Toeplitz Operators on Weighted Bergman Spaces over Tubular Domains
Lvchang Li Jiaqing Ding Haichou Li

	


	




Toeplitz Operators on Weighted Bergman Spaces over Tubular Domains
    Lvchang Li College of Mathematics and Informatics,
		South China Agricultural University,
		Guangzhou,
		510640,
		China
		 Email: 20222115006@stu.scau.edu.cn.,	Jiaqing DingCollege of Mathematics and Informatics,
		South China Agricultural University,
		Guangzhou,
		510640,
		China
	Email: 20222115001@stu.scau.edu.cn.,	Haichou Li  Corresponding author,
		College of Mathematics and Informatics,
		South China Agricultural University,
		Guangzhou,
		510640,
		China
		 Email: hcl2016@scau.edu.cn.  Li is supported by NSF of China (Grant No. 12326407 and 12071155). 

    
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	120mm
		
Abstract

		In this paper, we mainly study the necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators on weighted Bergman spaces over a tubular domains by using the Carlson measures on tubular domains. We also give some related results about Carlson measures.		
			
		Key words:  weighted Bergman spaces; Toeplitz operator; tubular domain; Carleson measure
	



Toeplitz Operators on Weighted Bergman Spaces over Tubular Domains
    Lvchang Li College of Mathematics and Informatics,
		South China Agricultural University,
		Guangzhou,
		510640,
		China
		 Email: 20222115006@stu.scau.edu.cn.,	Jiaqing DingCollege of Mathematics and Informatics,
		South China Agricultural University,
		Guangzhou,
		510640,
		China
	Email: 20222115001@stu.scau.edu.cn.,	Haichou Li  Corresponding author,
		College of Mathematics and Informatics,
		South China Agricultural University,
		Guangzhou,
		510640,
		China
		 Email: hcl2016@scau.edu.cn.  Li is supported by NSF of China (Grant No. 12326407 and 12071155). 

    
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§ INTRODUCTION

       Bergman's book <cit.> systematically discusses a Hilbert space of square-integrable analytic functions on a 
  domain for the first time, now known as the Bergman space defined on a domain. The Bergman space is a closed subspace of the familiar L^p space. When p=2, the Bergman space is a Hilbert space. A useful tool for studying the Bergman space is the reproducing kernel, which plays a very important role. For related theories on the reproducing kernel, please refer to relevant literatures<cit.>.

Another important tool in the study of operators on function spaces is the Carleson measure, which was initially introduced by Carleson <cit.> to address the Corona problem. Nowadays, Carleson measures play a crucial role in studying the boundedness and compactness of operators, especially for Toeplitz operators, see <cit.>. Regarding the further applications of Carleson measures in operator theory on function spaces, refer to <cit.>.
The theory of Toeplitz operators on the unit disk and the unit ball in the Bergman spaces has been extensively studied by many authors, such as <cit.>. Subsequently, many authors have also extended the domains to bounded symmetric domains<cit.>, strongly pseudoconvex domains<cit.>, pseudoconvex domains<cit.>, and so on.

 However, researches on the theory of the Bergman spaces on unbounded domains is scarce. In particular, when n=1, the Bergman spaces on the upper half-plane lacks many good properties of the Bergman spaces over the unit disk, such as the well-known constant functions and monomial functions not being in the Bergman spaces over the  upper half-plane. 

In the present paper, we are interested in the case of the higher dimensional unbounded domains, such as  the tubular domains. More specifically, this paper will mainly study the boundedness and compactness of Toeplitz operators on a certain class of tubular domains in ℂ^n and their relationship with Carleson measures. These tubular domains may share some similarities with the well-known second kind of Siegel upper half-space, but the second kind of Siegel upper half-space are not tubes in ℂ^n. Instead, the tubes are larger than the second kind of Siegel upper half-space, hence they have corresponding research value.

 Deng et al. <cit.> computed the reproducing kernel of the Bergman spaces on such tubes using Laplace transform methods, laying the groundwork for subsequent theoretical researches. Liu et al. <cit.> provided some basic properties of the Bergman spaces on these tubes.
Si et al. <cit.> studied the boundedness and compactness of Toeplitz operators on the Bergman space of the second kind of Siegel upper half-space and their relationship with Carleson measures. This paper will mainly study the theory of Toeplitz operators on the Bergman spaces over tubes with the help of Carleson measure, which is a  powerful tool and an interesting object to study.

The structure of the paper is as follows: The second section provides an overview of fundamental terminology. The third section presents essential lemmas and their proofs. In the fourth section, we obtain the characterization of Carleson measures on the tubular domains. Moving forward, in the fifth section, we can find a dense subspace of Bergman spaces over tubular domains, which is crucial for establishing the boundedness of Toeplitz operators. Finally, the last section comprehensively explores the characterizations of boundedness and compactness of Toeplitz operators on Bergman spaces over tubular domains, with detailed discussions on Theorem <ref> and Theorem <ref>.




§ PRELIMINARIES



   Let ℂ^n be the n dimensional complex Euclidean space. For any two points z=( z_1,⋯ ,z_n ) and w=( w_1,⋯ ,w_n ) in ℂ^n, we write
    z·w̅:=z_1w̅_1+⋯ +z_nw̅_n,
 
    z^2=z· z:=z_1^2+z_2^2+⋯ +z_n^2
 and 
    | z |:=√(z·z̅)=√(| z_1 |^2+⋯ +| z_n |^2).
  

