Weighted integral representations of harmonic functions in the unit disc by means of Mittag-Leﬄer type kernels

. For weighted L p -classes of functions harmonic in the unit disc, we obtain a family of weighted integral representations with weight function of the type | w | 2 ϕ · (1 − | w | 2 ρ ) β .


Introduction
It is well-known that the Cauchy integral formula has numerous applications in complex analysis.This formula makes it possible to reproduce values of holomorphic functions inside of a domain by integration of function along the boundary of the domain. First results are contained in [1,2], where the values of holomorphic functions inside of a domain were obtained by integration of functions over the whole domain. In [3,4] for the weighted spaces H p (α) (1 ≤ p < ∞, α > −1) of functions f holomorphic in the unit disc D and satisfying the condition D |f (ζ)| p (1 − |ζ| 2 ) α dudv < +∞, ζ = u + iv, the following result was established: Theorem 1. Each function f ∈ H p (α) has the integral representation These representations had numerous applications in the theory of factorization of meromorphic functions in the unit disc (see [3,4] as well as [5]).
The spaces above were introduced in [6]. Moreover, for these spaces an analogue of representations (1) and (2) were written out by means of special reproducing kernels adapted to new weight functions (see [6] and [7]): where z ∈ D and ζ ∈ D.
2. For all z ∈ D and ζ ∈ D, 3. S β,ρ,ϕ (z, ζ) can be majorated by a positive convergent series uniformly in z ∈ K ⊂ D and ζ ∈ D, where K is a compact set.
4. For a fixed ζ ∈ D, S is holomorphic in z ∈ D. For a fixed z ∈ D, S is antiholomorphic in ζ ∈ D and continuous in ζ ∈ D.
5. For all z ∈ D and ζ ∈ D, where E ρ ( · ; µ) is the well-known Mittag-Leffler type entire function. Moreover, the function under the sign of the integral is majorated by a positive integrable function uniformly in z ∈ K ⊂ D and ζ ∈ D, where K is a compact set.
The corresponding generalization of (1) and (2) is formulated as follows (see [6], [7]): when p > 1, and put µ = (ϕ + 1)/ρ. Then each function f ∈ H p α,ρ,γ (D) has the following representation: and In the present paper, we prove an analogous result of the theorem above for harmonic functions from the corresponding weighted L p -spaces in the unit disc D.

Necessary Estimates
First of all, we intend to strengthen some assertions of the Theorem 2.
. Then there exists a positive function Φ(ζ) ∈ L 1 (D) such that uniformly in z ∈ K D and in β and ϕ satisfying the conditions when p = 1 and the conditions when p > 1.
Proof. Repeating the argument from the proof of Theorem 6, we can assume additionally, that β ∈ R, ϕ ∈ R and β > 0.
Since u is a harmonic function, there exists a holomorphic function f in D such that u = Ref in D. Fix an arbitrary z 0 ∈ D and denote f r (ζ) = f (rζ). Obviously, f r ∈ H p α,ρ,γ (D). As u(rζ) = Ref (rζ), from (10) we obtain where χ r (ζ) is the characteristic function of the disc {ζ : |ζ| < r} and I r stands for the right-most integral in the expression above. Assume also that 0 < r 0 ≤ r < 1 for some r 0 . We intend to let r ↑ 1 in the both sides of u(r · z 0 ) = I r .
If p > 1, using Holder inequality, we get Remark. In [9]- [13], one can find various interesting results relating to the weighted integral representations of harmonic functions.