Products of Toeplitz and Hankel operators on the Bergman space in the polydisk

. In this paper we obtain a condition for analytic square integrable functions f, g which guarantees the boundedness of products of the Toeplitz operators T f T ¯ g densely deﬁned on the Bergman space in the polydisk. An analogous condition for the products of the Hankel operators H f H ∗ g is also given.


Introduction.
Let D be the open unit disk in the complex plane C. For a fixed positive integer n ≥ 2, the unit polydisk D n is the Cartesian product of n copies of D. By dA we will denote the Lebesgue volume measure on D n , normalized so that A(D n ) = 1.
The Bergman space A 2 = A 2 (D n ) is the space of all analytic functions on D n such that For w = (w 1 , w 2 , . . ., w n ) ∈ D n the reproducing kernel in A 2 is the function K w given by If •, • is the inner product in L 2 (D n ), then for every function f ∈ A 2 we have f, K w = f (w), w ∈ D n .
In the special case when f = K w , we obtain So, the normalized reproducing kernel for A 2 is Now we quote the definition of the Toeplitz operator.The orthogonal projection P from L 2 (D n ) onto A 2 is defined by For a function f ∈ L ∞ and h ∈ A 2 the Toeplitz operator T f is given by Similarly, the Hankel operator H f acting on A 2 is defined as and P is the projection mentioned above.It is clear that H f h ∈ A 2 ⊥ .Both operators T f and H f can be defined when the symbol f belongs to the space L 2 (D n ).In that case the Toeplitz and Hankel operators are densely defined on the Bergman space A 2 , that is on H ∞ .
Let w i , i = 1, 2, . . ., n, belong to the unit disk D. For each w i we define an automorphism ϕ w i of D by Then the map whenever such integrals make sense.

Problem and results.
As we mentioned, the Toeplitz operator may be considered when the index f belongs to the space L 2 (D n ).If f ∈ A 2 , then by the definition of the Toeplitz operator, we have The main problem in this note is what conditions must be satisfied by functions f, g ∈ A 2 to guarantee that the product of the Toeplitz operators T f T ḡ is bounded on the Bergman space A 2 in the polydisk D n .We provide a sufficient condition for boundedness of such products.Similarly, we give a sufficient condition to ensure that the product of the Hankel operators H f H * g is bounded on the space (A 2 ) ⊥ , where H * is the adjoint of H.
In [9] Stroethoff and Zheng established the following necessary condition for boundedness of the products T f T ḡ on the unit disk D.

Theorem 1. Let f and g be in
In the same paper the authors also gave a little stronger sufficient condition.
Theorem 2. Let f and g be in A 2 .If there is a positive constant ε such that sup There is a conjecture that the necessary condition is also a sufficient condition for boundedness.But in view of a counter-example of Nazarov [6] for Toeplitz products on the Hardy space, it may not be possible to prove that this necessary condition is also sufficient.
Stroethoff and Zheng [12] showed the analogous results on the Bergman spaces of the polydisk [11], weighted Bergman space of the unit disk [13] and the unit ball [12].Next, Miao in [4] gave an interesting way to transfer Theorem 1 and Theorem 2 to the space A p α , 1 < p < ∞, α > −1, of the unit ball.Recently, Michalska and Sobolewski [5] improved a sufficient condition on boundedness of T f T ḡ on A p α .A similar problem concerns the products of the Hankel operators H f H * g .Such operators are densely defined on space (A 2 ) ⊥ .The following condition for the Hankel products on the unit disk was established by Stroethoff and Zheng in [9].
The same authors showed that this necessary condition is, like for T f T ḡ, very close to being sufficient.
Their theorems were extended to the weighted Bergman spaces of the unit ball by Lu and Liu [2] and for the Bergman space of the polydisk by Lu and Shang [3].
In this paper we provide a sufficient condition for the boundedness of the operators where ϕ w is the automorphism of D n and z = (z 1 , z 2 , . . ., z n ).The following theorems are the main results in this paper.
After sending this paper for publication we found that Theorem 5 is contained in a result obtained in [1].

Proofs.
A very important role in our considerations is played by the formula for the inner product in A 2 introduced in [11].Let α = {α 1 , α 2 , . . ., α m } be a nonempty subset of {1, 2, . . ., n} with α 1 < α 2 < . . .< α m .We define the measure on D n by where m is the cardinality of α.Let us set D j h = ∂h/∂z j and For f, g ∈ A 2 we have (1) where α runs over all subsets of {1, 2, . . ., n}.We start with some lemmas which we will apply to prove the main theorems.
for all w ∈ D n .
Proof.First we show the inequality for α = ∅. .
In the case α = {1, 2, . . ., n}, we have Following the previous calculations, we obtain the desired inequality.It remains to consider the case when α is a proper subset of {1, 2, . . ., n}.Then where the last inequality follows from .
Proof.The proof will proceed in three steps as above.Suppose first that α = ∅.Then

In view of [8, Proposition 1] we may write
Thus, by Hölder's inequality, we obtain .
By the change-of-variable formula z → ϕ w (z) and using that |1 − zj w j | ≤ 2, we have .

Proof of Theorem
By (1), we get Using Lemma 1, we obtain dµ α (w). Since we get dA(w).
It remains to prove that the integral is convergent independently of z.Indeed, the change-of-variable formula ζ = ϕ z (w) and the fact that We need only to show that According to Theorem 1.7 in [14], we have Therefore By the change-of-variable formula, Clearly, for ε ∈ (0, 1) the integrals I i are bounded by a constant which is independent of z.Finally, we conclude that which proves the theorem.
Proof of Theorem 6.To prove the theorem we need to use Lemma 2 and the method used in the proof of Theorem 5.The details are left to the reader.
Now, we propose one additional theorem concerning products of Toeplitz and Hankel operators T f H * g .The following result can be proved in much the same way as Theorem 5 and Theorem 6. .
Since the last integral is convergent, our claim follows.