Lipschitz continuity for harmonic functions and solutions of the $\bar{\alpha}$-Poisson equation

We study Lipschitz continuity for solutions of the $\bar{\alpha}$-Poisson equation in planar cases. We also review some recently obtained results. As corolary we can restate results for harmonic and gradient harmonic functions.


Introduction and preliminaries
The weighted Laplacian operator L ρ is defined by L ρ = D z (ρD z ) and L * ρ = D z (ρD z ).If the weight function ρ is chosen to be ρ α (z) = (1 − |z| 2 ) −α (α > −1) in the unit disk U, we call L ρ the standard weighted Laplacian operator and write it by L α for simplicity, and L α will be notation for L * ρ .Our main result is: Theorem 8. Assume that g ∈ C(D) is such that (1 − |z| 2 ) α g is bounded and u ∈ V D→Ω [g] with the representation u(w) = v(w) + G α [g](w).If α > 0, and v is Lipshitz, then u is also Lipshitz continuous on D.
We can restate this result by means of certain solutions to α−Poisson's equation.First we consider some basic properties of α−harmonic mappings.In particular, we improve Chen and Kalaj result [6].Behm [2] found Green function and solved the Dirichlet boundary value problem of the α-Poisson equation.Our method is based on Theorem 7 which gives estimate of the Green potential G α of g and the local C 2 -coordinate method flattering the boundary [23].
At the begining of this paper we will introduce basic notation together with definition of so called α−Laplacian and α−harmonic functions.Also, definition and properties of α−Poisson's kernel and α−Poisson's integral are stated, as a very important technical assets used in our research.More information about this notion can be found in Olofsson's and Wittsten's paper [29].In the proceeding we recall definition of Green function for α−Laplacian which is thoroughly invesigated in Behm [2].Formulation and solution of Dirichlet boundary value problem for α−Poisson's equation, proven in Chen and Kalaj paper [3], is shown in Theorem 1.In paper [7] Chen used this result to prove the boundary characterizations of a Lipschitz continuous αharmonic mappings, and proved Theorem 2.
As an introductory result of this paper, we loose assumption on boundary value of α-harmonic mapping v, which is written in part (d) of Theorem 2 and attain Theorem 5. Proof of this theorem is based on Hardy space technique which can be found in the first author's monography [27], and Theorem 6 proven in first author's and A. Khalfallah's paper [13].The second improvement of Theorem 2 consider the condition on g = −L α u.This result is proven in Theorem 7, and uses various estimates which we establish in Sunsection 2.3.
1.1.α−harmonic mappings.Linear and semilinear equations can be treated together.We take a(x, y)u x + b(x, y)u y = c(x, y, u), where a, b, c are C 1 functions of their arguments.The operator a(x, y)u x + b(x, y)u y on the left hand side of this equation represents differentiation in a direction (a, b) at the point (x, y) in (x, y)-plane.Let us consider a curve, whose tangent at each point (x, y) has the direction (a, b).Coordinates (x(s), y(s)) of a point on this curve satisfy (1) dx/ds = a(x, y), dy/ds = b(x, y) or (2) dy/dx = b(x, y)/a(x, y).
Two complex derivatives ∂ ∂z = D z and ∂ ∂z = D z of u are written by respectively, where z = x + iy.
The weighted Laplacian operator L ρ is defined by If the weight function ρ is chosen to be ρ α (z) = (1 − |z| 2 ) −α in the unit disk U, we call L ρ the standard weighted Laplacian operator and write it by L α for simplicity, here α is a real number with α > −1.It is clear that It can be easily checked that L α u = 0 iff L * α u = 0. Set p = u z and q = u z .Since u z = u z and u z = u z , we find zu z and zu z are conjugate, and also u zz = q z and u zz = q z = q z and therefore u zz and u zz are conjugate.If we set d(z) = 1 − |z| 2 , then ρ α = d −α and by easy computation we find By [38], solutions of yu x − xu y = 0 is u = f (x 2 + y 2 ).Since ρu z = ρg(r)z, we find ρg(r)r 2 = zF (z) = c and hence F = 0 and u z = 0. Thus u = c.
If a function u ∈ C 2 (U) satisfies the α-harmonic equation then we call it an α-harmonic mapping.In the case α = 0, α-harmonic mappings are just Euclidean harmonic mappings.For L α notation ∆ α is also used in the literature.
1.2.α-Poisson's integral.In [29], Olofsson and Wittsten showed that if an α-harmonic function f satisfies lim then it has the form of a Poisson type integral in D, where is the α-harmonic (complex valued) Poisson kernel in D. In the case α = 0 we have classiacal Poisson's kernel for harmonic function and we write it as P instead of P 0 .Also, we write