The set 𝔹_n={ z∈ℂ^n:| z |<1 } will be called the unit ball of ℂ^n. 

The tubular domain T_B of ℂ^n with base B, is  defined as follows:

 
    T_B={ z=x+iy ∈ℂ^n | x∈ℝ^n,   y∈ B ⊆ℝ^n  },
   
 where
    B={( y',y_n )=( y_1,⋯ ,y_n-1, y_n) ∈ℝ^n| y'^2:=y_1^2+⋯ +y_n-1^2<y_n . }.
 

We define the spaces L_α^p( T_B ), which is composed of all Lebesgue measurable functions f on T_B, and its norm 
    ‖ f ‖ _p,α={∫_T_B| f( z ) |^p dV_α( z )} ^1/p
 is finite, where dV_α(z)=( y_n-| y' |^2)^α dV(z), α>-1, dV(z) denotes the Lebesgue measure on ℂ^n. 

The Bergman spaces A_α^p( T_B ) on tube T_B  is a set composed of all holomorphic functions in L_α^p( T_B ). 

Since the valuation functional is bounded, so the Bergman space A_α^p( T_B ) is the closed subspace of L_α^p( T_B ). At the same time, we know that when 1≤ p<∞ the space A_α^p( T_B ) is a Banach space with the norm ‖·‖ _p,α. In particular, when p=2, A_α^2( T_B ) is a Hilbert space. 

 An very important orthogonal projection P_α from L_α^2( T_B ) to A_α^2( T_B ) is the following integral operator:
    P_αf( z ) =∫_T_BK_α( z,w ) f( w ) dV_α( w ),
 with the Bergman kernel 
    K_α( z,w ) =2^n+1+2α( n+1+α)/( α +1 ) π ^n( ( z'-w') ^2-2i( z_n-w_n) ) ^-n-α -1.
  

For convenience, we introduce the following notation:
 
    ρ( z,w ) =1/4( ( z'-w') ^2-2i( z_n-w_n) )
 and let ρ( z ) :=ρ( z,z ) =y_n-y'^2. 

With the above notion ρ( z,w ), the weighted Bergman kernel of T_B becomes
 
    K_α( z,w ) =( n+α +1 )/2^n+1π ^n( α +1 ) ρ( z,w ) ^n+α +1.


Recall ρ( z ) =y_n-y'^2 and let ∂ T_B:={ z∈ℂ^n  : ρ( z ) =0 } denote the boundary of T_B. Then 
    T_B:=T_B∪∂ T_B∪{∞}
 is the one-point compactification of T_B.

 Also, let ∂T_B:=∂ T_B∪{∞} . Thus, z→∂T_B 
means ρ( z ) → 0 or | z |→∞ . 

We denote by C_0( T_B ) the space of complex-valued continuous functions f on T_B such that f( z ) → 0 as z→∂T_B.


For a positive Borel measure μ on T_B, we define a function μ̃ on T_B by 
    μ̃( z ) :=∫_T_B| k_z( w ) |^2dμ( w ),     z∈ T_B,
 where, for fixed z∈ T_B, 
    k_z(w):= K(z,w)/√(K(z,z)),    w∈ T_B.



For z∈ T_B and r>0, we define the averaging function 
    μ̂_r( z ) :=μ( D( z,r ) )/V_α( D( z,r ) ).


Let μ be a positive Borel measure  on T_B and p>0. We say that μ is Carleson measure for the Bergman Space A_α^p(T_B), if there exists a positive constant C > 0 such that

    ∫_T_B| f(z) |^p   dμ(z) ≤ C ‖ f ‖_p,α^p

for every f ∈ A_α^p(T_B).

A positive Borel measure μ as a vanishing Carleson measure, if for any bounded sequence f_k in A_α^p(T_B) that converges uniformly to 0 on every compact subset of T_B, we have 
    lim_k→∞∫_T_B| f_k |^p dμ=0.


First, we review the Bergman metric on domains in ℂ^n. Let K(z,w) be the kernel of T_B. We define the complex matrix 

    𝐁(z)=(b_ij(z))_1≤ i,j≤ n=1/n+1 ( ∂ ^2/∂z̅_̅i̅∂ z_jlnK(z,z) )_1≤ i,j≤ n

as the Bergman matrix of T_B. 

For a smooth curve γ: [ 0,1 ] → T_B, we define 

    l(γ )=∫_0^1⟨𝐁(γ (t))γ '(t),γ '(t)⟩ dt.


Based on the definition of  l(γ ), we can define the Bergman metric β on T_B as follows:

    β (z,w)=inf{l(γ ):γ (0)=1, γ (1)=w}.


Let D( z,r ) denote the Bergman metric ball at z with radius r, that is 
 
    D(z,r)={w ∈ T_B:  β(z,w)<r}.



We will use the important  transform :𝔹_n→ T_B given by

    (z)= ( √(2)z'/1+z_n,  i1-z_n/1+z_n-iz' · z'/(1+z_n)^2 ),    z∈𝔹_n

and it is not hard to calculate that 

    ^-1(w)= ( 2iw'/i+w_n+i/2w'· w', i-w_n-i/2w'· w'/i+w_n+i/2w'· w' ),    w∈ T_B.

