Big Hankel operators on Hardy spaces of strongly pseudoconvex domains
aa r X i v : . [ m a t h . C V ] F e b BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLYPSEUDOCONVEX DOMAINS
BO-YONG CHEN AND LIANGYING JIANGA
BSTRACT . In this article, we investigate the (big) Hankel operators H f on Hardyspaces of strongly pseudoconvex domains with smooth boundaries in C n . We alsogive a necessary and sufficient condition for boundedness of the Hankel operator H f on the Hardy space of the unit disc, which is new in the setting of one variable.
1. I
NTRODUCTION
Let
Ω = { z ∈ C n : ρ ( z ) < } be a strongly pseudoconvex domain with smoothboundary, where ρ ( z ) ∈ C ∞ ( C n ) is a strictly plurisubharmonic defining function of Ω , with dρ = 0 in some neighborhood of ∂ Ω .Let dσ denote the surface measure on ∂ Ω induced by the Lebesgue measure of C n . For < p < ∞ , we denote L p ( ∂ Ω) the usual L p space on ∂ Ω with respect tothe measure dσ . Let H p (Ω) denote the Hardy space of holomorphic functions on Ω ,whose norm is given by k f k pH p := sup ε> Z ∂ Ω ε | f | p dσ ε , where Ω ε = { z ∈ C n : ρ ( z ) < − ε } and dσ ε is the surface measure on ∂ Ω ε . It iswell-know that any f ∈ H p (Ω) has a radial limit at almost all points on ∂ Ω (cf. [42]).Thus one can identify H p (Ω) as a closed subspace of L p ( ∂ Ω) . Let H ∞ (Ω) denote thespace of bounded holomorphic functions in Ω .Let P : L ( ∂ Ω) → H (Ω) be the orthogonal projection via the Szeg¨o kernel S ( z, w ) , that is P g ( z ) = Z ∂ Ω S ( z, ζ ) g ( ζ ) dσ ( ζ ) for g ∈ L ( ∂ Ω) . We may regard this operator as a singular integral operator on ∂ Ω , for S ( z, w ) is C ∞ on ∂ Ω × ∂ Ω \ ∆ (where ∆ is the diagonal of ∂ Ω × ∂ Ω ). We also definethe Poisson-Szeg¨o kernel by P ( z, w ) = S ( z, z ) − | S ( z, w ) | for z ∈ Ω and w ∈ ∂ Ω (see [42]).For f ∈ L ( ∂ Ω) , the (big) Hankel operator H f with symbol f is the operator from H (Ω) into ( H (Ω)) ⊥ defined by H f g = ( I − P )( f g ) , g ∈ H (Ω) . We may identify H f with the operator ( I − P ) M f P on the Hilbert space L ( ∂ Ω) ,where M f is the multiplication operator given by ( M f g )( z ) = f ( z ) g ( z ) , z ∈ Ω . If f is bounded on ∂ Ω , then clearly H f is bounded with k H f k ≤ k f k ∞ . In general, H f isonly densely defined, whose domain contains H ∞ (Ω) . Key words and phrases.
Hankel operator, Hardy space, strongly pseudoconvex domain.
This operator is a generalization of the Hankel operator in the case of the unit disc D , which is different from the (small) Hankel operator h f defined by h f g = P ( f g ) : H (Ω) → H (Ω) , where H (Ω) is the space of complex conjugate of functions in H (Ω) which are zero at the origin of C n . The theory of Hankel operators of the unitdisc is classical. There are several books dealing with Hankel operators on the Hardyspace H ( D ) , see [32], [35], [36] and [48] for details.For the unit disc, the characterization of bounded Hankel operators on the Hardyspace is due to Nehari [30]. The compactness is given by Hartman [20]. Schatten classHankel operators is characterized by Peller [33], [34] and Semmes [40].The extension of these results to higher dimensional domains for (big) Hankel oper-ators is proved to be resistant in the case of the unit ball B n . A fundamental result dueto Coifman, Rochberg and Weiss [13] asserts that [ M f , P ] = M f P − P M f is boundedon L ( ∂B n ) for f ∈ BM O , and [ M f , P ] is compact if f ∈ V M O . The commutatorresult implies that H f = [ M f , P ] P is bounded on L ( ∂B n ) for every f ∈ BM O and H f is compact if f ∈ V M O . In fact, by virtue of the relation [ M f , P ] = H f − H ∗ f , the study of the commutator [ M f , P ] is equivalent to the problem concerning both H f and H f .The study of H f alone is usually more difficult than corresponding problems con-cerning H f and H f simultaneously. Since H f = H f − P f , properties of the Hankeloperator H f for general f ∈ L ( ∂ Ω) are recovered by properties of f − P f .For f ∈ L ( ∂B n ) , it was showed in [46] that H f is bounded on the Hardy space H ( B n ) if and only if f − P f ∈ BM O , and H f is compact if and only if f − P f ∈ V M O . Moreover, in [15], the authors proved that H f belongs to the Schatten class S p , n < p < ∞ , if and only if f lies in the Besov space B p . Other characterizations forthe Hankel operator H f on the Hardy space H ( B n ) were given in [16], [45] and [47].In particular, Amar [2] established links between the Hankel operator H f on theHardy space H ( B n ) and the ∂ of a Stokes extension of f in B n , which is also themotivation of our work. In addition, we mention the work of Li and Luecking (see[20-22] and [28]), where the theory of ∂ -operator is used to give complete character-izations of boundedness, compactness and Schatten ideals of Hankel operators actingon Bergman spaces of strongly pseudoconvex domains.In this article, we try to give some characterizations for boundedness of Hankeloperators on the Hardy space H (Ω) , based on the theory of BM O , V M O , and ∂ -operator, when Ω is a bounded strongly pseudoconvex domain in C n . Our main resultis the following. Theorem 1.1.
Let Ω be a bounded strongly pseudoconvex domain with smooth bound-ary and f ∈ L ( ∂ Ω) . If f ∈ BM O ( ∂ Ω) , then the Hankel operator H f is boundedfrom H (Ω) into L ( ∂ Ω) . The idea for proving this result comes from Amar’s ∂ -method and the characteriza-tion of BM O given by Varopoulos [44]. This method is different from the proofs ofcorresponding results in the setting of the unit disc and the unit ball. Particularly, inthe proof of Theorem 1.1, we get a solution to Gleason’s problem on the Hardy space H p (Ω) for ≤ p ≤ ∞ . IG HANKEL OPERATORS ON THE HARDY SAPCE 3
In fact, the proof of Theorem 1.1 shows that H f is bounded on H (Ω) if f admitsa decomposition f = f + f with f ∈ L ∞ ( ∂ Ω) and f having an extension e f in Ω ,such that | ∂ e f | dV and √− ρ | ∂ e f ∧ ∂ρ | dV are Carleson measures in Ω , where dV is theLebesgue measure on Ω . We conjecture that this condition is also necessary.Based on this observation, we give the following ∂ -characterization for Hankel op-erators on the Hardy space H ( D ) of the unit disc. Theorem 1.2. If f ∈ L ( ∂D ) , then the following properties are equivalent.(1) H f is bounded from H ( D ) into L ( ∂D ) .(2) The function F ( z ) = inf h ∈ H ( D ) π Z | f − h | P z ( θ ) dθ (1 . is bounded.(3) The measure | ∂f∂z | log | z | dxdy is a Carleson measure, where f ( z ) = R f ( θ ) P z ( θ ) dθ is the Poisson integral of f . Moreover, applying Berndtsson’s L -estimations for solutions of ∂ b -equation, weshall answer the question about the boundedness of Hankel operators on the Hardyspace H (Ω) , when Ω is a general pseudoconvex domain. We shall also use Donnelly-Fefferman’s L -estimates to give a new proof for the ∂ -characterization of H f on theBergman space A (Ω) . This result has been proved in Li [25], by using the integralrepresentation of a solution to ∂ -equation.This paper is divided into six sections. After some necessary preliminaries in Section2, we solve the Gleason’s problem on the Hardy space H p (Ω) . In Section 3, we give theproof of Theorem 1.1 and generalize this result to Hardy spaces H p (Ω) for p > . Thecompactness of these Hankel operators shall be discussed in this section. In Section 4,we try to investigate the boundedness of Hankel operators when acting on Hardy spacesof general pseudoconvex domains. Similarly, we use a different method to prove thecorresponding result on the Bergman space of strongly pseudoconvex domain. Section5 devotes to give a new characterization for Hankel operator on the Hardy space in thecase of the unit disc. In last section, we show that both H f and H f are bounded onthe Hardy space H (Ω) if and only if f ∈ BM O ( ∂ Ω) , when Ω is a bounded stronglypseudoconvex domain, and /S ( · , z ) is holomorphic and bounded in Ω for fixed z ∈ Ω .2. G EOMETRY OF STRONGLY PSEUDOCONVEX DOMAINS
The geometry of strongly pseudoconvex domains is well understood (see [23] and[37]). In this section, we collect some notations and results needed later.Let d ( p, q ) be the Koranyi distance between points on ∂ Ω and B ( p, r ) = { ζ ∈ ∂ Ω : d ( ζ , p ) < r } the corresponding balls. The tent Q ( p, r ) over B ( p, r ) is Q ( p, r ) = { z ∈ Ω : d ( z ′ , p ) + d ( z ) < r } , where d ( z ) = d ( z, ∂ Ω) is the Euclidean distance and z ′ is the projection of z on ∂ Ω .A positive measure dµ on Ω is a Carleson measure if k µ k = sup p ∈ ∂ Ω ,r> σ ( B ( p, r )) Z Q ( p,r ) dµ = C B.-Y. CHEN AND L. JIANG is finite. It is well-known [22] that dµ is a Carleson measure if and only if Z Ω | φ | p dµ ≤ C k φ k pH p , φ ∈ H p (Ω) . (2 . Also, we call dµ a vanishing Carleson measure if lim j →∞ Z Ω | φ j | p dµ = 0 for any sequence { φ j } , which is bounded in H p (Ω) and converges to uniformly onany compact subset of Ω as j → ∞ .For ≤ α ≤ , we denote by W α the interpolation spaces between the space W of Carleson measures and the space W of finite measures in Ω . If < α < , thenthe measure dµ is in W α if and only if dµ = φdτ , where dτ ∈ W , φ ∈ L p ( dτ ) and /p = 1 − α (see [3]). In view of (2.1), if φ ∈ H p (Ω) and dτ ∈ W , then φdτ ∈ W α and k φdτ k W α . k φ k H p k dτ k W . (2 . Let ω be a ∂ -closed (0 , form in Ω . If u is a function defined on ∂ Ω that can beobtained as the boundary value of a solution to ∂u = ω , then we write ∂ b u = ω (see [41], [3] or [8]). Following Skoda [41], this equation can be given by Z ∂ Ω uh = Z Ω ω ∧ h. (2 . for any smooth ∂ -closed form h of bidegree ( n, n − .In [3], Amar and Bonami gave the following on the smoothly bounded stronglypseudoconvex domains, which is very important for proving our main result. Theorem A.
Suppose that ω is a smooth ∂ -closed (0 , form on Ω and p > . If thecoefficients of ω and ∂ρ ∧ ω √− ρ belong to the space W α with /p = 1 − α , then there is asolution of ∂ b u = ω continuous on Ω such that k u k L p ( ∂ Ω) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | ω | + | ω ∧ ∂ρ |√− ρ (cid:19) dV (cid:13)(cid:13)(cid:13)(cid:13) W α for a constant C > . The space
BM O ( ∂ Ω) is defined in terms of the Koranyi distance as the space offunctions such that k f k BMO := sup p ∈ ∂ Ω ,r> σ ( B ( p, r )) Z B ( p,r ) | f ( ζ ) − f B ( p,r ) | dσ ( ζ ) < ∞ , where f B ( p,r ) is the average of f over the ball B ( p, r ) . Moreover, V M O ( ∂ Ω) is definedas the subspace of BM O which is the closure of the continuous functions in the
BM O topology. We also put
BM OA = BM O ∩ O (Ω) and
V M OA = V M O ∩ O (Ω)
When Ω is strongly pseudoconvex, the reproducing property of the Poisson-Szeg¨okernel gives f ( z ) = Z ∂ Ω f ( ζ ) P ( z, ζ ) dσ ( ζ ) for f ∈ H (Ω) IG HANKEL OPERATORS ON THE HARDY SAPCE 5 (see [42]). It follows from [27] that f ∈ BM OA (Ω) if and only if k f k BMOA = sup (cid:26)Z ∂ Ω | f ( ζ ) − f ( z ) | P ( z, ζ ) dσ ( ζ ); z ∈ Ω (cid:27) < ∞ , by using similar arguments as in the book of Garnett [18] and the John-NirebergLemma. In fact, as in [18], one may show that these characterizations also hold for BM O ( ∂ Ω) and V M O ( ∂ Ω) , the only difference is that f ( z ) should be replaced by e f ( z ) = Z ∂ Ω f ( ζ ) P ( z, ζ ) dσ ( ζ ) , the Poisson-Szeg¨o integral of f .By [29] there is a smooth function Φ( ζ , z ) on Ω × Ω which is holomorphic in z forfixed ζ ∈ Ω and satisfies Re Φ ≥ − ρ ( ζ ) − ρ ( z ) + δ | ζ − z | and d ζ Φ | ζ = z = − d z Φ | ζ = z = − ∂ρ ( ζ ) . It is well-known that | Φ( ζ , z ) | is compatiblewith the Koranyi distance d ( ζ , z ) . Moreover, there is a form Q ( ζ , z ) = n P j =1 Q j ( ζ , z ) dζ j ,holomorphic in z , such that Φ( ζ , z ) = n P j =1 Q j ( ζ , z )( z j − ζ j ) and the following Cauchy-Fantappi`e formula holds F u ( z ) = c Z ∂ Ω u ( ζ ) Q ∧ ( ∂Q ) n − Φ( ζ , z ) n dσ ( ζ ) , z ∈ Ω . Clearly,
F u is holomorphic in Ω if u ∈ L p ( ∂ Ω) , ≤ p ≤ ∞ and F u = u if u is the boundary value of a holomorphic function. In fact, the kernel in the Cauchy-Fantappi`e formula is also called the Henkin-Ramirez reproducing kernel. Thus, F u has admissible boundary values a.e. if u ∈ L p ( ∂ Ω) , p > , and this operator maps L p ( ∂ Ω) into H p (Ω) , BM O into
BM OA , we refer to [21] or [37] for details. We alsonotice that the Cauchy-Fantappi`e formula for a holomorphic function u can be writtenas follows u ( z ) = Z ∂ Ω A ( ζ , z ) u ( ζ )Φ( ζ , z ) n dσ ( ζ ) , where A ( ζ , z ) is smooth and holomorphic in z .To conclude this section, we will solve the following Gleason’s problem in the set-ting of bounded strongly pseudoconvex domains. Let X be some class of holomorphicfunctions in a region Ω ⊂ C n . Gleason’s problem for X is the following: If a ∈ Ω and f ∈ X , do there exist functions g , . . . , g n ∈ X such that f ( z ) − f ( a ) = n X j =1 ( z j − a j ) g j ( z ) for all z ∈ Ω ? Gleason [19] originally asked this problem for
Ω = B n , a = 0 and X = A ( B n ) ,the ball algebra, which was solved by Leibenson. If X is one of the spaces H p ( B n ) ( ≤ p ≤ ∞ ), A ( B n ) and Lip α ( < α ≤ ), then the problem was also solved (see[38]). The simplest solution is probably the one found by Ahern and Schneider [1], B.-Y. CHEN AND L. JIANG which also works for strongly pseudoconvex domains when X = Lip α or A (Ω) . Here,we shall solve the problem for X = H p (Ω) when Ω is strongly pseudoconvex.First of all, we recall the following two results due to Stout [43]. Theorem B. If f ∈ H p (Ω) , ≤ p ≤ ∞ and if γ is a Lipschitz continuous function on C n , then the function F ( z ) = Z ∂ Ω f ( ζ ) γ ( ζ ) Q ∧ ( ∂Q ) n − Φ( ζ , z ) n dσ ( ζ ) belongs to H p (Ω) . Note that F ( z ) = Z ∂ Ω f ( ζ )[ γ ( ζ ) − γ ( z )] Q ∧ ( ∂Q ) n − Φ( ζ , z ) n dσ ( ζ ) + γ ( z ) f ( z ) . Thus, in order to proving Theorem B, the author considered the operator T = T γ defined by T g ( z ) = Z ∂ Ω g ( ζ )[ γ ( ζ ) − γ ( z )] Q ∧ ( ∂Q ) n − Φ( ζ , z ) n dσ ( ζ ) , and obtained the following boundedness properties of the operator T . Theorem C. (1) If g ∈ L q ( ∂ Ω) , q ≥ n , then T g is bound on Ω . (2) If g ∈ L p ( ∂ Ω) , ≤ p < ∞ , then sup r< Z ρ = r | T g ( ζ ) | p dσ ( ζ ) < ∞ . Now we may solve Gleason’s problem for H p (Ω) . Proposition 2.1.
