ) , which along with (2.5) completes the proof. (cid:3) Lemma 5.3.
Suppose that < p < ∞ , n + 1 /p < α < n + 1 , γ > n + ( p − /p andthat µ ∈ M + is a Carleson measure for A ( U ) . Then (5.1) holds for all f ∈ S α and g ∈ S γ . Proof.
Let f ∈ S α and g ∈ S γ . In view of Theorem 4.1 and Lemma 5.2, we see that f ∈ A p ( U ), g ∈ A p ′ ( U ) and T µ f ∈ A p ( U ). Then both side of (5.1) are well defined.By Fubini’s theorem, h T µ f, g i = Z U (cid:18) Z U K ( z, w ) f ( w ) dµ ( w ) (cid:19) g ( z ) dV ( z )= Z U f ( w ) (cid:18) Z U K ( w, z ) g ( z ) dV ( z ) (cid:19) dµ ( w )= Z U f ( w ) g ( w ) dµ ( w ) , where the last equality follows from [5, Theorem 2.1]. The interchange of the orderof integration is justified as follows. By (5.3) and (5.4),with ǫ ∈ (0 , Z U (cid:18) Z U | K ( z, w ) f ( w ) g ( z ) | dµ ( w ) (cid:19) dV ( z ) ≤ C Z U | ρ ( z, i ) | α + γ (cid:18) | ρ ( z, i ) | ǫ ρ ( z ) ǫ (cid:19) dV ( z ) ≤ C (cid:18) Z U dV ( z ) | ρ ( z, i ) | α + γ + Z U ρ ( z ) − ǫ | ρ ( z, i ) | α + γ − ǫ dV ( z ) (cid:19) , which is finite, in view of (2.5). The proof of the lemma is complete. (cid:3) Corollary 5.4.
Suppose that µ ∈ M + is a Carleson measure for A q ( U ) for some q > . Then T µ is densely defined and extends to a bounded operator on A p ( U ) for any p > . Moreover, (5.1) holds for all f ∈ A p ( U ) and g ∈ A p ′ ( U ) , where p ′ = p/ ( p − . Now we are in the position to prove Theorems 1.1 and 1.2.
Proof of Theorem 1.1.
Combining Theorem 3.1 with Corollary 2.6, we see that (ii),(iii) and (iv) are equivalent. Also, that (iv) implies (i) is immediate from Corollary5.4. So we only need to prove that (i) implies (ii).Assume that T µ is bounded on A p ( U ). For every z ∈ U , by Lemma 2.2, h T µ k z , k z i is well defined. Lemma 2.2 also yields the following identity: k K z k p k K z k p ′ = CK ( z, z ) , where C is a positive constant depending on p and n . Hence |h T µ k z , k z i| ≤ k T µ K z k p k K z k p ′ K ( z, z ) = C (cid:13)(cid:13)(cid:13)(cid:13) T µ (cid:18) K z k K z k p (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ C k T µ k . On the other hand, again by [5, Theorem 2.1], we have h T µ k z , k z i = h T µ k z , K z i p K ( z, z ) = ( T µ k z )( z ) p K ( z, z ) = e µ ( z ) . Hence, e µ is a bounded function on U . (cid:3) OSITIVE TOEPLITZ OPERATORS 17
Proof of Theorem 1.2.
Combining Theorem 3.2 with Corollary 2.6, we see that (ii),(iii) and (iv) are equivalent. Therefore, it will suffice to prove the implications (i) ⇒ (ii) and (iv) ⇒ (i).(i) ⇒ (ii). Assume that T µ is compact on A p ( U ) for some p >
1. As is shown inthe proof of Theorem 1.1, | e µ ( z ) | ≤ C (cid:13)(cid:13)(cid:13)(cid:13) T µ (cid:18) K z k K z k p (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) p for all z ∈ U . This, together with Lemma 2.12 and the compactness of T µ , implies e µ ∈ C ( U ).(iv) ⇒ (i). Assume that µ is a vanishing Carleson measure for A q ( U ). Then µ also is a vanishing Carleson measure for A p ′ ( U ), where p ′ := p/ ( p − (cid:8) k g k L p ′ ( µ ) : k g k p ′ = 1 (cid:9) is finite. Also, by Theorem 1.1, T µ is bounded on A p ( U ). Therefore, by Corollary 5.4, we have k T µ f k p = sup (cid:8) |h T µ f, g i| : k g k p ′ = 1 (cid:9) = sup (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U f gdµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : k g k p ′ = 1 ≤ k f k L p ( µ ) sup (cid:8) k g k L p ′ ( µ ) : k g k p ′ = 1 (cid:9) for any f ∈ A p ( U ). If f j → A p ( U ), then the compactness of i p impliesthat k f j k L p ( µ ) →
0, and hence k T µ f j k p →
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E-mail address : [email protected]@mail.ustc.edu.cn