The mapping  is a biholomorphic map from 𝔹_n to T_B and also a key tool for this paper.

In Krantz's book <cit.>, there is the following proposition <cit.>:

Let _1,  _2⊆ℂ^n be domains and f: _1→ _2 a biholomorphic mapping. Then f induces an isometry of Bergman metrics:
    β _ _1( z,w ) =β _ _2( f( z ) ,f( w ) )
 for all z,w∈ _1.

Hence, taking _1=T_B and _2=𝔹_n, we have:
    β _T_B( z,w ) =β _𝔹_n( Φ ^-1( z ) ,Φ ^-1( w ) ) =tanh ^-1( | φ _Φ ^-1( z )( Φ ^-1( w ) ) | ) .

A computation shows that

    β_T_B(z, w)=tanh ^-1√(1-ρ(z) ρ(w)/|ρ(z, w)|^2) .

Throughout the paper we use C to denote a positive constant whose value may change from line to line but does not depend on the functions being considered. The notation A≲ B means that there is a positive constant C such that A≤ CB, and the notation A≃ B means that A≲ B and B≲ A.


§ MAIN LEMMAS

   To prove our main results, we need the following key lemmas, where Lemmas <ref>-<ref> are from <cit.>. They play a crucial role as instrumental lemmas in the present paper.


	There exists a positive integer N such that for any 0<r≤1 we can find a sequence {a_k} in T_B with the following properites:

	
	(1) T_B=⋃_k=1^∞D( a_k,r );
	
	(2) The sets D(a_k,r/4) are mutually disjoint;
	
	(3)Each point z∈ T_B belongs to at most N of the sets D(a_k,2r).
	




	For any r>0, the inequalities 
	
    | ρ( z,u ) |≃| ρ( z,v ) |

	hold for all z,u,v∈ T_B with β (u,v) <r .



	For any z∈ T_B and r>0 we have 
    V_α( D( z,r ) ) ≃ρ( z ) ^n+α +1.




	Let a,b and c∈ℝ, 1⩽ p<∞ and 
    Tf( z ) =ρ( z ) ^a∫_T_Bρ( w ) ^b/ρ( z,w ) ^cf( w ) dV( w )
 and 
    Sf( z ) =ρ( z ) ^a∫_T_Bρ( w ) ^b/| ρ( z,w ) |^cf( w ) dV( w ).
 Then the following conditions are equivalent for any real α.
	
	(1) The operator S is bounded on L_α^p( T_B ).
	
	(2) The operator T is bounded on L_α^p( T_B ).
	
	(3) The parameters satisfy -pα<α+1<p(b+1) and c=n+α+b+1.

		When p=∞, condition (3) should be a>0, b>-1, and c=n+a+b+1.
	



	Let r, s>0, t>-1, and r+s-t>n+1, then 
	
    ∫_T_Bρ (w)^t/ρ (z,w)^rρ (w,u)^sdV(w)=C_1(n,r,s,t)/ρ (z,u)^r+s-t-n-1

	for all z,u ∈ T_B, where
	
    C_1(n,r,s,t)=2^n+1π^nΓ (1+t)Γ (r+s-t-n-1)/Γ (r)Γ (s).

	In particular, let s,t∈ℝ, if t>-1,s-t>n+1, then 
    ∫_T_Bρ( w ) ^t/| ρ( z,w ) |^sdV( w ) =C_1( n,s,t )/ρ( z ) ^s-t-n-1,
 Otherwise, the above equation is infinity.



	
	The following properties hold for holomorphic mappings from 𝔹_n to T_B:	
	
	(1) The real Jacobian of  at z∈ T_B is 
	
    J_R((z))=2^n+1/ | 1+z_n |^2(n+1).

	
	(2) The real Jacobian of ^-1 at z∈ T_B is 
	
    (J_R^-1)(z)=1/4| ρ (z,𝐢 )|^2(n+1).

	
	(3)  The identity 
	
    1-⟨^-1(z),^-1(w)⟩= ρ (z,w)/ρ (z,𝐢 )ρ (𝐢,w)

	holds for all z,w∈ T_B, where 𝐢=(0',i). 
 And moreover, 
	
    1-  |^-1(z) |^2= ρ (z)/ | ρ (z,𝐢) |^2, 1+[^-1(z)]_n = 1/ρ (z,𝐢).

	
	(4) The identity 
	
    ρ (z,w)= ρ ( (ξ ), (η ))= 1-⟨ξ ,η⟩/(1+ξ_n)(1+η_n)

	holds for all z,w∈ T_B, where ξ =^-1(z), η =^-1(w).
	



	The properties mentioned above are derived from <cit.>, and here we only present the unproven property (2).
	
