Hausdorff operators on Bergman spaces of the upper half plane

In this paper we study Hausdorff operators on the Bergman spaces $A^{p}(\mathbb{U})$ of the upper half plane.


introduction
Given a σ-finite positive Borel measure µ on (0, ∞), the associated Hausdorff operator H µ , for suitable functions f is given by where U = {z ∈ C : Im z > 0} is the upper half plane. Its formal adjoint, the quasi-Hausdorff operator H * µ in the case of real Hardy spaces H p (R) is Moreover for appropriate functions f and measures µ they satisfy the fundamental identity: where f denotes the Fourier transform of f . The theory of Hausdorff summability of Fourier series started with the paper of Hausdorff [Ha21] in 1921. Much later Hausdorff summability of power series of analytic functions was considered in [Si87] and [Si90] on composition operators and the Cesáro means in Hardy H p spaces. General Hausdorff matrices were considered in [GaSi01] and [GaPa06]. In [GaPa06] the authors studied Hausdorff matrices on a large class of analytic function spaces such as Hardy spaces, Bergman spaces, BMOA, Bloch etc. They characterized those Hausdorff matrices which induce bounded operators on these spaces.
Results on Hausdorff operators on spaces of analytic functions were extended in the Fourier transform setting on the real line, starting with [LiMo00] and [Ka01]. There are many classical operators in analysis which are special cases of the Hausdorff operator if one chooses suitable 1 measures µ such as the classical Hardy operator, its adjoint operator, the Cesáro type operators and the Riemann-Liouville fractional integral operator. See the survey article [Li013] and the references there in. In recent years, there is an increasing interest on the study of boundedness of the Hausdorff operator on the real Hardy spaces and Lebesque spaces (see for example [An03], [BaGo19], [FaLi14], [LiMo01] and [HuKyQu18]).
Motivated by the paper of Hung et al. [HuKyQu18] we describe the measures µ that will induce bounded operators on the Bergman spaces A p (U) of the upper half-plane. Next Theorem summarizes the main results (see Theorems 3.5 and 3.7 ): Theorem 1.1. Let 1 ≤ p < ∞ and µ be an σ-finite positive measure

preliminaries
To define single-valued functions, the principal value of the argument it is chosen to be in the interval (−π, π]. For 1 ≤ p < ∞, we denote by L p (dA) the Banach space of all measurable functions on U such that where dA is the area measure. The Bergman space A p (U) consists of all holomorphic functions f on U that belong to L p (dA). Sub-harmonicity yields a constant C > 0 such that [ChKoSm17]). In particular, this shows that each point evaluation is a continuous linear functional on A p (U). The duality properties of Bergman spaces are well known in literature see [Zh90] and [BaBoMiMi16]. It is proved that for 1 < p < ∞, 1 under the duality pairing,

Main results
In what follows, unless otherwise stated, µ is a positive σ-finite measure on (0, ∞). We start by giving a condition under which H µ is well defined.
is a well defined holomorphic function on U.
Proof. For f ∈ A p (U) using (4) we have Thus H µ f is well defined, and is given by an absolutely convergent integral, so it is holomorphic.
Lemma 3.2. Let λ > 0 and δ > 0. If g λ,δ (z) = |z + δi| − 2+λ p , then Proof. Using polar coordinates for the integral over U we find Denote by I the last double integral. Then On the other hand, and the assertion follows.
Note that if Re f ε or Im f ε have constant sign on some sector A, then for every z ∈ A.
Lemma 3.4. Let 1 ≤ p < ∞ and suppose that H µ is bounded on A p (U). Then there are ε(p) and k(p) positive constants such that for every ε in (0, ε(p)].
Proof. We will consider three cases for the range of p. Note that if z is in a truncated sector S then z/t belongs to the corresponding sector A for every t > 0.
We will consider H * µ on A p (U) and suppose for a moment that it is well defined for functions in A p (U). Let λ(t) = t −1 , t > 0, then λ maps (0, ∞) onto (0, ∞) and is measurable. Set f (tz) = f z (t) then where dν = dλ * (µ)(t) and λ * (µ) is the push-forward measure of µ with respect to λ. We can now apply the results of the first part of the paper to have: