Two classes of integral operators over the Siegel upper half-space
aa r X i v : . [ m a t h . C V ] A p r TWO CLASSES OF INTEGRAL OPERATORS OVER THESIEGEL UPPER HALF-SPACE
CONGWEN LIU, YI LIU, PENGYAN HU, AND LIFANG ZHOU
Abstract.
We determine exactly when two classes of integral operators arebounded on weighted L p spaces over the Siegel upper half-space. Introduction
This short note is motivated by the work of Kures and Zhu [7], in which theauthors characterized the boundedness of two classes of integral operators inducedby Bergman type kernels on weighted Lebesgue spaces on the unit ball B of C n .Fix three real parameters a, b, c and define two integral operators T a,b,c and S a,b,c by T a,b,c f ( z ) := (1 − | z | ) a Z B (1 − | w | ) b (1 − h z, w i ) c f ( w ) dν ( w )and S a,b,c f ( z ) := (1 − | z | ) a Z B (1 − | w | ) b | − h z, w i| c f ( w ) dν ( w ) , where dν is the volume measure on B , normalized so that ν ( B ) = 1. Also, for anyreal parameter α we define dν α ( z ) := (1 − | z | ) α dν ( z ).Kures and Zhu [7] obtained the following two theorems. Theorem A.
Suppose < p < ∞ . Then the following conditions are equivalent: (i) The operator T a,b,c is bounded on L p ( B , dν α ) . (ii) The operator S a,b,c is bounded on L p ( B , dν α ) . (iii) The parameters satisfy ( − pa < α + 1 < p ( b + 1) c ≤ n + 1 + a + b. Theorem B.
The following conditions are equivalent: (i)
The operator T a,b,c is bounded on L ( B , dν α ) . (ii) The operator S a,b,c is bounded on L ( B , dν α ) . Mathematics Subject Classification.
Primary 32A35, 47G10; Secondary 32A26, 30E20.
Key words and phrases.
Siegel upper half-space; Bergman type operators; weighted L p spaces;boundedness.The first author was supported by the National Natural Science Foundation of China grants11571333, 11471301; the fourth author was supported by Natural Science Foundation of Zhejiangprovince grant (No. LQ13A010005), the Scientific Research and Teachers project of HuzhouUniversity(No. RP21028) and partially by the National Natural Science Foundation of Chinagrant 11571105. (iii) The parameters satisfy ( − a < α + 1 < b + 1 c = n + 1 + a + b. or ( − a < α + 1 ≤ b + 1 c < n + 1 + a + b. Actually, these two theorems were proved in [7] under the additional assump-tion that c is neither 0 nor a negative integer. Recently, Zhao [11] removed thisextra requirement as well as generalized these two theorems by characterizing theboundedness of T a,b,c and S a,b,c , from L p ( B , dν α ) to L q ( B , dν β ).The case c = n + 1 + a + b of Theorems A is well known and being extensivelyused, see for example [12, Theorem 2.10]. It is also worthy to mention that, recently,a variant of Theorem A played a crucial role in the proof of the corona theorem forthe Drury-Arveson Hardy space, see [2, Lemma 24].In this note we consider the counterparts of Theorems A and B for two classesof integral operators over the Siegel upper half-space. The situation turns out tobe quite different in this setting.Before stating our main result, we introduce some definitions and notation.We fix a positive integer n throughout this paper and let C n = C ×· · ·× C denotethe n -dimensional complex Euclidean space. For any two points z = ( z , · · · , z n )and w = ( w , · · · , w n ) in C n , we write h z, w i := z ¯ w + · · · + z n ¯ w n and | z | := p h z, z i . The open unit ball in C n is the set B := { z ∈ C n : | z | < } . For z ∈ C n , we also use the notation z = ( z ′ , z n ) , where z ′ = ( z , . . . , z n − ) ∈ C n − and z n ∈ C . The Siegel upper half-space in C n is the set U := (cid:8) z ∈ C n : Im z n > | z ′ | (cid:9) . It is biholomorphically equivalent to the unit ball B in C n , via the Cayley transformΦ : B → U given by ( z ′ , z n ) (cid:18) z ′ z n , i − z n z n (cid:19) , and so it is also referred to as the unbounded realization of the unit ball in C n .We denote by dV the Lebesgue measure on C n . For any real parameters a , b ,and c , we consider two integral operators as follows. T a,b,c f ( z ) := ρ ( z ) a Z U ρ ( w ) b ρ ( z, w ) c f ( w ) dV ( w )and S a,b,c f ( z ) := ρ ( z ) a Z U ρ ( w ) b | ρ ( z, w ) | c f ( w ) dV ( w ) , where ρ ( z, w ) := i w n − z n ) − h z ′ , w ′ i . WO CLASSES OF INTEGRAL OPERATORS 3 and ρ ( z ) := ρ ( z, z ) = Im z n − | z ′ | . These operators are modelled on the weightedBergman projections on U . Recall that the Bergman projection P on U is given by P f ( z ) = n !4 π n Z U f ( w ) ρ ( z, w ) n +1 dV ( w ) = n !4 π n T , ,n +1 f ( z ) , z ∈ U . See, for instance, [4, Proposition 5.1].For real parameter α , we define dV α ( z ) := ρ ( z ) α dV ( z ) . As usual, for p >
0, the space L p ( U , dV α ) consists of all Lebesgue measurablefunctions f on U for which k f k p,α := (cid:26) Z U | f ( z ) | p dV α ( z ) (cid:27) /p is finite.Our main result gives necessary and sufficient conditions for the boundedness ofthe operators S a,b,c and T a,b,c on L p ( U , dV α ) in terms of parameters a, b, c , and α . Theorem 1.
Suppose α ∈ R and ≤ p ≤ ∞ . Then the following conditions areequivalent: (i) The operator T = T a,b,c is bounded on L p ( U , dV α ) . (ii) The operator S = S a,b,c is bounded on L p ( U , dV α ) . (iii) The parameters satisfy the conditions (1) ( − pa < α + 1 < p ( b + 1) ,c = n + 1 + a + b. When p = ∞ , these conditions should be interpreted as (2) ( a > , b > − ,c = n + 1 + a + b. Note that Condition (iii) in Theorem 1 is different from the corresponding onesin Theorems A and B. In particular, unlike T a,b,c and S a,b,c , both T a,b,c and S a,b,c are unbounded whenever c = n + 1 + a + b . This is due to the unboundedness ofthe Siegel upper half-space and the homogeneity of the operators T a,b,c and S a,b,c .The proof follows the same main lines as in [7]. However, the computations hereare more subtle. For instance, in the proof of the necessity for the boundednessof T a,b,c , we cannot simply choose polynomials to serve as test functions as in [7],since polynomials do not belong to L p ( U , dV α ). Instead, we consider the functionsof the form ρ ( z ) t / ρ ( z, w ) s , with appropriate choices of the parameters involved.This leads to more complicated calculations than those arising in the unit ballsetting. Hence, an essential role is played by the following lemma, which might beof independent interest. Key Lemma.
Suppose that r, s > , t > − and r + s − t > n + 1 . Then (3) Z U ρ ( w ) t ρ ( z, w ) r ρ ( w, u ) s dV ( w ) = C ( n, r, s, t ) ρ ( z, u ) r + s − t − n − holds for all z, u ∈ U , where (4) C ( n, r, s, t ) := 4 π n Γ(1 + t )Γ( r + s − t − n − r )Γ( s ) . The formula (3), with implicit constant C ( n, r, s, t ), is not new; it is a specialcase of [1, Lemma 2.2’]. The novelty here is to find the explicit expression (4) of C ( n, r, s, t ).The rest of the paper is organized as follows: In Section 2 we recall some basicmaterials about M¨obius transformations and the Cayley transform. Section 3 isdevoted to the proof of Key Lemma. Our main result, Theorem 1 will be provedin Sections 4. Finally, in Section 5, two examples are given to illustrate the use ofTheorem 1. 2. Preliminaries
We begin by recalling that the Cayley 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 easy to check that the identity(5) ρ (Φ( η ) , Φ( ξ )) = 1 − h η, ξ i (1 + η n )(1 + ξ n )holds for all η, ξ ∈ B , and the real Jacobian of Φ at ξ ∈ B is(6) ( J R Φ) ( ξ ) = 4 | ξ n | 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 η given by ϕ η ( ξ ) := η − P η ξ − (1 − | η | ) Q η ξ − h ξ, η i , where η ∈ B , P η is the orthogonal projection onto the space spanned by η , and Q η ξ = ξ − P η ξ .It is easily shown that the mapping ϕ η satisfies ϕ η (0) = η, ϕ η ( η ) = 0 , ϕ η ( ϕ η ( ξ )) = ξ. Furthermore, for all ξ, ζ ∈ B ,1 − h ϕ η ( ξ ) , ϕ η ( ζ ) i = (1 − | η | )(1 − h ξ, ζ i )(1 − h ξ, η i )(1 − h η, ζ i ) , (7)and in particular, 1 − h ϕ η ( ξ ) , η i = 1 − | η | − h ξ, η i . (8)Finally, an easy computation shows that(9) 1 − h ϕ η ( ξ ) , ζ i = (1 − h ξ, ϕ η ( ζ ) i )(1 − h η, ζ i )1 − h ξ, η i holds for all ξ, η ∈ B .The best general reference here is [9, Chapter 2]. WO CLASSES OF INTEGRAL OPERATORS 5
The following lemma, usually called Schur’s test, is one of the most commonlyused results for proving the L p -boundedness of integral operators. See, for example,[13, Theorem 3.6]. Lemma 2.
Suppose that ( X, µ ) is a σ -finite measure space and Q ( x, y ) is a non-negative measurable function on X × X and T is the associated integral operator T f ( x ) = Z X Q ( x, y ) f ( y ) dµ ( y ) . Let < p < ∞ and q = p/ ( p − . If there exist a positive constant C and a positivemeasurable function g on X such that Z X Q ( x, y ) g ( y ) q dµ ( y ) ≤ Cg ( x ) q for almost every x in X and Z X Q ( x, y ) g ( x ) p dµ ( x ) ≤ Cg ( y ) p for almost every y in X , then T is bounded on L p ( X, µ ) with k T k ≤ C . The proof of Key Lemma
We begin with two lemmas.
Lemma 3.
Suppose that r, s > , t > − and r + s − t > n + 1 . Then Z B (1 − | ξ | ) t dV ( ξ )(1 − h η, ξ i ) s (1 − h ζ, ξ i ) n +1+ t − s (1 − h ξ, ζ i ) n +1+ t − r = C ( n, r, s, t )4(1 − h η, ζ i ) n +1+ t − r holds for any η ∈ B and ζ ∈ S .Proof. We may further assume that r + s > n + 1 + t ); if we prove the lemma inthis special case, the general case follows by analytic continuation.According to [8, Lemma 2.3], the identity Z B (1 − | ξ | ) t dV ( ξ )(1 − h η, ξ i ) s (1 − h ̺ζ, ξ i ) n +1+ t − s (1 − h ξ, ̺ζ i ) n +1+ t − r = π n Γ(1 + t )Γ( n + 1 + t ) ∞ X j =0 ( s ) j ( n + 1 + t − r ) j ( n + 1 + t ) j j ! × F (cid:20) n + 1 + t − s, n + 1 + t − r + jn + 1 + t + j ; ̺ (cid:21) ( ̺ h η, ζ i ) j (10)holds for all ̺ ∈ [0 , η ∈ B and ζ ∈ S . Note that (cid:12)(cid:12) the integrand in (10) (cid:12)(cid:12) ≤ r + s − n +1+ t ) (1 − | ξ | ) t | − h η, ξ i| s , since r + s > n + 1 + t ). Letting ̺ →
1, by the dominated convergence theoremand using the well-known formula F (cid:20) a, bc ; 1 (cid:21) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) , Re ( c − a − b ) > , C. LIU, Y. LIU, P. HU, AND L. ZHOU we obtain Z B (1 − | ξ | ) t dV ( ξ )(1 − h η, ξ i ) s (1 − h ζ, ξ i ) n +1+ t − s (1 − h ξ, ζ i ) n +1+ t − r = π n Γ(1 + t )Γ( n + 1 + t ) ∞ X j =0 ( s ) j ( n + 1 + t − r ) j ( n + 1 + t ) j j ! × F (cid:20) n + 1 + t − s, n + 1 + t − r + jn + 1 + t + j ; 1 (cid:21) h η, ζ i j = π n Γ(1 + t )Γ( r + s − t − n − r )Γ( s ) ∞ X j =0 ( n + 1 + t − r ) j j ! h η, ζ i j = C ( n, r, s, t )4(1 − h η, ζ i ) n +1+ t − r , as desired. (cid:3) Lemma 4.
Suppose that r, s > , t > − and r + s − t > n + 1 . Then Z B (1 − | ω | ) t dV ( ω )(1 − h η, ω i ) r (1 − h ω, ζ i ) s (1 + ω n ) n +1+ t − s (1 + ω n ) n +1+ t − r = C ( n, r, s, t )4 (1 + η n ) s − n − − t (1 + ζ n ) r − n − − t (1 − h η, ζ i ) n +1+ t − r − s (11) holds for all η, ζ ∈ B .Proof. We make the change of variables ω = ϕ η ( ξ ) in the integral, where ϕ η is theM¨obius transformation of the unit ball, as defined in Section 2, as well as apply theformulas (8) and (9). After simplification, we obtain Z B (1 − | ω | ) t dV ( ω )(1 − h η, ω i ) r (1 − h ω, ζ i ) s (1 + ω n ) n +1+ t − s (1 + ω n ) n +1+ t − r = (1 − | η | ) n +1+ t − r (1 − h η, ζ i ) − s (1 + η n ) s − n − − t (1 + η n ) r − n − − t × Z B (1 − | ξ | ) t dV ( ξ )(1 − h ξ, ϕ η ( ζ ) i ) s (1 − h ξ, ϕ η ( − e n ) i ) n +1+ t − s (1 − h ϕ η ( − e n ) , ξ i ) n +1+ t − r . By Lemma 3 and the formula (7), this equals(1 − | η | ) n +1+ t − r (1 − h η, ζ i ) − s (1 + η n ) s − n − − t (1 + η n ) r − n − − t × C ( n, r, s, t )4 (1 − h ϕ η ( − e n ) , ϕ η ( ζ ) i ) r − n − − t = (1 − | η | ) n +1+ t − r (1 − h η, ζ i ) − s (1 + η n ) s − n − − t (1 + η n ) r − n − − t × C ( n, r, s, t )4 (cid:26) (1 − | η | )(1 + ζ n )(1 + η )(1 − h η, ζ i ) (cid:27) r − n − − t which establishes the formula. (cid:3) Now we turn to the proof of Key Lemma.
WO CLASSES OF INTEGRAL OPERATORS 7
By the change of variables w = Φ( ξ ) in the integral and using (5), we obtain Z U ρ ( w ) t ρ ( z, w ) r ρ ( w, u ) s dV ( w )= Z B ρ (Φ( ξ )) t ρ ( z, Φ( ξ )) r ρ (Φ( ξ ) , u ) s | ξ n | n +1) dV ( ξ )= 4(1 + [Φ − ( z )] n ) r (1 + [Φ − ( u )] n ) s × Z B (1 − | ξ | ) t dV ( ξ )(1 − h Φ − ( z ) , ξ i ) r (1 − h ξ, Φ − ( u ) i ) s (1 + ξ n ) n +1+ t − s (1 + ¯ ξ n ) n +1+ t − r . In view of (11), this equals C ( n, r, s, t ) (cid:0) − ( z )] n (cid:1) r (cid:16) − ( u )] n (cid:17) s (cid:0) − ( z )] n (cid:1) s − n − − t × (cid:16) − ( u )] n (cid:17) r − n − − t (cid:0) − h Φ − ( z ) , Φ − ( u ) i (cid:1) n +1+ t − r − s = C ( n, r, s, t ) ( (1 + [Φ − ( z )] n )(1 + [Φ − ( u )] n )1 − h Φ − ( z ) , Φ − ( u ) i ) r + s − n − − t = C ( n, r, s, t ) ρ ( z, u ) n +1+ t − r − s where we used (5) to obtain the last equality. The proof is complete.We single out a special case of Key Lemma as the following lemma, which willbe used repeatedly. Lemma 5.
Let s, t ∈ R . Then we have (12) Z U ρ ( w ) t | ρ ( z, w ) | s dV ( w ) = C ( n, s, t ) ρ ( z ) s − t − n − , if t > − and 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/ . Proof.