	Simple calculations show that 
    ρ (z,𝐢)=1/4(z'^2-2iz_n+2) ,

	so 
	
    ( J_C ^-1) ( z )    =| 2ρ( w,𝐢) -w_1^2/2ρ( w,𝐢) ^2   -w_1w_2/2ρ( w,𝐢) ^2   ⋯   -w_1w_n-1/2ρ( w,𝐢) ^2   iw_1/2ρ( w,𝐢) ^2
    -w_2w_1/2ρ( w,𝐢) ^2   2ρ( w,𝐢) -w_2^2/2ρ( w,𝐢) ^2   ⋯   -w_2w_n-1/2ρ( w,𝐢) ^2   iw_2/2ρ( w,𝐢) ^2
    ⋮   ⋮   ⋱   ⋮   ⋮
    -w_n-1w_1/2ρ( w,𝐢) ^2   -w_n-1w_2/2ρ( w,𝐢) ^2   ⋯   2ρ( w,𝐢) -w_n-1^2/2ρ( w,𝐢) ^2   iw_n-1/2ρ( w,𝐢) ^2
    -w_1/2ρ( w,𝐢) ^2   -w_2/2ρ( w,𝐢) ^2   ⋯   -w_n-1/2ρ( w,𝐢) ^2   i/2ρ( w,𝐢) ^2
    |
       =i/2ρ( z,𝐢)^n+1.

	Therefore J_R( ( z ) ) =| J_C( ( z ) ) |^2=1/ 4| ρ( z,𝐢) |^2( n+1 )
	.




The following lemma plays a key role in estimating the inequality.

	For any z,w∈T_B, we have 
	
    2| ρ( z,w ) |≥max{ρ( z ) ,ρ( w ) }.




	Let δ_t(w)=(tw',t^2w_n) be the nonisotropic dilation for t>0, w∈ T_B. 
	
	For any w ∈ T_B and each fixed z ∈ T_B, consider the holomorphic mapping 
    h_z( w ) :=( w'-z',w_n-Rez_n-iw'z'+i| z' |^2/2+iz'·z'/4+iw'· z'/4).
 
	
	It is evident that h_z(w) is a holomorphic automorphism of T_B. Thus, the mapping σ _z:=δ _ρ (z)^-1/2∘ h_z is a holomorphic automorphism of T_B. Simple calculations reveal that σ _z(z)=𝐢:=(0',i) and 
    (J_ℂσ_z)(w)=ρ (z)^-n+1/2,
 where (J_ℂσ_z)(w) denotes the complex Jacobian of σ_z at w. 
	
	Hence, we obtain 
	
    K_α( z,w )    =( J_ℂσ _z ) ( z ) K_α( σ _z( z ) ,σ _z( w ) ) ( J_ℂσ _z ) ( w ) 
       =K_α( 𝐢,σ _z( w ) )   ρ( z ) ^-n-1.


	It's worth noting that 
	
    | K_α( 𝐢,w ) |≤( n+α +1 ) 2^α/π ^n( α +1 )

	for all w∈ T_B. By substituting the expression for K_α( z,w ) into the above inequality, considering the arbitrariness of α and n, and rearranging the positions of z and w, we arrive at the desired result.




The following lemma <ref> is commonly encountered in the operator theory on function spaces, illustrating that the growth of functions in Bergman spaces is controlled.

	On the Bergman space A_α^p( T_B ), every valuation functional is a bounded linear functional. Specifically, for each function f∈ T_B, we have 
	
    | f( z ) |^p ≤C/ρ( z ) ^n+α+1∫_D( z,r )| f( w ) |^pdV_α( w ),

	where 0<p<∞, α>-1, r>0 and C is a positive constant .

	
		Let f∈ H( T_B ). Then f∘∈ H(𝔹_n ). Note that D( 𝐢,r ) =( B( 0,R ) ) with R=tanh( r ). By the subharmonicity of | f |^p and variable transformation, we have 
		
    | f( ( 0 ) ) |^p    ≤1/V_α( D( 0,R ) )∫_B( 0,R )| f( ( ξ) ) |^pdV_α( ξ)
       = 1/V_α( D( 0,R ) )∫_D( 𝐢,r )| f( w ) |^p1/| ρ( w,𝐢) |^2( n+α +1 )dV_α( w ).

		
		Since f( ( 0 ) ) =f( 𝐢) and 2ρ( w,𝐢) ≥ 1, there exists a positive constant C such that 
		
    | f( 𝐢) |^p ≤ C∫_D( 𝐢,r )| f( w ) |^pdV_α( w ).
 
		Replacing f by f∘σ _z^-1 in the above inequality, we obtain
		
    | f( z ) |^p ≤C/ρ( z ) ^n+α +1∫_D( z,r )| f( w ) |^pdV_α( w ).

		This completes the proof of the lemma.
	



As is well known, the reproducing kernel is an element of the Bergman space A_α^p(T_B), hence its norm is finite. Lemma <ref> below clarifies the size of the reproducing kernel.


	For 1<p<∞ and α>-1, for each z∈ T_B, the Bergman kernel function K_α,z(w) belongs to A_α^p(T_B), and its norm is Cρ( z ) ^-( n+α+1 ) /p', where p' is the conjugate exponent of p and C is a positive constant depending only on n,α and p.



	The above conclusion can be obtained through straightforward calculation and the lemma <ref>, so we omit the proof.




The following lemma <ref> provides an equivalent condition for weak convergence of functions in Bergman spaces. Although it is well-known, its proof has not been found, thus, we provide one below.

