Characterizing the RIFs that induce bounded composition operators on the Bergman space of the bidisc

1 Introduction and formulation of the problem

Let.

where β ≥ −1, denote the weighted Bergman spaces on the bidisc. Consider Φ = (ϕ, ψ), where Φ ∈ O ( D 2 , D 2 ) and assume that ϕ , ψ ∈ O ( D 2 , D ) are induced by Rational Inner Functions with singularities. Let p ∈ C [ z 1 , z 2 ] a polynomial in two complex variables, restricted on the bidisc. We assume that the bidegree of p satisfies deg p = ( n , m ) ∈ N 2 . We further assume that this polynomial has zeros on the bitorus T 2 . Then, RIFs on the bidisc D 2 are of the form ϕ = λ z 1 N z 2 M p ̃ p , where p ̃ ( z 1 , z 2 ) = z 1 n z 2 m p 1 z 1 ̄ , 1 z 2 ̄ ̄ , for z 1 , z 2 ∈ D 2 , and N, M are some positive integers. This characterization is due to Rudin, see [1]. As it will be pointed out in the references later on, it is evident that RIFs may posses singularities on the distinguished boundary T 2 , contrary to what happens on the one (complex) dimensional case. Kosiński and Bayart in [2], 3] respectively, studied the composition operator acting on the Hardy and Bergman space of the bidisc, whenever the self-map of the bidisc is induced by smooth symbols. In the paper [4], the author studied the case where the self map of the bidisc is induced by RIFs. Specifically, it was shown that whenever Φ = (ϕ, ϕ) and ϕ is a Rational Inner Function with a singularity, then C Φ : A 2 ( D 2 ) → A 2 ( D 2 ) is not bounded. An example which showcases this, is the Rational Inner Mapping

for ( z 1 , z 2 ) ∈ D 2 . In the sequel, we will provide an alternative proof for this example.

Contrary to the one-dimensional setting, where the only Rational Inner Functions are the Finite Blaschke Products, in two complex dimensions, Rational Inner Functions may have singularities. These singularities might occur on the distinguished boundary T 2 , as it is evident from the above example. A striking result of Greg Knese shows that Rational Inner Functions have non-tangential limit everywhere on T 2 , even on the singularities that might occur. The value of this non-tangential limit is unimodular. For more about Rational Inner Functions and their properties, like the membership of their derivatives on the Hardy and Dirichlet spaces of the bidisc, or the behavior of their zero sets, the reader can consult the articles [5], [6], [7], [8], [9], [10] and the references therein.

Having the essential background, let us now state the problem in question:

2 Recent results

In the papers [4] and [11] the problem of boundedness of the composition operator acting on the weighted Bergman spaces of the bidisc was introduced. Before we pass to the statements of the main results, let us briefly mention a Carleson measure criterium which is the main tool in the proofs of these Theorems.

Recall that a two-dimensional Carleson box is defined as.

where ζ = ( ζ 1 , ζ 2 ) ∈ T 2 , δ ̃ = ( δ 1 , δ 2 ) ∈ ( 0,2 ) 2 . It is well understood that the Volume of the Carleson box in two dimensions behaves like V a ( S ( ζ , δ ) ) ≍ δ 1 a + 2 δ 2 a + 2 . The above lemma stated in [12] was the main tool in the papers of Kosiński [2], Bayart [3],4],11],13]. The main results regarding composition operators induced by RIFs are the following.

The second Theorem is.

Note here that non-boundedness here is somehow the expected result, in the sense that the self map collapses on the diagonal of D 2 and this makes the volume of the inverse image of the Carleson box under Φ behave similarly to the volume of a one dimensional disc.

3 An example

In the following example we estimate the volume of the sub-level sets { z ∈ D 2 : | ϕ ( z ) − 1 | < δ } for ϕ(z) being the Knese function, namely ϕ ( z ) = 2 z 1 z 2 − z 1 − z 2 2 − z 1 − z 2 , z 1 , z 2 ∈ D .