Let Ω be a bounded strongly pseudoconvex domain with smoothboundary. If f ∈ H p (Ω) , ≤ p ≤ ∞ , and a ∈ Ω , then there exist functions g , . . . , g n ∈ H p (Ω) such that f ( z ) − f ( a ) = n X j =1 ( z j − a j ) g j ( z ) . Proof.
We follow the arguments used in solving Gleason’s problem for the case X = A (Ω) . As in Theorem 4.2 of [37], the reproducing property of the Henkin-Ramirezkernel gives f ( z ) − f ( a ) = Z ∂ Ω f ( ζ ) (cid:20) A ( ζ , z )Φ( ζ , z ) n − A ( ζ , a )Φ( ζ , a ) n (cid:21) dσ ( ζ )= Z ∂ Ω f ( ζ ) (cid:20) A ( ζ , z )Φ( ζ , a ) n − A ( ζ , a )Φ( ζ , z ) n Φ( ζ , z ) n Φ( ζ , a ) n (cid:21) dσ ( ζ ) for f ∈ H p (Ω) and z, a ∈ Ω .By [31], p. 148, there are functions L j , j = 1 , . . . , n , which are C ∞ on ( ∂ Ω) × Ω × Ω and holomorphic in z and a , such that A ( ζ , z )Φ( ζ , a ) n − A ( ζ , a )Φ( ζ , z ) n = n X j =1 ( z j − a j ) L j ( ζ , z, a ) . IG HANKEL OPERATORS ON THE HARDY SAPCE 7
It follows that f ( z ) − f ( a ) = n X j =1 ( z j − a j ) Z ∂ Ω f ( ζ ) L j ( ζ , z, a )Φ( ζ , z ) n Φ( ζ , a ) n dσ ( ζ ) . (2 . Define T j f ( z, a ) = Z ∂ Ω f ( ζ ) L j ( ζ , z, a )Φ( ζ , z ) n Φ( ζ , a ) n dσ ( ζ ) . Clearly, T j f is holomorphic on Ω × Ω . If we show that T j f ( z, a ) ∈ H p (Ω) for fixed a , we were done.Note that Corollary 3.10 in [37] gives A ( ζ , z ) = h ( ζ ) + O ( | ζ − z | ) , where h ( ζ ) = 0 for all ζ ∈ ∂ Ω . Set χ ( ζ , z, a ) = L j ( ζ , z, a ) h ( ζ )Φ( ζ , a ) n . We have χ ( ζ , z, a ) = χ ( ζ , ζ , a ) + O ( | ζ − z | ) . A calculation shows that L j ( ζ , z, a )Φ( ζ , z ) n Φ( ζ , a ) n = L j ( ζ , z, a ) h ( ζ )Φ( ζ , a ) n · h ( ζ )Φ( ζ , z ) n = χ ( ζ , z, a ) A ( ζ , z )Φ( ζ , z ) n + χ ( ζ , z, a ) O ( | ζ − z | )Φ( ζ , z ) n = χ ( ζ , ζ , a ) A ( ζ , z )Φ( ζ , z ) n + O ( | ζ − z | ) A ( ζ , z )Φ( ζ , z ) n + χ ( ζ , z, a ) O ( | ζ − z | )Φ( ζ , z ) n . This implies T j f ( z, a ) = Z ∂ Ω f ( ζ ) χ ( ζ , ζ , a ) A ( ζ , z )Φ( ζ , z ) n dσ ( ζ )+ Z ∂ Ω f ( ζ ) O ( | ζ − z | ) A ( ζ , z )Φ( ζ , z ) n dσ ( ζ ) + Z ∂ Ω f χ O ( | ζ − z | )Φ( ζ , z ) n dσ ( ζ ) . (2 . Next, we will need a technique of Stout [43] to estimate (2.5), which is differentfrom that in the proof of Theorem 4.2 in [37]. Since χ ( ζ , ζ , a ) ∈ C ∞ , it follows fromTheorem B that F ( z ) = Z ∂ Ω f ( ζ ) χ ( ζ , ζ , a ) A ( ζ , z )Φ( ζ , z ) n dσ ( ζ ) belongs to H p (Ω) , ≤ p ≤ ∞ .Since the functions A and χ are smooth, we may use similar ideas to estimate thesecond term in (2.5). It remains to consider the last term. Set T f ( z ) = Z ∂ Ω f χ O ( | ζ − z | )Φ( ζ , z ) n dσ ( ζ ) and T ( r ) f = T f | { ρ = r } when r is close to . Once we prove the following claim k T ( r ) f k L ∞ ( { ρ = r } ) ≤ C k f k L ∞ ( ∂ Ω) and k T ( r ) f k L ( { ρ = r } ) ≤ C k f k L ( ∂ Ω) for a constant C > , the Riesz-Thorin theorem implies that if f ∈ L p ( ∂ Ω) , < p < ∞ , then k T ( r ) f k L p ( { ρ = r } ) ≤ C k f k L p ( ∂ Ω) . Since the constant C is independent of r , we obtain that T f ∈ L p ( ∂ Ω) for < p < ∞ . B.-Y. CHEN AND L. JIANG
To verify the claim, we first assume f ∈ L ∞ ( ∂ Ω) . Since χ is smooth, we have | T f ( z ) | . k f k ∞ (cid:18)Z ∂ Ω O ( | ζ − z | ) | Φ( ζ , z ) | n dσ ( ζ ) (cid:19) for any z ∈ Ω . In the proof of Theorem C, the author has showed that the aboveintegral is finite. The proof of Theorem C also yields that Z ρ = r O ( | ζ − z | ) | Φ( ζ , z ) | n dσ ( z ) are bounded uniformly in ζ , r , where ζ ∈ ∂ Ω and r is close to . Thus, if f ∈ L ( ∂ Ω) ,we get that Z ρ = r | T f ( z ) | dσ ( z ) are bounded uniformly in r near . Thus the claim is verified.Combining the above arguments with (2.5), we find that T j f ( · , a ) ∈ H p (Ω) for ≤ p ≤ ∞ . Taking g j ( z ) = T j f ( z, a ) , we obtain the desired result from (2.4). (cid:3) Proposition 2.1 gives a decomposition for H (Ω) . Similar as the proof of Lemma2.5 in [2], we may easily deduce the following lemma. Lemma 2.2.
The space H (Ω) ⊥ can be identified with the space of ∂ b -closed forms ofbidegree ( n, n − satisfying h ∈ H (Ω) ⊥ → H ∈ L n,n − ( ∂ Ω) , ∂ b H = 0 and Z ∂ Ω φ · hdσ = Z ∂ Ω φ · H , ∀ φ ∈ L ( ∂ Ω) . As an easy consequence of this lemma and Stokes’ theorem, we obtain the followingresult which generalizes a result of Amar [2] in the unit ball.
Proposition 2.3.
Suppose that Ω is a bounded strongly pseudoconvex domain withsmooth boundary. Let f ∈ L ( ∂ Ω) and e f be any Stokes’ extension of f in Ω . Then ∂ b ( H f g ) = g · ∂ e f , ∀ g ∈ H ∞ (Ω) . Proof.
The argument is similar to Proposition 2.4 in [2]. For convenience, we give asimple proof here.Let g ∈ H ∞ (Ω) and u = H f g = f g − P ( f g ) . Applying Lemma 2.2 and Stokes’theorem, for any h ∈ H (Ω) ⊥ , we get Z ∂ Ω u · hdσ = Z ∂ Ω [ f g − P ( f g )] · hdσ = Z ∂ Ω f g · hdσ = Z ∂ Ω f g · H = Z Ω g · ∂ e f ∧ H . This combined with (2.3) yields ∂ b u = g · ∂ e f , which completes the proof. (cid:3) Remark.
In fact, if g ∈ H ∞ (Ω) and f ∈ L ( ∂ Ω) , then P ( f g ) ∈ H (Ω) . Now,applying the definition of ∂ b in Page 166 of [12], we easily deduce that H f g = f g − P ( f g ) is a solution of ∂ b u = g · ∂ e f for the extension e f of f . IG HANKEL OPERATORS ON THE HARDY SAPCE 9
3. H
ANKEL OPERATORS ON H ARDY SPACES OF STRONGLY PSEUDOCONVEXDOMAINS WITH SMOOTH BOUNDARYS
Before proving Theorem 1.1, we shall introduce one more notion. Let f ∈ C (Ω) ,we define the nonisotropic gradiant near the boundary of Ω as | Df | = | ν ( f ) | + | µ ( f ) | + | ρ | − / n − X j =1 | µ j ( f ) | (see [44]), where ν is a normalized vector field in some neighborhood of ∂ Ω that isnormal and directed inwards to ∂ Ω at every point ζ ∈ ∂ Ω , µ is a normalized fieldthat is orthogonal to the holomorphic tangent space at every point ζ ∈ ∂ Ω , and fields µ , . . . , µ n − are constructed to form an orthonormal basis at ζ .In [13], the authors used the theory of singular integral operators to show that H f isbounded on the Hardy space H ( B n ) if f ∈ BM O ( ∂B n ) . Amar [2] applied the theoryof ∂ to prove that H f is bounded on H ( B n ) if f admits a Stokes’ extension e f to B n such that both | ∂ e f | dV and √− ρ | ∂ e f ∧ ∂ρ | dV are Carleson measures. Motivated by thisresult and the work of Varopoulos [44] about the BM O theory, we may generalize theresult of [13] to the bounded strongly pseudoconvex domains.
Proof of Theorem 1.1.
Let f ∈ BM O ( ∂ Ω) . By Theorem 2.1.1 of [44], f admits adecomposition f = f + f , such that f ∈ L ∞ ( ∂ Ω) and f has an extension e f in Ω ,which satisfies that lim λ → e f ( ζ + λν ( ζ )) = e f ( ζ ) , ∀ ζ ∈ ∂ Ω , and | D e f | dV is a Carlesonmeasure.Since f ∈ L ∞ ( ∂ Ω) , it is easy to see that the multiplication operator M f is boundedfrom H (Ω) into L ( ∂ Ω) , which implies that H f = ( I − P ) M f is bounded. Thus, itsuffices to prove the boundedness of H f .Now, f ∈ L ( ∂ Ω) and f ∈ L ∞ ( ∂ Ω) give that f ∈ L ( ∂ Ω) . Since | D e f | dV is a Carleson measure, we infer from Lemma 2.5.1 in [44] that both | ∂ e f | dV and √− ρ | ∂ e f ∧ ∂ρ | dV are Carleson measures in Ω . Let g ∈ H (Ω) . Applying (2.2) to | g · ∂ e f | dV and √− ρ | g · ∂ e f ∧ ∂ρ | dV , we deduce that the coefficients of g · ∂ e f and g · ∂ e f ∧ ∂ρ/ √− ρ belong to the space W / . Thus there is, according to Theorem A,a solution of the equation ∂ b u = g · ∂ e f which satisfies u ∈ L ( ∂ Ω) . Moreover, usingTheorem A and (2.2), we obtain k u k L ( ∂ Ω) . k| g · ∂ e f | dV k W / + (cid:13)(cid:13)(cid:13)(cid:13) √− ρ | g · ∂ e f ∧ ∂ρ | dV (cid:13)(cid:13)(cid:13)(cid:13) W / . k g k H (cid:18) k| ∂ e f | dV k W + (cid:13)(cid:13)(cid:13)(cid:13) √− ρ | ∂ e f ∧ ∂ρ | dV (cid:13)(cid:13)(cid:13)(cid:13) W (cid:19) . In view of Proposition 2.3, we have ∂ b ( H f g ) = g · ∂ e f when g ∈ H ∞ ( ∂ Ω) . Usingthe uniqueness of the solution orthogonal to H (Ω) , we get k H f g k L ( ∂ Ω) ≤ k u k L ( ∂ Ω) . k g k H (cid:18) k| ∂ e f | dV k W + (cid:13)(cid:13)(cid:13)(cid:13) √− ρ | ∂ e f ∧ ∂ρ | dV (cid:13)(cid:13)(cid:13)(cid:13) W (cid:19) . Since H ∞ ( ∂ Ω) is dense in H (Ω) , we see that H f is bounded on H (Ω) and k H f k . (cid:18) k| ∂ e f | dV k W + (cid:13)(cid:13)(cid:13)(cid:13) √− ρ | ∂ e f ∧ ∂ρ | dV (cid:13)(cid:13)(cid:13)(cid:13) W (cid:19) . It follows that H f = H f + H f is bounded from H (Ω) into L ( ∂ Ω) . (cid:3) From the proof of Theorem 1.1, we may also give the following characterization forthe compactness of H f on the Hardy space H (Ω) . Theorem 3.1.