	Suppose { f_j } is a sequence in A_α^p( T_B ) with 1<p<∞. Then f_j→ 0 weakly in A_α^p( T_B ) as j→∞ if and only if { f_j } is bounded in A_α^p( T_B ) and converges to 0 uniformly on each compact subset of T_B.
 


	(1) We first prove the necessary part of the lemma. 
	
	Suppose { f_j } converges to 0 weakly in A_α^p( T_B ) as j→∞, then { f_j } is bounded in A_α^p( T_B ) according to the uniform boundedness principle.
	
	 Based on the above lemma <ref>, { f_j } is uniformly bounded on every compact subset of A_α^p( T_B ) and thus is a normal family.
	 
	  Note that { f_j } converges to 0 pointwise according to the property of the reproducing kernel as j→∞, so { f_j } converges uniformly to 0 on every compact subset of T_B as j→∞.
	
	(2) Now we prove the sufficiency part of the lemma.
	
	 Suppose sup_j‖ f_j ‖ _p,α<∞ and { f_j } converges uniformly to 0 on every compact subset of T_B as j→∞. 
	For any ε >0 and any g∈ A_α^q( T_B ) (where 1/p+1/q=1), there exists a compact subset K of T_B such that 
	
    ( ∫_T_B\ K| g( z ) |^qρ( z ) ^αdV( z )) ^1/q<ε/2M
    	.
 Then, through the Holder inequality, we have
	
    | ( f_j,g ) |    ≤| ∫_Kf_j( z ) g( z ) ρ( z ) ^αdV( z )|
       +| ∫_T_B\ Kf_j( z ) g( z ) ρ( z ) ^αdV( z )|
       ≤ V( k ) ^1/p‖ g( z ) ‖ _q,αsup_z∈ K| f_j( z ) |
       +‖ f_j( z ) ‖ _p,α( ∫_T_B\ K| g( z ) |^qρ( z ) ^αdV( z )) ^1/q.

	Since { f_j } uniformly converges to 0 on K as j→∞, there exists N>0 such that when j≥ N, the second term of the above formula is less than ε/2.
	
	Therefore, ( f_j,g )→ 0 holds when j→∞, and then { f_j } converges weakly to 0 in A_α^p( T_B ) as j→∞. The proof is complete.
	



	As | z | approaches infinity, ρ( z,𝐢) also tends to infinity.



	We consider two cases.
	
	Case 1: When z_n is fixed and | z_k | →∞ for k = 1, 2, …, n-1, we have
	
    4| ρ( z,𝐢) |    ≥ 4Reρ( z,𝐢) 
       =∑_k=1^n-1Rez_k^2+2Imz_n+2 
       ≥∑_k=1^n-1Rez_k^2+2∑_k=1^n-1( Imz_k ) ^2+2 
       =∑_k=1^n-1( Rez_k ) ^2+∑_k=1^n-1( Imz_k ) ^2+2 
       =∑_k=1^n-1| z_k |^2+2.

This implies | ρ( z,𝐢) | →∞ as | z_k | →∞ for k = 1, 2, …, n-1.

	Case 2: When the previous n-1 variables are fixed (assuming they are all zero without loss of generality), then
	
    2| ρ( z,𝐢) |≥| z_n |-1,
	
	which shows that | ρ( z,𝐢) | →∞ as | z_n | →∞.
	
	Therefore, from cases 1 and 2, we can obtain that | ρ( z,𝐢) | →∞ as | z | →∞, which completes the proof.



In the Bergman spaces on the unit disk, it is known that as | z | approaches the boundary, the normalized reproducing kernel weakly converges to 0 in A_α^p( T_B ), as referenced in <cit.>. The following lemma <ref>  informs us that in the Bergman spaces on  tubes, the normalized reproducing kernel also possesses this property.


	For 1<p<∞ and α>-1, we have K_α,z‖ K_α,z‖ _p,α^-1→ 0 weakly in A_α^p( T_B ) as z→∂T_B. 

	
		According to Lemma <ref>, our proof is to demonstrate the uniform convergence to 0 of K_α,z‖ K_α,z‖ _p,α^-1 on each Q_j:= D( 𝐢,j ).

		By applying Lemma <ref> and Lemma <ref>, we can establish the existence of a constant C>0 such that 
		
    sup_w∈ Q_j| K_α,z( w )/‖ K_α,z‖ _p,α|≤ Cρ( z ) ^(n+α+1)/p'/| ρ( z,𝐢) |^n+α+1
 
		for all z∈ T_B. Given that 2ρ( z,𝐢) =1/2| z'^2-2iz_n+2 |≥1 for all z∈ T_B, we deduce 
		
    sup_w∈ Q_j| K_α,z( w )/‖ K_α,z‖ _p,α|≤ Cρ( z ) ^(n+α+1)/p',

		which indicates that K_α,z‖ K_α,z‖ _p,α^-1→ 0 uniformly on Q_j as z→∂ T_B. 
		
		Furthermore, through the utilization of Lemma <ref> and Lemma <ref>, we ascertain that 
		
    sup_w∈ Q_j| K_α,z( w )/‖ K_α,z‖ _p,α|≤C/| ρ( z,𝐢) |^( n+α+1 ) /p→ 0,

		which demonstrates that K_α,z‖ K_α,z‖ _p,α^-1→ 0 uniformly on Q_j as | z |→∞. Thus, the lemma is successfully proven.
	