An introductory result.
As a starting point of our investigation we used Theorem 2 which can be found in Chen's paper [7].This theroem gives some rather strong assumption on g = −L α u (g ∈ C(D)) as well as for boundary values of u (condition (d) of Theorem 2), which are proven to be sufficient for Lipshitz continuity of u.
In order to formulate basic result we need to introduce some preliminary notes.Let V D→Ω [g] denote the family of solutions of v : where g ∈ C(D), f ∈ L 1 (D) is the limit of v(re iθ ) as r tends to 1 − , and v is a sense-preserving diffeomorphism.
For the case wherein the boundary function f vanishes, Behm [2] solved the above Dirichlet boundary value problem of the α−Poisson equation.In paper [3], Chen and Kalaj combined the representation theorem given by Olofsson and Wittsten [29] with the one given by Behm.They obtained the following theorem.
and the Green function G α (z, w) of the adjoint Laplacian L α is given by In [7] Chen provided the boundary characterizations of a Lipschitz continuous α-harmonic mapping as follows Define f (t) = f (e it ) and where z = e it .In order to prove the main result of this paper we will need to prove two refinement of the result above.

Refinement of part (d) in Theorem 2.
As we can see in [13], for p ∈ (0, ∞], the generalized Hardy space H p G (D) consists of all measurable functions from D to C such that M p (r, f ) exists for all r ∈ (0, 1), and f p < ∞, where and The classical Hardy space H p (D) (resp.h p (D)) is the set of all elements of H p G (D) which are analytic (resp.harmonic) on D.
denotes the Hilbert transformations of ψ.
Recall that h(t) = h(e it ).The following property of the Hilbert transform is also sometimes taken as the definition: Note that, if ψ is 2π-periodic, absolutely continuous on [0, 2π] (and therefore Hence, since r ∂ ∂r h is the harmonic conjugate of ∂ ∂θ h, we find Using the above outline we can derive: Let h be gradient harmonic or α-harmonic, α > 0, on U and h ∈ h 1 (U).
The second part of the last theorem is a direct consequence of the first author's result, together with A. Khalfallah, which is stated below.
In this subsection we will prove that instead of g ∈ C(D) we can use assume that g ∈ C(D) can be such that |g This fact will play an important part in the proof of our main result.
The following two estimates can be obtained by direct investigation of the Green function G α , and can be found in [7].
In order to start with our work, we will prove the following two tehnical lemmas.
since the last integral converges, we have desired result.
can easily be veryfied.
The following two lemmas are crucial for the main result of this section: Lemma 3.There exists c 3 > 0 such that for every |w| < 1.
Proof.Using (9) we get Since we can use coordinate change s = w |w| z, we can use Lemma 2 to get our result.
Let ζ = ϕ w (z) = w−z 1−wz .Then, for each w ∈ D, ϕ w is a conformal mapping of D satisfying the following identities: Lemma 4.There exists c 4 > 0 such that Proof.By using supstitution s = w |w| ζ, and s = ρe it we get for some c 4 > 0 which does not depent of 0 r < 1.
We are now ready to formulate the main result of this section, which is generalisation of Lemma 3.4 in Chen's paper [7].The proof of this result follows directly from Lemma 3 and Lemma 4.

Theorem 2 (
[7]).Assume that g ∈ C(D) and u ∈ V D→Ω [g] with the representation u(w) = v(w) + G α [g](w).If α 0, then the following conditions are equivalent: (a) u is a (K, K')-quasiconformal mapping and | ∂ ∂r v| ≤ L on D, and L is a constant.(b) u is Lipschitz continuous with the Euclidean metric.(c) v is Lipschitz continuous with the Euclidean metric.
(d) f is absolutely continuous on T, f ′ ∈ L ∞ and S[f ′ ] is bounded on D.

Theorem 7 .Theorem 8 .
Let g ∈ C(D) be such that |g(z)| M (1 − |z| 2 ) −α , z ∈ D for some M > 0 and let α > 0 be arbitrary.Assume that G α [g] is the Green potential of g given byG α [g](w) = D G α (z, w)g(z) dx dy.Then ∂ ∂w G α [g] and ∂ ∂w G α [g]are both bounded in the unit disc D. As a ditect consequence of Theorem 7 and Theorem 5 we have the main result of this paper.Assume that g ∈ C(D) is such that (1 − |z| 2 ) α g is bounded and u ∈ V D→Ω [g] with the representation u(w) = v(w) + G α [g](w).If α > 0, and v is Lipshitz, then u is also Lipshitz continuous on D.
2Key words and phrases: Poisson kernel, harmonic functions.