It remains to show that the integral is finite if and only if t > − s − t > n + 1.Before proceeding, we recall the definition of the Heisenberg group and somebasic facts which can be found in [10, Chapter XII].We denote by H n − the Heisenberg group, that is, the set C n − × R = { [ ζ, t ] : ζ ∈ C n − , t ∈ R } endowed with the group operation[ ζ, t ] · [ η, s ] = [ ζ + η, t + s + 2Im h ζ, η i )] . To each element h = [ ζ, t ] of H n − , we associate the following (holomorphic) affineself-mapping of U :(13) h : ( z ′ , z n ) ( z ′ + ζ, z n + t + 2 i h z ′ , ζ i + i | ζ | ) . C. LIU, Y. LIU, P. HU, AND L. ZHOU
It is easy to check that(14) ρ ( h ( z ) , h ( w )) = ρ ( z, w )for any z, w ∈ U and any h ∈ H n − .For fixed z ∈ U , we put h = [ − z ′ , − Re z n ] ∈ H n − . It is easy to check that h ( z ) = ρ ( z ) i , where i = (0 ′ , i ), and ρ ( h ( z ) , w ) = i w n − ρ ( z ) i )for all w ∈ U . Using (14) and making the change of variables w h ( w ) in theintegral, we see that Z U ρ ( w ) t | ρ ( z, w ) | s dV ( w ) = Z U ρ ( w ) t | ρ ( h ( z ) , w ) | s dV ( w )= 2 s Z U (Im w n − | w ′ | ) t | w n + ρ ( z ) i | s dV ( w ) . By Fubini’s theorem, this equals2 s Z Im w n > | w n + ρ ( z ) i | s ( Z | w ′ | < (Im w n ) / (Im w n − | w ′ | ) t dm n − ( w ′ ) ) dm ( w n )= 2 s ( Z Im w n > (Im w n ) n − t | w n + ρ ( z ) i | s dm ( w n ) )( Z | w ′ | < (1 − | w ′ | ) t dm n − ( w ′ ) ) , which is finite if and only if t > − s − ( n − t ) > (cid:3) The proof of Theorem 1 (ii) ⇒ (i): Obvious. (i) ⇒ (iii): Suppose that T is bounded on L p ( U , dV α ). Case 1: p = ∞ . Note that the constant function cannot serve as a test functionat this moment, since T ( z ) ≡
0. Instead, we consider the function f z ( w ) := ρ ( z, w ) c | ρ ( z, w ) | c , w ∈ U . Each f z is a unit vector in L ∞ ( U ) and( T f z )( z ) = ρ ( z ) a Z U ρ ( w ) b | ρ ( z, w ) | c dV ( w )for every z ∈ U . Since | ( T f z )( z ) | ≤ k T k ∞→∞ for all z ∈ U , where k T k ∞→∞ denotesthe operator norm of T acting on L ∞ ( U ), by Lemma 5, we have b > − ,c > n + 1 + b,c − n − − b = a, which is clearly nothing but (2). WO CLASSES OF INTEGRAL OPERATORS 9
Case 2: p = 1 . Note that the boundedness of T on L ( U , dV α ) implies the bound-edness of T ∗ on L ∞ ( U ), where T ∗ is the adjoint of T . It is easy to see that(15) T ∗ f ( z ) = ρ ( z ) b − α Z U ρ ( w ) a + α ρ ( z, w ) c f ( w ) dV ( w ) . So we can apply the previous case to T ∗ to obtain a + α > − ,c > n + 1 + ( a + α ) ,c − n − − ( a + α ) = b − α, which implies ( − a < α + 1 < b + 1 ,c = n + 1 + a + b. Case 3: < p < ∞ . We first show that c >
0. In order that
T f be always well-defined for f ∈ L p ( U , dV α ), it is necessary and sufficient that Z U ρ ( w ) bq + α | ρ ( z, w ) | cq dV ( w ) < + ∞ for all z ∈ U , where q : p/ ( p −
1) is the conjugate exponent of p . Again by Lemma5, this happens if and only if ( bq + α > − ,cq − bq − α > n + 1 . Summing up the two inequalities, we get c > n/q > β >
0, we put f β ( z ) := ρ ( z ) t ρ ( z, β i ) s , z ∈ U , where s, t are real parameters satisfying the conditions s > , (C.1) t > max (cid:26) − αp , − − b (cid:27) , (C.2) s − t > max (cid:26) n + 1 + αp , n + 1 + b − c (cid:27) . (C.3)By Lemma 5, Conditions (C.1)–(C.3) guarantee that f β ∈ L p ( U , dV α ) and(16) k f β k pp,α = C ( n, α, p, s, t ) β n +1+ α − p ( s − t ) , where C ( n, α, p, s, t ) := 4 π n Γ( pt + 1 + α )Γ( p ( s − t ) − n − − α )Γ ( ps/ . Also, in view of Conditions (C.1)–(C.3) and that c >
0, we can apply Key Lemmato obtain (
T f β )( z ) = ρ ( z ) a Z U ρ ( w ) b + t ρ ( z, w ) c ρ ( w, β i ) s dV ( w )= C ( n, b, c, s, t ) ρ ( z ) a ρ ( z, β i ) c − b − n − s − t , where C ( n, b, c, s, t ) := 4 π n Γ( b + t + 1)Γ( c − b − n − s − t )Γ( c )Γ( s ) . Since
T f β ∈ L p ( U , dV α ), again by Lemma 5, it is necessary that pa + α > − , (17) p ( c − a − b − n −
1) + p ( s − t ) − n − − α > . Moreover, we have(18) k T f β k pp,α = C ( n, α, p, b, c, s, t ) β n +1+ α − p ( s − t )+ p ( n +1+ a + b − c ) , where C ( n, α, p, b, c, s, t ) equals C ( n, b, c, s, t ) p × π n Γ(1 + pa + α )Γ( p ( c − a − b − n − s − t ) − n − − α )Γ ( p ( c − b − n − s − t ) / . Since T is bounded on L p ( U , dV α ), there is a positive constant C , independent of β , such that k T f β k p,α ≤ C k f β k p,α for all β ∈ (0 , ∞ ). Taking (16) and (18) intoaccount, we can find another positive constant C ′ , independent of β , such that β n +1+ α − p ( s − t )+ p ( n +1+ a + b − c ) ≤ C ′ β n +1+ α − p ( s − t ) for all β ∈ (0 , ∞ ). But this is true only when c = n + 1 + a + b .Having proved that c = n + 1 + a + b and − pa < α + 1, we proceed to show that α + 1 < p ( b + 1). Note that the boundedness of T on L p ( U , dV α ) is equivalent tothe boundedness of T ∗ on L q ( U , dV α ), where T ∗ is the adjoint of T , as is given by(15). Applying (17) to T ∗ , we conclude that α + 1 > − q ( b − α ) , which is exactly the same as α + 1 < p ( b + 1) . (iii) ⇒ (ii): The cases p = 1 and p = ∞ are direct consequences of Lemma 5.In the case 1 < p < ∞ , the proof appeals to Schur’s test. Let Q ( z, w ) = ρ ( z ) a ρ ( w ) b − α | ρ ( z, w ) | n +1+ a + b . WO CLASSES OF INTEGRAL OPERATORS 11 and g ( z ) = ρ ( z ) − (1+ α ) / ( pq ) , where q = p/ ( p − Z U Q ( z, w ) g ( w ) q ρ ( w ) α dV ( w )= ρ ( z ) a Z U ρ ( w ) b − (1+ α ) /p | ρ ( z, w ) | n +1+ a + b dV ( w )= ρ ( z ) a π n Γ(1 + b − (1 + α ) /p )Γ( a + (1 + α ) /p )Γ (( n + 1 + a + b ) / ρ ( z ) − a − (1+ α ) /p = 4 π n Γ(1 + b − (1 + α ) /p )Γ( a + (1 + α ) /p )Γ (( n + 1 + a + b ) / g ( z ) q holds for every z ∈ U . Similarly, Z U Q ( z, w ) g ( z ) p ρ ( z ) α dV ( z ) = 4 π n Γ(1 + b − (1 + α ) /p )Γ( a + (1 + α ) /p )Γ (( n + 1 + a + b ) / g ( w ) p holds for every w ∈ U . Hence, by Lemma 2, S is bounded on L p ( U , dV α ) with k S k ≤ π n Γ( a + (1 + α ) /p )Γ(1 + b − (1 + α ) /p )Γ (( n + 1 + a + b ) / Applications
We present two examples to illustrate the use of our main result.In order to state the first example we need to introduce more notation. It isknown that the Bergman kernel function K Ω induces a Riemannian metric on adomain Ω in C n . The infinitesimal Bergman metric is defined by g Ω i,j ( z ) = 1 n + 1 ∂ log K Ω ( z, z ) ∂z i ∂ ¯ z j , i, j = 1 , , . . . , n, and the complex matrix B ( z ) = (cid:0) g Ω i,j ( z ) (cid:1) ≤ i,j ≤ n is called the Bergman matrix of Ω. For a C