§ CARLESON MEASURES ON T_B

   As mentioned in the introduction, the importance of Carleson measures in studying the analytical properties of Toeplitz operators is significant. In the following, we will introduce two important theorems related to Carleson measures, which provide powerful support for the proof of the main results theorem <ref>mianth1 and <ref>mianth2.
 
The following theorem <ref>thCraleson presents some equivalent characterizations of Carleson measures on tubes, and this theorem is derived from <cit.>.
<cit.>
Suppose 0<p<∞,r>0,α>-1 and μ is a positive Borel measure on the T_B. Then the following conditions are equivalent.



	(1) μ is a Carleson measure for A_α^p( T_B ).

	(2) There exists a constant C>0 such that 
    ∫_T_Bρ( z ) ^n+α+1/| ρ( z,w ) |^2( n+α+1 )dμ( w )≤ C
 for all z∈ T_B

	(3) There exists a constant C>0 such that 
    μ( D( z,r ) ) ≤ Cρ( z ) ^n+α+1
 for all z∈ T_B.

	(4) There exista a constant C>0 such that 
    μ( D( a_k,r ) ) ≤ Cρ( a_k ) ^n+α+1
 for all k≥1, where { a_k } is an r-lattice in the Bergman metric.



The proof of Theorem <ref> has been provided in <cit.>, and we will not repeat it here.


Following that, we provide some equivalent characterizations of vanishing Carleson measures. These characterizations illustrate the property of being a vanishing Carleson measure for A_α^p( T_B ), which depends neither on p nor on r in the following theorem <ref>thVanishing Carleson.


	Suppose 1<p<∞,r>0,α>-1 and μ is a positive Borel measure on the T_B. Then the following conditions are equivalent.


	
		(1) μ is a vanishing Carleson measure for A_α^p( T_B ).
		
		(2) The measure μ satisfies 
    lim_a→∂T_B∫_T_Bρ( a ) ^n+α+1/| ρ( z,a ) |^2( n+α+1 )dμ( z )=0.


		(3) The measure μ has the property that 
    lim_a→∂T_Bμ( D( a,r ) )/ρ( a ) ^n+α+1=0.


		(4) For {a_k} an r-lattice in the Bergman metric, we have 
    lim_k→∞μ( D( a_k,r ) )/ρ( a_k ) ^n+α+1=0.




	
	(1) ⇒ (2): According to Lemmas <ref> and <ref>, take 
	
    g_a(z)=( ρ( a ) ^n+α +1/| ρ( z,a ) |^2( n+α +1 )) ^1/p.

	By Lemma  <ref> and <ref>, we see that g_a(z) converges weakly to 0 in A_α^p( T_B ) as a→∂T_B. Therefore,
	
    ∫_T_Bρ( a ) ^n+α+1/| ρ( z,a ) |^2( n+α+1 )dμ( z )=C∫_T_B| g_a( z ) |^pdμ( z )→ 0
 
	as a→∂T_B, which shows that  (2)  holds.

	
	(2) ⇒ (3):
	Obviously, the following equation holds:
	
    lim_a→∂T_B∫_T_Bρ( a ) ^n+α+1/| ρ( z,a ) |^2( n+α+1 )dμ( z )=0.
 
	By Lemma <ref>, | ρ( z,a ) | and ρ( z ) are comparable when z∈ D(a,r). Therefore, μ has that protery.

	
	(3) ⇒ (4):
	We know that a_k→∂T_B as k→∞ if {a_k} is an r-lattice in the Bergman metric. So the conclusion is trivial.

	
	(4) ⇒ (1):
	If the equality holds, we show that the inclusion map i_p from A_α^p( T_B ) into L^p(T_B,dμ) is compact. 
	
	To this end, we assume that {f_j} is a sequence in A_α^p( T_B ) that converges to 0 uniformly on compact subsets of T_B and ‖ f_j ‖ _p,α≤ M for some positive constant M.
	
	 By assumption, given ε >0 there exists a positive integer N_0 such that 
    μ( D( a_k,r ) )/ρ( a_k ) ^n+α+1<ε,     k≥ N_0.

By Lemma <ref>, there is a constant C>0 such that 

    ∑_k=N_0^∞∫_D( a_k,r )| f_j( z ) |^pdμ( z )
       ≤ C∑_k=N_0^∞μ( D( a_k,r ) )/ρ( a_k ) ^n+α+1∫_D( a_k,2r )| f_j( z ) |^pdV_α( z )
       ≤ε CN∫_T_B| f_j( z ) |^pdV_α( z )≤ε CNM^p

	for all j, where C,N and M are all independent of ε. Since 
    lim_j→∞∑_k=1^N_0-1∫_D( a_k,r )| f_j( z ) |^pdμ( z )=0,
 
	we have 
	
    lim_j→∞   sup∫_T_B| f_j( z ) |^pdμ( z )
       ≤lim_j→∞ sup[ ∑_k=1^N_0-1∫_D( a_k,r )| f_j( z ) |^pdμ( z )+∑_k=N_0^∞∫_D( a_k,r )| f_j( z ) |^pdμ( z )] 
    	
       ≤ε CNM^p.

	Since ε is arbitrary, we know that μ is a vanishing Carleson measure. The proof of the theorem is complete.	






§ DENSE SUBSPACES OF A_Α^P( T_B )

   For the case of unit disk 𝔻, we know that the space H^∞(𝔻) of bounded analytic functions  is dense in the Bergman spaces A_α^p(𝔻). Since the Toeplitz operator is an integral operator, it is always well-defined, making the Toeplitz operator densely defined in A_α^p(𝔻). 

However, in the case of tubes, there is not dense property as good as in a unit disk. In order to study the theory of Toeplitz operators on the Bergman space A_α^p( T_B ) over tubular domains T_B,  we have to first find a dense subspaces of A_α^p( T_B ), and then obtain the Toeplitz operators densely defined on A_α^p( T_B ).

 To do this, we first introduce the following definition: 

Let ℳ_+ be the set of all positive Borel measure μ such that 
    ∫_T_Bdμ( z )/| ρ( z,𝐢) |^t<∞
 for some t>0.


Given t ∈ℝ, we denote by S_t the vector spaces of functions f holomorphic in T_B satisfying 
    sup_z∈ T_B| ρ( z,𝐢) |^t| f( z ) |<∞ .



	If 1≤ p<∞, α>-1, t>n+(α+1)/p and μ∈ℳ_+, then the Toeplitz
	operator T_μ is densely defined on A_α^p(T_B).

	
		To prove this theorem, we need to find a dense subspace S_t of A_α^p(T_B), such that for any f∈ S_t, the following equation holds: 
    ∫_T_B|K_α( z,w ) f( w ) |dμ( w ) <∞ .

		Now we prove the density of S_t in A_α^p(T_B). Let f_j=f·χ_Q_j for j=1,2,⋯, where Q_j=D(𝐢,j) and χ_Q_j is the characteristic function of Q_j. Clearly, ‖ f_j-f ‖_p,α→ 0 as j→∞.
		Given λ >-1, let P_λ be the integral operator given by
		
    P_λg(z) = c_λ∫_T_Bρ(w)^λ/ρ(z,w)^n+1+λg(w)dV(w),  z∈ T_B,

		where c_λ=2^n+1+2λΓ(n+1+λ)/π^nΓ(1+λ). It was shown in <cit.> that P_λ is a bounded projection from L_α^p(T_B) onto A_α^p(T_B), provided that λ>(α+1)/p-1.
		Taking λ=t-n-1, by Holder's inequality, we obtain
		
    | P_t-n-1f_j(z) | ≤ c_t-n-1‖ f ‖_p,αV_α(Q_j)^1/p'sup_w∈ Q_jρ(w)^t-n-α -1/| ρ(z,w) |^t

		for all z∈ T_B. Combined with Lemma <ref>, we only need to consider the upper bound of
		
    sup_w∈ Q_jρ(w)^t-n-α-1/| ρ(z,w) |^t.

		Since the Q_j are fixed, a simple calculation yields
		
    | P_t-n-1f_j(z) | ≤C‖ f ‖_p,α/| ρ(z,𝐢) |^t

		for all z∈ T_B, where C>0 is a constant independent of z. 

		Thus 
    P_t-n-1f_j(z) ∈ S_t.

		Since f∈ A_α^p(T_B) and P_α-n-1 is a bounded projection from L_α^p(T_B) onto A_α^p(T_B), we have
		
    ‖ P_α -n-1f_j(z) -f ‖_p,α=‖ P_α -n-1( f_j(z) -f ) ‖_p,α≤‖ P_α -n-1‖‖ f_j-f ‖_p,α→ 0

		as j→∞. This implies that S_t is dense in A_α^p(T_B).
		
		
		According to Lemma <ref>, we know that there exists a constant C>0 such that 
		
    ∫_T_B|K_α( z,w ) f( w ) |dμ( w )    ≲1/ρ( z ) ^n+α +1∫_T_B| ρ( w,𝐢) |^t| f( w ) |/| ρ( w,𝐢) |^tdμ( w )
       ≲1/ρ( z ) ^n+α +1∫_T_Bdμ( w )/| ρ( w,𝐢) |^t
       <∞
 
		holds. This completes the proof of the theorem.	

	





§ CHARACTERIZATION OF TOEPLITZ OPERATORS

   Building on the foundational results previously, we can now to  prove the following result:

	Suppose that r>0, 1<p<∞, 0<q<∞, α>-1 and that μ∈ℳ_+. Then the following conditions are equivalent:


		(1) T_μ is bounded on A_α^p(T_B).
		
		(2) μ is a bounded function on T_B.
		
		(3) μ_r is a bounded function on T_B.
		
		(4) μ is a Carleson measure for A_α^q(T_B).




(1) ⇒ (2):
We have the well-known conclusion 
    < T_μk_z,k_z > =∫_T_B| k_z( w ) |^2dμ( w )=μ̃( z ),
 And according to the conditions and lemma <ref> we have
    | < T_μk_z,k_z > |≤‖ T_μk_z ‖ _p,α‖ k_z ‖ _p',α≤‖ T_μ‖‖ K_z ‖ _p,α‖ K_z ‖ _p',α/K( z,z )=C‖ T_μ‖.
 
Therefore, μ̃(z) is a bounded function on T_B.