Let f ∈ L ( ∂ Ω) . If f admits a decomposition f = f + f with f ∈ C ( ∂ Ω) and e f being an extension of f in Ω , such that | ∂ e f | dV and √− ρ | ∂ e f ∧ ∂ρ | dV are vanishing Carleson measures in Ω , then the Hankel operator H f is compact from H (Ω) into L ( ∂ Ω) .Proof. For f ∈ C ( ∂ Ω) ⊂ L ∞ ( ∂ Ω) , we know that H f is bounded on H (Ω) . Using aconsequence of the Kohn’s solution of ∂ b -Neumann problem (cf. [17]), we see that H φ is compact for φ ∈ C ∞ ( ∂ Ω) . Since any φ ∈ C ( ∂ Ω) can be approximated uniformlyby smooth ones, it follows from Lemma 1 in [39] that H φ = ( I − P ) M φ : H ( ∂ Ω) → L ( ∂ Ω) is compact for φ ∈ C ( ∂ Ω) , where H ( ∂ Ω) denotes the closure of the C ∞ -functions on ∂ Ω which can be extended holomorphically to Ω . This means that H f is also compacton H (Ω) .Now consider the function f . In the proof of Theorem 1.1, we have shown thatthere exists a positive constant C such that k H f g k L ( ∂ Ω) ≤ C (cid:18) k| g · ∂ e f | dV k W / + (cid:13)(cid:13)(cid:13)(cid:13) √− ρ | g · ∂ e f ∧ ∂ρ | dV (cid:13)(cid:13)(cid:13)(cid:13) W / (cid:19) for any g ∈ H (Ω) . Let { g j } be a bounded sequence in H (Ω) and converge to uniformly on any compact subset of Ω as j → ∞ . Since | ∂ e f | dV and √− ρ | ∂ e f ∧ ∂ρ | dV are vanishing Carleson measures, we infer from (2.1) and (2.2) that k| g j · ∂ e f | dV k W / → and (cid:13)(cid:13)(cid:13)(cid:13) √− ρ | g j · ∂ e f ∧ ∂ρ | dV (cid:13)(cid:13)(cid:13)(cid:13) W / → as j → ∞ . This yields that lim j →∞ k H f g j k L ( ∂ Ω) = 0 . Thus H f is compact on H (Ω) .So we complete the proof. (cid:3) In [44], Varopoulos found that f ∈ BM O ( ∂ Ω) admits a decomposition f = f + f , such that f ∈ L ∞ ( ∂ Ω) and f has an extension e f in Ω , which satisfying that lim λ → e f ( ζ + λν ( ζ )) = e f ( ζ ) for any ζ ∈ ∂ Ω and | D e f | dV is a Carleson measure. Thisfollows from a well-known result of Carleson, i.e. if f ∈ BM O ( ∂ Ω) then there existssome Carleson measure dµ in Ω such that f ( ζ ) − Z Ω P ( z, ζ ) dµ ( z ) ∈ L ∞ ( ∂ Ω) . IG HANKEL OPERATORS ON THE HARDY SAPCE 11
For f ∈ V M O ( ∂ Ω) it is natural to make the following conjecture: Conjecture 1. If f ∈ V M O ( ∂ Ω) , then there exists some vanishing Carleson measure dµ in Ω such that f ( ζ ) − Z Ω P ( z, ζ ) dµ ( z ) ∈ C ( ∂ Ω) . Remark.
Once this conjecture is true, one may use similar arguments as Theo-rem 2.1.1 and Lemma 2.5.1 in [44] to obtain the following decomposition: if f ∈ V M O ( ∂ Ω) , then f = f + f with f ∈ C ( ∂ Ω) and e f being an extension of f , suchthat the measures | ∂ e f | dV and √− ρ | ∂ e f ∧ ∂ρ | dV are vanishing Carleson measures in Ω .As a consequence, Theorem 3.1 implies that H f is compact from H (Ω) into L ( ∂ Ω) for f ∈ V M O ( ∂ Ω) .Now, for < p < ∞ and s > − , we recall that the weighted Bergman spaces A ps (Ω) consist of all holomorphic functions f on Ω satisfying k f k p,s = (cid:18)Z Ω | f | p ( − ρ ) s dV (cid:19) /p < ∞ . For s = − , we define A p − (Ω) to be the Hardy space H p (Ω) . Let Ω be a stronglypseudoconvex domain in C n , Theorem 5.4 in [6] gives that O (Ω) is dense in A ps (Ω) for < p < ∞ and s ≥ − . This implies that H ∞ (Ω) is dense in H p (Ω) for < p < ∞ .Since the projection P via the Szeg¨o kernel S ( z, w ) is bounded from L p ( ∂ Ω) into H p (Ω) if p > , we may define Hankel operators H f on Hardy spaces H p (Ω) . On theother hand, Theorem A and (2.2) hold for all p > . Therefore, we might use the sameproof to obtain the following. Theorem 3.2.
Let Ω be a bounded strongly pseudoconvex domain with smooth bound-ary in C n . Assume f ∈ L p ( ∂ Ω) and p > . If f ∈ BM O ( ∂ Ω) , then the Hankeloperator H f is bounded from H p (Ω) into L p ( ∂ Ω) . Finally, we try to give a necessary condition for Hankel operator H f to be boundedon the Hardy space H (Ω) , which shall build a bridge to solve the boundedness con-jecture pointed out in Section 5. Theorem 3.3.
Let Ω be a smoothly bounded strongly pseudoconvex domain. If theHankel operator H f is bounded from H (Ω) into L ( ∂ Ω) , then the following inequality inf h ∈ H ( B ( w,r )) σ ( B ( w, r )) Z B ( w,r ) | f ( ζ ) − h ( ζ ) | dσ ( ζ ) < ∞ holds for any w ∈ ∂ Ω and r > .Proof. Let Φ( z, w ) be the function introduced in Section 2 and v ( z, w ) = Φ( z, w ) − ρ ( z ) . It follows that | v ( z, w ) | ∼ d ( z ) + d ( w ) + d ( z ′ , w ′ ) . Set g z ( ζ ) = d ( z ) n/ v ( z, ζ ) n , z ∈ Ω , ζ ∈ Ω . Since v ( z, ζ ) is holomorphic in ζ for fixed z , and the well-known estimate Z ∂ Ω dσ ( ζ ) | v ( z, ζ ) | n + α . d ( z ) α for α > , we have sup z ∈ Ω k g z k H ≤ C for some constant C > . If H f is bounded, then sup z ∈ Ω k H f ( g z ) k L ( ∂ Ω) ≤ C k H f k . It follows that ∞ > sup z ∈ Ω Z ∂ Ω | f g z − P ( f g z ) | dσ = sup z ∈ Ω Z ∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) f ( ζ ) − g z ( ζ ) P ( f g z )( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) | g z ( ζ ) | dσ ( ζ ) . Let w ∈ ∂ Ω and B ( w, r ) be a ball on ∂ Ω in the Koranyi distance. Then for ζ ∈ B ( w, r ) and e w = w − rv (1) , where { v (1) , · · · , v ( n ) } is the orthonormal basis at w with v (1) transversal to the boundary, we see that | g e w ( ζ ) | = d ( e w ) n | v ( e w, ζ ) | n & σ ( B ( w, r )) . Since g e w ( ζ ) P ( f g e w )( ζ ) is holomorphic on ∂ Ω , we may choose a holomorphic function h on B ( w, r ) satisfying σ ( B ( w, r )) Z B ( w,r ) | f ( ζ ) − h ( ζ ) | dσ ( ζ ) . Z ∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) f ( ζ ) − g e w ( ζ ) P ( f g e w )( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) | g e w ( ζ ) | dσ ( ζ ) . As a consequence, we obtain the desired inequality. (cid:3)
4. H
ANKEL OPERATORS ON H ARDY SPACES OF GENERAL PSEUDOCONVEXDOMAINS
Let Ω be a bounded domain with C boundary in C n . A positive measure dµ on Ω is called a Carleson measure if there exists a constant C > such that Z Ω | g | dµ ≤ C Z ∂ Ω | g | dσ, ∀ g ∈ H (Ω) . The starting point is the following result due to Berndtsson:
Theorem D [cf. [9], see also [11]].
Let Ω be a bounded pseudoconvex domain with C boundary in C n and let ρ be a negative C function on Ω . Suppose there exists a C psh function ψ on Ω which is continuous on Ω and satisfies Θ := ( − ρ ) ∂ ¯ ∂ψ + ∂ ¯ ∂ρ > . (4.1) Let u be the L ψ (Ω) − minimal solution of the equation ¯ ∂u = v . Then Z ∂ Ω | u | e − ψ dσ/ |∇ ρ | ≤ Z Ω | v | e − ψ dV. IG HANKEL OPERATORS ON THE HARDY SAPCE 13
Applying this result, we shall give the following characterization for the bounded-ness of Hankel operators.
Theorem 4.1.
Let Ω be a bounded pseudoconvex domain with C boundary in C n andlet f ∈ L ( ∂ Ω) . Suppose the following conditions hold: (1) there exist a negative C function ρ on Ω and a C psh function ψ on Ω whichis continuous on Ω such that (4 . holds. (2) there is a decomposition f = f + f , where f ∈ L ∞ ( ∂ Ω) and f admits aStokes extension ˜ f to Ω such that | ¯ ∂ ˜ f | dV is a Carleson measure on Ω .Then H f : H (Ω) → L ( ∂ Ω) is bounded.Proof. Clearly H f : H (Ω) → L ( ∂ Ω) is bounded. For given g ∈ H (Ω) , H f g isnothing but the L ( ∂ Ω) − minimal solution of the equation ¯ ∂ b u = ¯ ∂ ( ˜ f g ) = g · ¯ ∂ ˜ f . It follows from Theorem D that Z ∂ Ω | H f g | dσ/ |∇ ρ | ≤ Z ∂ Ω | u | dσ/ |∇ ρ | = lim r → − (1 − r ) Z Ω | u | ( − ρ ) − r dV . lim r → − (1 − r ) Z Ω | u | ( − ρ ) − r e − ψ dV . Z ∂ Ω | u | e − ψ dσ/ |∇ ρ | . Z Ω | g | | ¯ ∂ ˜ f | e − ψ dV . Z Ω | g | | ¯ ∂ ˜ f | dV . Z ∂ Ω | g | dσ, since | ¯ ∂ ˜ f | dV is a Carleson measure. The proof is complete. (cid:3) Let Ω be a bounded strongly pseudoconvex domain in C n and let ρ ≥ − /e be adefinition function which is strictly psh on Ω . Given ε > , let ψ = − [ − log( − ρ )] − ε . A straightforward calculation shows that if ε ≤ / then ∂ ¯ ∂ψ = ε [ − log( − ρ )] − ε − ∂ ¯ ∂ [ − log( − ρ )] − ε (1 + ε )[ − log( − ρ )] − ε − ∂ log( − ρ ) ∧ ¯ ∂ log( − ρ ) ≥ ε − log( − ρ )] − ε − ∂ ¯ ∂ [ − log( − ρ )] ≥ ε − log( − ρ )] − ε − ρ − ∂ρ ∧ ¯ ∂ρ, so that Θ = ( − ρ ) ∂ ¯ ∂ψ + ∂ ¯ ∂ρ ≥ C (cid:20) ∂ ¯ ∂ρ + ∂ρ ∧ ¯ ∂ρ ( − ρ )[ − log( − ρ )] ε (cid:21) . Set N = | ¯ ∂ρ | − P j ∂ρ∂ ¯ z j · ∂∂ ¯ z j . For any (0 , − form ω on Ω , we set ω N = h ω, N i · ¯ ∂ρ | ¯ ∂ρ | and ω T = ω − ω N . It follows that | ¯ ∂ ˜ f | dV is a Carleson measure provided that | ( ¯ ∂ ˜ f ) T | dV and ( − ρ )[ − log( − ρ )] ε | ( ¯ ∂ ˜ f ) N | dV are Carleson measures. Consequently, we obtain Corollary 4.2.
Let Ω be a bounded strongly pseudoconvex domain in C n and ρ astrictly psh defining function on Ω . Let f ∈ L ( ∂ Ω) . Suppose f = f + f , where f ∈ L ∞ ( ∂ Ω) and f admits a Stokes extension ˜ f such that | ( ¯ ∂ ˜ f ) T | dV and ( − ρ )[ − log( − ρ )] ε | ( ¯ ∂ ˜ f ) N | dV are Carleson measures for some ε > . Then H f : H (Ω) → L ( ∂ Ω) is bounded. Similar ideas work for more general domains, e.g. pseudoconvex domains of finiteD’Angelo type (compare [11], Proposition 5.2).In the setting of strongly pseudoconvex domains, the theory of the ∂ -operator hasbeen used to investigate Hankel operators H f on the Bergman space A (Ω) . Theboundedness, compactness, essential norms and Schatten ideals of H f have been com-pletely characterized (see [20-22] and [5]). In the proofs of these results, an essentialtool is the integral representation of a solution to the ∂ -equation.To conclude this section, we shall apply the L -estimates of ∂ -equation due toDonnelly-Fefferman [14] to give a new proof for the boundedness of H f on A (Ω) .Let Ω be a bounded pseudoconvex domian and let ψ be a C plurisubharmonicfunction on Ω . For any (0 , form f , let | f | ∂∂ψ := n X i,j =1 ψ i j f i f j , where ( ψ i j ) = ( ψ i j ) − . According to an estimate of Donnelly and Fefferman, one cansolve ∂u = f with the following estimate Z Ω | u | e − ϕ dV ≤ C Z Ω | f | ∂∂ψ e − ϕ dV for some numerical constant C , whenever ψ satisfies ∂∂ψ ≥ ∂ψ ∧ ∂ψ and ϕ isplurisubharmonic.We first consider the case of the unit ball. Theorem 4.3.
Let f ∈ L ( B n ) ∩ C ( B n ) and ρ = | z | − . If | ρ · ∂f | + | ρ | / | ∂f ∧ ∂ρ | ≤ C for a constant C > . Then the Hankel operator H f = ( I − P B ) M f is bounded from A ( B n ) into L ( B n ) , here P B is the Bergman projection. IG HANKEL OPERATORS ON THE HARDY SAPCE 15
Proof.
Take ψ = − log( − ρ ) . A calculation shows that ∂∂ψ = − ∂∂ρρ + ∂ρ ∧ ∂ρρ ≥ ∂ψ ∧ ¯ ∂ψ and ∂∂ψ = | z | n P i,j =1 z i z j dz i ∧ dz j (1 − | z | ) + n P i =1 dz i ∧ dz j − | z | n P i,j =1 z i z j dz i ∧ dz j − | z | . Write ∂∂ψ = P ψ i j dz i ∧ dz j . Let A = | z | ( z i z j ) and B = I − A , where I is theidentity matrix. It is easy to check A = A and AB = 0 . Thus we get X ψ i j dz i ∧ dz j = (1 − | z | ) | z | n X i,j =1 z i z j dz i ∧ dz j +(1 − | z | ) (cid:18) n X i =1 dz i ∧ dz j − | z | n X i,j =1 z i z j dz i ∧ dz j (cid:19) . Now, for ∂u = n P i =1 ∂u∂z i dz i , we compute that | ∂u | ∂∂ψ = (1 − | z | ) | z | n X i,j =1 z i z j ∂u∂z i ∂u∂z j +(1 − | z | ) (cid:18) n X i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂z i (cid:12)(cid:12)(cid:12)(cid:12) − | z | n X i,j =1 z i z j ∂u∂z i ∂u∂z j (cid:19) = − (1 − | z | ) n X i,j =1 z i z j ∂u∂z i ∂u∂z j + (1 − | z | ) | z | n X i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂z i (cid:12)(cid:12)(cid:12)(cid:12) +(1 − | z | ) n X i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂z i (cid:12)(cid:12)(cid:12)(cid:12) = (1 − | z | ) n X i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂z i (cid:12)(cid:12)(cid:12)(cid:12) + (1 − | z | ) (cid:18) | z | n X i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂z i (cid:12)(cid:12)(cid:12)(cid:12) − n X i,j =1 z i z j ∂u∂z i ∂u∂z j (cid:19) = (1 − | z | ) | ∂u | + (1 − | z | ) (cid:12)(cid:12)(cid:12)(cid:12) ∂u ∧ ∂ | z | (cid:12)(cid:12)(cid:12)(cid:12) = ρ | ∂u | + | ρ | · | ∂u ∧ ∂ρ | . Hence, for any solution u orthogonal to A ( B n ) , the estimate of Donnelly and Feffer-man yields that Z B n | u | dV ≤ C Z B n | ∂u | ∂∂ψ dV = C Z B n (cid:18) ρ | ∂u | + | ρ | · | ∂u ∧ ∂ρ | (cid:19) dV. Next, for any g ∈ A ( B n ) , consider the equation ∂u = ∂ ( f g ) = g · ∂f. It is obvious that g · ∂f is ∂ -closed and u = ( I − P )( f g ) = H f g is the minimal solutionto this equation. Thus, the above discussion gives that Z B n | H f g | dV ≤ C Z B n | g · ∂f | ∂∂ψ dV = C Z B n | g | (cid:18) | ρ · ∂f | + | ρ | · | ∂f ∧ ∂ρ | (cid:19) dV ≤ C C Z B n | g | dV, since | ρ · ∂f | + | ρ | / | ∂f ∧ ∂ρ | ≤ C . It follows that H f is bounded from A ( B n ) into L ( B n ) . Moreover, we obtain the following norm estimate k H f k ≤ C sup z ∈ B n (cid:18) | ρ · ∂f | + | ρ | · | ∂f ∧ ∂ρ | (cid:19) / . So the proof is complete. (cid:3)
As pointed out on page 89 of [4], if f is a (0 , form defined in the strongly pseu-doconvex domain Ω = { ρ < } , we also have | f | ∂∂ log(1 / − ρ ) = 1 B (cid:18) ρ | f | β + | ρ | · | f ∧ ∂ρ | β (cid:19) , where B = − ρ + | ∂ρ | β ∼ and | · | β denotes the norm induced by the metric form β = i ∂∂ρ , which is equivalent to the Euclidean metric, since ρ is strictly plurisubharmonic.Therefore, using similar ideas as Theorem 4.3, we get the following result. Theorem 4.4.
Let Ω be a strongly pseudoconvex domain with smooth boundary and f ∈ L (Ω) ∩ C (Ω) . If | ρ · ∂f | + | ρ | / | ∂f ∧ ∂ρ | ≤ C for a constant C > . Then the Hankel operator H f = ( I − P B ) M f is bounded from A (Ω) into L (Ω) . This result is just Theorem 3.5 of [25], where the authors also showed that this con-dition is necessary. Now, we once again find the relationship between Hankel operatorsand ∂ -theory. Moreover, using the inequality k H f g k A (Ω) . Z Ω | g | (cid:18) | ρ · ∂f | + | ρ | · | ∂f ∧ ∂ρ | (cid:19) dV, we may easily deduce the compactness of H f on A (Ω) as Theorem 3.6 in [25]. IG HANKEL OPERATORS ON THE HARDY SAPCE 17
5. P
ROOF OF T HEOREM (1) ⇒ (2) . For any z ∈ D , the normalized reproducing kernel k z ∈ H ( D ) and k k z k H ( D ) = 1 . So we have π Z | f k z − P ( f k z ) | dθ = k H f ( k z ) k H ( D ) ≤ C for a positive constant C . It is easy to see that k z P ( f k z ) ∈ H ( D ) and P z ( θ ) = 1 − | z | | e iθ − z | = | k z | . Thus we get inf h ∈ H π Z | f − h | P z ( θ ) dθ ≤ π Z | f − k z P ( f k z ) | | k z | dθ = 12 π Z | f k z − P ( f k z ) | dθ ≤ C , which implies (2). (2) ⇒ (3) . There exists a constant M > , such that F ( z ) ≤ M for any z ∈ D . For z = 0 , we have F (0) = inf h ∈ H π Z | f − h | dθ ≤ M. Choose h ∈ H ( D ) which satisfies π Z | f − h | dθ = inf h ∈ H π Z | f − h | dθ. It is clear that h is the orthogonal projection of f onto H ( D ) and f − h is conjugateanalytic. Moreover, we have f (0) − h (0) = 0 . Hence, |∇ ( f − h ) | = 2 | ∂ ( f − h ) /∂z | = 2 | ∂f /∂z | . Combining this with the Littlewood-Paley indetity (see, p. 230), we obtain π Z | f − h | dθ = 1 π Z Z D |∇ ( f − h ) | log 1 | z | dxdy = 2 π Z Z D (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log 1 | z | dxdy ≤ M. (5 . For fixed z ∈ D , let h be chosen to attain the infimum in (1.1) with respect P z ( θ ) dθ . Applying similar arguments as in the proof of Theorem 3.5 on page 372of [18], we see that f − h is conjugate analytic and f ( z ) − h ( z ) = 0 , because inthe Hilbert space L ( P z ( θ ) dθ ) , the function h is the orthogonal projection of f onto H ( D ) . This means |∇ ( f − h ) | = 2 | ∂ ( f − h ) /∂z | = 2 | ∂f /∂z | . Thus the Littlewood-Paley indetity gives π Z | f − h | P z ( θ ) dθ = 2 π Z Z D (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12)(cid:12) − z zz − z (cid:12)(cid:12)(cid:12)(cid:12) dxdy ≤ M. (5 . Notice that (1 − | z | ) 1 − | z | | − z z | = 1 − (cid:12)(cid:12)(cid:12)(cid:12) z − z − z z (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) − z zz − z (cid:12)(cid:12)(cid:12)(cid:12) . Combining this inequality with (5.2), we have a constant C > such that Z Z | z | > / − | z | | − z z | · (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log 1 | z | dxdy ≤ C Z Z | z | > / (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) (1 − | z | ) 1 − | z | | − z z | dxdy ≤ C Z Z D (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12)(cid:12) − z zz − z (cid:12)(cid:12)(cid:12)(cid:12) dxdy ≤ C M π.