curve γ : [0 , → Ω, the Bergmanlength of γ is defined by ℓ ( γ ) := Z h B ( γ ( t )) γ ′ ( t ) , γ ′ ( t ) i dt. If z, w ∈ Ω, then their Bergman distance is δ Ω ( z, w ) := inf { ℓ ( γ ) : γ (0) = z, γ (1) = w } , where the infimum is taken over all C curves from z to w . If Ω , Ω are twodomains in C n and ψ is a biholomorphic mapping of Ω onto Ω , then δ Ω ( z, w ) = δ Ω ( ψ ( z ) , ψ ( w )) for all z, w ∈ Ω . Hence, δ U ( z, w ) = δ B (Φ − ( z ) , Φ − ( w )) = tanh − (cid:0)(cid:12)(cid:12) ϕ Φ − ( z ) (Φ − ( w ) (cid:12)(cid:12)(cid:1) . Furthermore, a computation shows that(19) δ U ( z, w ) = tanh − s − ρ ( z ) ρ ( w ) | ρ ( z, w ) | . Let a, b and c be real numbers. We consider the operator S ca,b f ( z ) := ρ ( z ) a Z U ρ ( w ) b δ U ( z, w ) c | ρ ( z, w ) | n +1+ a + b f ( w ) dV ( w ) . It is a modification of the integral operator S a,b,c in Theorem 1, with an extraunbounded factor δ ( z, w ) c in the integrand. Theorem 6.
Suppose α ∈ R and ≤ p < ∞ . If − pa < α + 1 < p ( b + 1) and c ≥ then the operator S ca,b is bounded on L p ( U , dV α ) .Proof. Pick ǫ > − p ( a − cǫ ) < α + 1 < p ( b + 1 − cǫ ). Since log x < x ǫ holds for any x > ǫ >
0, it follows from (19) that δ U ( z, w ) . log 4 | ρ ( z, w ) | ρ ( z ) ρ ( w ) . | ρ ( z, w ) | ǫ ρ ( z ) ǫ ρ ( w ) ǫ . It follows that | S ca,b ( f )( z ) | . | S a,b,n +1+ a + b ( | f | )( z ) | + ρ ( z ) a − cǫ Z U ρ ( w ) b − cǫ | ρ ( z, w ) | n +1+ a + b − cǫ | f ( w ) | dV ( w )= | S a,b,n +1+ a + b ( | f | )( z ) | + | S ˜ a, ˜ b,n +1+˜ a +˜ b ( | f | )( z ) | , where ˜ a = a − cǫ and ˜ b = b − cǫ . The desired result then follows from Theorem1. (cid:3) We denote by A pα ( U ) the Bergman space, that is, the closed subspace of L p ( U , ρ α )consisting of holomorphic functions on U . As usual, we write ∂ n := ∂/ ( ∂z n ). Thefollowing result plays an important role in the study of the Besov spaces over theSiegel upper half-space. Theorem 7.
Suppose < p < ∞ , α > /p − and N ∈ N . Then ∂ Nn is a boundedlinear operator from A pα ( U ) into A pα + pN ( U ) .Proof. According to [3, Theorem 2.1], if f ∈ A pα ( U ) with p and α satisfying theassumption of the theorem, then f ( z ) = c α Z U f ( w ) ρ ( w ) α ρ ( z, w ) n +1+ α dV ( w ) , where c α := Γ( n + 1 + α )4 π n Γ(1 + α ) . It follows that (cid:12)(cid:12) ρ ( z ) N ∂ Nn f ( z ) (cid:12)(cid:12) . ρ ( z ) N Z U | f ( w ) | ρ ( w ) α | ρ ( z, w ) | n +1+ α + N dV ( w )= S N,α,n +1+ α + N ( | f | )( z ) . WO CLASSES OF INTEGRAL OPERATORS 13
By Theorem 1, this implies k ∂ Nn f k p,α + pN = k ρ N ∂ Nn f k p,α . k f k p,α , as asserted. (cid:3) Acknowledgement.
We are grateful to an anonymous referee for several valuablesuggestions and especially for pointing out a gap in the proof of Theorem 1 inthe original version of this paper. We also wish to thank Professor H. TurgayKaptanoglu for constructive comments and for bringing the paper by Ruhan Zhaoto our attention.
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E-mail address : [email protected] School of Mathematical Sciences, University of Science and Technology of China,Hefei, Anhui 230026, People’s Republic of China.
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