(2) ⇒ (3):
By the definition of μ̂_r and Theorem <ref>, the expected results are obtained by the following identity
    V_α( D( z,r ) ) =Kρ( z ) ^n+α +1,
 where K is a constant independent of z.


(3) ⇒ (4):
This result can be obtained from the fact that the Carleson measure in Theorem <ref> is not dependent on p, so it is trivial.


(4) ⇒ (1):
Since μ is a Carlson measure for A_α^q(T_B), and according to the Carlson measure does not depend on the index we know μ is a Carlson measure for A_α^1(T_B). Combining T_μ f is well defined on S_t, there exists a constant C>0 such that

    ∫_T_B |K_α(z,w) f(w)| dμ(w)  ≤     C ∫_T_B |K_α(z,w) f(w)| dV_α(w)
    ≤     C∫_T_Bρ( w ) ^αdV( w )/|ρ( z,w ) |^n+α +1|ρ( w,𝐢) |^t.


According to the holomorphic automorphism from 𝔹_n to T_B, let 
    w=( ξ),z=( η),
 and make a variable transformation of the above integral. we obtain


    ∫_T_Bρ( w ) ^αdV( w )/|ρ( z,w ) |^n+α +1|ρ( w,𝐢)|^t
       =2^n+1| 1+η _n |^n+α +1∫_𝔹_n( 1-| ξ|^2 ) ^αdV(ξ)/| 1-< η ,ξ> |^n+α +1|1+ξ _n|^n+α +1-t.

By <cit.>, the last integral is dominated by a constant multiple of

    1/|1+η_n|^n+α+1-t(1+log|1+η_n|/1-|η|^2),

Thus, there exists a constant C>0 such that

    ∫_T_Bρ( u ) ^αdV( u )/|ρ( z,u ) |^n+α +1|ρ( u,𝐢) |^t ≤     C |1+η_n|^t(1+ log1/1-|η|^2)
     =    C/|ρ(z,𝐢)|^t(1+log|ρ(z,𝐢)|^2/ρ(z))

holds. 

By the elementary inequality log x<x^ε, where 
    x>1,0<ε <min{( α +1 ) /p,t-( α +n+1 ) /p},
 we have  
    ∫_T_B |T_μf(z)|^p dV_α(z)  ≤  C( ∫_T_BdV_α(z)/|ρ(z,𝐢)|^pt +∫_T_Bρ(z)^-pϵ/|ρ(z,𝐢)|^p(t-2ϵ)dV_α(z)).
 This shows that T_μ is densely defined on S_t and can be extended to be a bounded operator on A_α^p(T_B).	


This completes the proof of Theorem <ref>.




The above theorem <ref>mianth1 characterizes the boundedness of the Toeplitz operator T_μ, and now we will characterize the compactness of T_μ in the following theorem <ref>mianth2.



	Suppose that r>0, 1<p,q<∞, α>-1 and that μ∈ℳ_+. Then the following conditions are equivalent:


		(1) T_μ is compact on A_α^p(T_B).
		
		(2) μ belongs to C_0(T_B).
		
		(3) μ_r belongs to C_0(T_B).
		
		(4) μ is a vanishing Carleson measure for A_α^q(T_B).




	(1) ⇒ (2):	
	Assume that T_μ is a compact operator on A_α^p(T_B) for some p>1.
	
	Since 
    | μ̃( z ) |≤ C‖ T_μ( K_z/‖ K_z ‖ _p,α) ‖ _p,α
 for all z∈ T_B. Combining the compactness of T_μ and lemma <ref>, implies μ̃∈ C_0(T_B).

	
	(2) ⇒ (3):
	By the definition of μ̂_r and Theorem <ref>, the expected results are obtained by the following identity
    V_α( D( z,r ) ) =Kρ( z ) ^n+α +1,
 where K is a constant independent of z.	

	
	(3) ⇒ (4):
	This conclusion can be drawn from the fact that the vanishing Carleson measure in Theorem <ref> is not dependent on p, so it is trivial.

	
	(4) ⇒ (1):
	Since μ is a vanishing Carlson measure for A_α^q(T_B), and according to the vanishing Carlson measure does not depend on the index we know μ is a vanishing Carlson measure for A_α^p'(T_B). By Theorem <ref>, we know that T_μ is bounded on f∈ A_α^p(T_B).
	
	For any f∈ A_α^p(T_B), we have
	
    ‖ T_μf ‖ _p,α   =sup{|< T_μf,g > || ‖ g ‖ _p',α=1  in  A_α^p'( T_B ) . }
       =sup{| ∫_T_Bfgdμ|| ‖ g ‖ _p',α=1  in  A_α^p'( T_B ) . }
       ≤‖ f ‖ _L^p( μ)sup{‖ g ‖ _L^p'( μ)| ‖ g ‖ _p',α=1  in  A_α^p'( T_B ) . } 
    	,

	where the rationality of the second equation uses the Fubini's theorem. Since μ is a vanishing Carlson measure on A_α^p'(T_B), the second term of the last inequality is finite. 
	
	Now if f_j → 0 weakly in A_α^p(T_B), then the compactness of the inclusion mapping implies that f_L^p(μ)→ 0. It follows that ‖ T_μf_j ‖ _p,α→ 0 and hence T_μ is compact.  
	
	This completes the proof of Theorem <ref>.





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