On the other hand, we infer from (5.1) that
Z Z | z |≤ / − | z | | − z z | · (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log 1 | z | dxdy ≤ C Z Z | z |≤ / (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log 1 | z | dxdy ≤ C M π. for a constant C > . All these arguments yield that sup z ∈ D Z Z D − | z | | − z z | · (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log 1 | z | dxdy ≤ C for a positive constant C . In view of Lemma 3.3 in Chapter VI of [18], we see that | ∂f∂z | log | z | dxdy is a Carleson measure. (3) ⇒ (1) . For any g ∈ H ( D ) , we see that f g − P ( f g ) is conjugate analyticand f (0) g (0) − P ( f g )(0) = 0 , because ( H ( D )) ⊥ = H ( D ) . If | ∂f∂z | log | z | dxdy is aCarleson measure, then the Littlewood-Paley identity implies k H f ( g ) k H ( D ) = 12 π Z | f g − P ( f g ) | dθ = 1 π Z Z D (cid:12)(cid:12)(cid:12)(cid:12) ∇ ( f g − P ( f g )) (cid:12)(cid:12)(cid:12)(cid:12) log 1 | z | dxdy = 2 π Z Z D | g | · (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log 1 | z | dxdy . k g k H ( D ) . IG HANKEL OPERATORS ON THE HARDY SAPCE 19
Thus, H f is bounded on H ( D ) . This completes the proof of Theorem 1.2. (cid:3) Remark.
Recall that f ∈ BM O ( ∂D ) if and only if dλ f = |∇ f ( z ) | log 1 | z | dxdy is a Carleson measure by Theorem 3.4 of [18]. Let dν f = (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) log 1 | z | dxdy. Since |∇ f ( z ) | = 2( | ∂f /∂z | + | ∂f /∂z | ) and ∂f /∂z = ∂f /∂z , we have λ f =2( ν f + ν f ) . In particular, when f is real-valued, λ f and ν f are equivalent. Thus,Theorem 1.2 implies a well-known result, that is, H f is bounded on H ( D ) if and onlyif f ∈ BM O ( ∂D ) for real-valued f .On the other hand, observing Theorem 1.2 and the corresponding result on theBergman space A (Ω) , we have the following conjecture for the boundedness of Han-kel operators on H (Ω) . Conjecture 2.
Suppose that Ω is a bounded strongly pseudoconvex domain withsmooth boundary and f ∈ L ( ∂ Ω) . Then the Hankel operator H f is bounded from H (Ω) into L ( ∂ Ω) , if and only if f admits a decomposition f = f + f with f ∈ L ∞ ( ∂ Ω) and f having an extension e f in Ω , which satisfies that | ∂ e f | dV and √− ρ | ∂ e f ∧ ∂ρ | dV are Carleson measures in Ω . H f AND H f ON H ARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS
For the unit disc D , it is well known that for f ∈ L ( ∂D ) , both H f and H f arebounded on H ( D ) if and only if f ∈ BM O ( ∂D ) . This result can be generalized tothe unit ball [45] and the key tool is the following inequality: k H f k z k + k H f k z k ≥ k f k z k + |h f k z , k z i| , where k z is the normalized reproducing kernel, which can be proved by using auto-morphisms of the ball. In this section, we denote the norm k · k := k · k H .Let us consider the corresponding result on the strongly pseudoconvex domain Ω .Since there might be no nontrivial holomorphic automorphisms for general stronglypseudoconvex domains in C n , we borrow some ideas from [7] to obtain the following. Theorem 6.1.
Let Ω be a strongly pseudoconvex domain with smooth boundary and f ∈ L ( ∂ Ω) . Suppose that for each fixed z ∈ Ω , /S ( · , z ) is holomorphic and boundedin Ω . Then the Hankel operators both H f and H f are bounded from H (Ω) into L ( ∂ Ω) if and only if f lies in BM O ( ∂ Ω) .Proof. Let S ( ζ , z ) be the Szeg¨o kernel and k z ( ζ ) = S ( ζ,z ) √ S ( z,z ) be the normalized kernel.For fixed ζ ∈ ∂ Ω , the function S ( z, ζ ) is holomorphic in Ω . Let b z ( ζ ) = S ( ζ,z ) S ( z,ζ ) . Thenfor any z ∈ Ω , we have h H f ( k z ) , b z H f ( k z ) i = h H f ( k z ) H f ( k z ) , b z i . Note that H f ( k z ) H f ( k z ) = ( I − P )( f k z )( I − P )( f k z )= | f | k z − f k z P ( f k z ) − f k z P ( f k z ) + P ( f k z ) P ( f k z ) . Since k z b z = | k z | , we obtain h| f | k z , b z i = k f k z k . From k z b z = k z , we see that h f k z P ( f k z ) , b z i = h P ( f k z ) , f k z b z i = h P ( f k z ) , f k z i = k P ( f k z ) k and similarly h f k z P ( f k z ) , b z i = k P ( f k z ) k . Since /S ( · , z ) is holomorphic and bounded in Ω for each fixed z ∈ Ω , the function k − z is well defined. Using the reproducing property of the Szeg¨o kernel, we have h P ( f k z ) P ( f k z ) , b z i = h P ( f k z ) P ( f k z ) k − z , k z i = P ( f k z )( z ) P ( f k z )( z ) S ( z, z ) − . Now, P ( z, ζ ) = S ( ζ,z ) S ( z,ζ ) S ( z,z ) is the Poisson-Szeg¨o kernel. Thus, we get P ( f k z ) = p S ( z, z ) Z ∂ Ω f ( ζ ) S ( ζ , z ) S ( z, ζ ) S ( z, z ) dσ ( ζ )= p S ( z, z ) Z ∂ Ω f ( ζ ) P ( z, ζ ) dσ ( ζ ) = p S ( z, z ) e f ( z ) , It follows that h P ( f k z ) P ( f k z ) , b z i = e f ( z ) e f ( z ) = | e f ( z ) | because of e f = e f . Therefore, the above discussions show that h H f ( k z ) , b z H f ( k z ) i = k f k z k − k P ( f k z ) k − k P ( f k z ) k + | e f ( z ) | . Recall that k H f ( k z ) k = k f k z k H − k P ( f k z ) k and k H f ( k z ) k = k f k z k −k P ( f k z ) k . This means h H f ( k z ) , b z H f ( k z ) i = k H f ( k z ) k + k H f ( k z ) k − k f k z k + | e f ( z ) | . Applying this identity and the Schwarz inequality, we obtain k f k z k − | e f ( z ) | ≤ (cid:18) k H f ( k z ) k + k H f ( k z ) k (cid:19) . (6 . On the other hand, in Section 2, we have talked about an equivalent norm for
BM O ,i.e. f ∈ BM O ( ∂ Ω) if and only if sup z ∈ Ω Z ∂ Ω | f ( ζ ) − e f ( z ) | P ( z, ζ ) dσ ( ζ ) < ∞ . IG HANKEL OPERATORS ON THE HARDY SAPCE 21
In fact, P ( z, ζ ) = | k z ( ζ ) | , so we get Z ∂ Ω | f ( ζ ) − e f ( z ) | P ( z, ζ ) dσ ( ζ )= Z ∂ Ω | f ( ζ ) | P ( z, ζ ) dσ ( ζ ) − e f ( z ) Z ∂ Ω f ( ζ ) P ( z, ζ ) dσ ( ζ ) − e f ( z ) Z ∂ Ω f ( ζ ) P ( z, ζ ) dσ ( ζ ) + | e f ( z ) | Z ∂ Ω P ( z, ζ ) dσ ( ζ )= k f k z k − | e f ( z ) | . As a consequence, if both H f and H f are bounded, then (6.1) gives Z ∂ Ω | f ( ζ ) − e f ( z ) | P ( z, ζ ) dσ ( ζ ) ≤ (cid:18) k H f ( k z ) k + k H f ( k z ) k (cid:19) < ∞ . This implies that f ∈ BM O ( ∂ Ω) .For another direction, Theorem 1.1 gives the desired result and the proof is complete. (cid:3) R EFERENCES [1] P. Ahern and R. Schneider,
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