Bloch spaces of holomorphic functions in the polydisk

This work is an introduction to anisotropic spaces of holomorphic functions, which have ω -weight and are generalizations of Bloch spaces to a polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. Some theorems on projection and diagonal mapping are proved. We establish a description of (Ap(ω))∗ (or (Hp(ω))∗ ) via the Bloch classes for all 0 < p ≤ 1.


Introduction
The aim of this paper is to extend the Bloch spaces to a polydisk so that the well-known properties of the Bloch spaces of one variable remain true.Moreover, our generalization gives results which are new also for function of one variable.We are interested, for instance, in theorems on projection and the description of (A p (α)) * via Bloch spaces.One can consider other generalisations of the Bloch space for polydisk (see for example [1]).
In the paper of F. Shamoyan [2], the spaces A p (α) have been generalized and investigated as H p (ω) spaces in a polydisk.The Bloch spaces play the same role for A p (α) as the BMOA spaces for the Hardy theory.Hence, it is very important to generalize the ω weighted Bloch spaces to a polydisk.
In Section 1, such classes are defined and an auxiliary lemma is proved.In Section 2, some theorems describing the properties of the introduced Bloch spaces are proved.In particular, we prove that they are invariant with respect to monomial multiplication and in the some case they are a Lipschitz class.In Section 3, we consider bounded operators in these classes and diagonal mappings.Section 4 is devoted to a description of (A p (ω)) * via the introduced Bloch spaces.

Preliminaries and basic constructions
Let be the unit polydisk in the n-dimensional complex plane C n and let be its torus.We denote by H(U n ) the set of holomorphic functions in U n and by H ∞ (U n ) the set of bounded holomorphic functions in U n .
For A p (ω) we assume that 0 (i) we define the fractional differentiation D β in the following way: (ii) the inverse operator D −β we define in the standard sense: In particular, The following properties of D are valid: It is easy to see that B ω is a Banach space with respect to the norm For B ω we assume that 0 < α ωj < 1, 1 ≤ j ≤ n.
The proof is evident.
) be the class of measurable functions on U n , for which To prove the main results, we need the following lemmas.
First we estimate the integral I 1 : It follows that (a + 1) and On the other hand, we have the inequality We now estimate I 2 .For r 2 ≥ 1/2, we have and it follows that Then the inequality gives us (4) Summing up, from ( 3) and ( 4) we get the proof of Lemma 2.6.
From the representation of M. M. Djrbashian (see [5]), we have Integrating with respect to z 1 , we obtain It is easy to see (using for example the Taylor expansion) that the second integral is 0. Integrating the last equality with respect to z 2 , . . ., z n , we obtain ( 5) Next, using the last representation, Lemma 2.6 and maximum modulus principle, we get

Main properties of B ω spaces
For ω j ∈ S , we say that ω satisfies the condition Ω if We consider the torus We take η j so that T j lies in the unit disc.Using the Cauchy formula for T n , we get It follows that As an example we can take the function Using the fact that h(z)z = z 0 (h(t)t) dt, h ∈ H(U ), we get Therefore, and the fact that f, g The following proposition shows that, in some cases, the spaces B ω are ω weighted Lipschitz classes (see [7]).Proposition 3.5.Let t αω j ≤ ω j (t) ≤ t γω j , 0 < α ωj , γ ωj < 1 , 1 ≤ j ≤ n then B ω = Λ a (ω), where Λ a (ω) denotes the weighted Lipschitz spaces for polydisk.
Proof.Using (1), it is not difficult to see that there exist C 1 , C 2 = 0 such that Integrating this inequality with respect to z k+1 , . . ., z n , we derive From the invariance of Λ a (ω) with respect to the monomial multiplication, it follows that f ∈ Λ a (ω).Let now f ∈ Λ a (ω).Using the characterisation and invariance with respect to the monomial multiplication of holomorphic Lipschitz classes again, we see that f ∈ B ω .

Projection theorems and diagonal mappings
For α = (α 1 , . . ., α n ), α j > 0, 1 ≤ j ≤ n, we introduce the operator Proof.We show that (2) is true for the function Using Lemma 2.6, we get Next we show that Q α is surjective: for any f ∈ B ω there exists a On the other hand, the function f (z) is holomorphic and belongs to It can be written as Thus, we get Consider now the projection it is not diffucult to prove the following result: and is a bounded.
The question is whether there is an inverse operator R α,β which maps B ω to L ∞ ω −1 and furthermore, if this is the case, whether The answer is positive.
Let us define the 'inverse' operator as We first show that To this end, let us calculate P α R α,β f (z) using the Fubini theorem: 1 and the open mapping theorem, we get To prove the other inequality in (7), we use the fact, that P α R α,β f (z) = f (z) for all f ∈ B ω .Then by Proposition 4.2, there exists C 0 > 0 such that (9) Taking C 1 = C −1 0 , we obtain the left-hand side inequality in (7).(b) The proof follows from (8) and (9).
Let us formulate the problem of diagonal mapping in the general case: let f ∈ X ⊂ H(U n ), then the function g(z) = f * (z) = f (z, . . ., z) is analytic on the unit disk.
Let us show that f ∈ B ω .In fact, Using Holder inequality and Lemma 2.6, we get

Linear continuous functionals on A p (ω)
We need to establish the following theorem before proving the duality result.

Theorem 3 . 3 .
Let ω satisfy the condition Ω.Then the classes B ω are invariant with respect to multiplication by monomials: