On a weighted harmonic Green function and a theorem of Littlewood

Journal Pre-proof On a weighted harmonic Green function and a theorem of Littlewood

Anders Olofsson

PII:

S0007-4497(19)30085-5

DOI:

https://doi.org/10.1016/j.bulsci.2019.102809

Reference:

BULSCI 102809

To appear in:

Bulletin des Sciences Mathématiques

Received date:

12 January 2018

Please cite this article as: A. Olofsson, On a weighted harmonic Green function and a theorem of Littlewood, Bull. Sci. math. (2019), 102809, doi: https://doi.org/10.1016/j.bulsci.2019.102809.

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ON A WEIGHTED HARMONIC GREEN FUNCTION AND A THEOREM OF LITTLEWOOD ANDERS OLOFSSON

Abstract. We continue our study of weighted harmonic functions. The emphasis is put on properties of conformal invariance and associated Green functions. A classical result of Littlewood about pointwise limits almost everywhere of Green potentials of measures in the disc is, as far as the unit disc is concerned, generalized to the full scale of Green potentials considered.

0. Introduction Let Ω be a simply connected planar domain not equal to the whole complex plane C and denote by ∂ and ∂¯ the standard complex partial derivatives over C. Following Paul Garabedian [17] we consider weighted Laplace differential operators of the form Δw,z = ∂z w(z)−1 ∂¯z , z ∈ Ω, where w : Ω → (0, ∞) is a positive weight function. The weight function corresponds to a weighted area element dAw (z) = w(z) dA(z),

z ∈ Ω,

in Ω, where dA(z) = dxdy/π, z = x + iy, is usual planar Lebesgue area measure normalized so that the open unit disc D has unit area. Let ϕ : D → Ω be a Riemann map, that is, a one-to-one analytic function in D such that ϕ(D) = Ω. In view of the change of variables formula it is natural to consider the induced area element ϕ∗ (dAw )(z) = w(ϕ(z))|ϕ (z)|2 dA(z),

z ∈ D,

in D. We suggest to consider weight functions w as above such that (0.1)

ϕ∗ (dAw )(z) = |ϕ (z)|2+α (1 − |z|2 )α dA(z),

z ∈ D,

for some real number α. This latter requirement (0.1) determines the weight function w completely and provides us with a natural scale of weight functions w = wΩ;α in Ω depending on a parameter α ∈ R. Notably, the case α = 0 corresponds to usual unweighted area and the case α = −2 corresponds to hyperbolic area induced by hyperbolic geometry. For Ω = D we obtain in this way the usual standard weight functions wα (z) = wD;α (z) = (1 − |z|2 )α , z ∈ D, Date: November 4, 2019. 2010 Mathematics Subject Classification. Primary: 31A05; Secondary: 35J08. Key words and phrases. Green function, standard weight function, weighted Laplace differential operator, conformal invariance. Research supported by GS Magnuson’s fund of the Royal Swedish Academy of Sciences. 1

2

ANDERS OLOFSSON

for D whereas also other choices of Ω are possible such as the open upper half-plane (see Proposition 1.1). Following Garabedian [17] we consider the weighted Laplace differential operator defined by ΔΩ;α;z = ∂z wΩ;α (z)−1 ∂¯z , z ∈ Ω, for α ∈ R. Notice that ΔΩ;0 = ∂ ∂¯ is one quarter of the usual Laplacian in Ω. Our first main result is that the weighted Laplacians ΔΩ;α defined above have an important property of conformal invariance with respect to weighted compositions with conformal maps. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and consider the weighted composition (0.2)

uϕ,α (z) = ϕ (z)−α/2 (u ◦ ϕ)(z),

z ∈ Ω1 ,

of a, say, twice continuously differentiable function u ∈ C 2 (Ω2 ) in Ω2 . Then (0.3)

ΔΩ1 ;α uϕ,α (z) = S(ϕ (z)−α/2 )|ϕ (z)|2 ((ΔΩ2 ;α u) ◦ ϕ)(z)

z is the reflection in the unit circle (see Theorems 1.1 for z ∈ Ω1 , where S(z) = 1/¯ and 1.2). This formula (0.3) generalizes a well-known invariance property of the usual Laplacian to a weighted context. A function u is called α-harmonic in Ω if u ∈ C 2 (Ω) and u satisfies the homogeneous partial differential equation ΔΩ;α u = 0 in Ω. Notice that in this language a 0-harmonic function in Ω is a harmonic function in Ω in the usual sense. By regularity theory we know that an α-harmonic function u in Ω is indefinitely differentiable in Ω, that is, u ∈ C ∞ (Ω). Notice that if u is α-harmonic in Ω2 and v = uϕ,α is as in (0.2), then v is α-harmonic in Ω1 . In order for the class of α-harmonic functions to have a sufficiently rich supply of functions with well-behaved boundary values we need to restrict the parameter range to α > −1 (see Theorem 3.1). The recent study of α-harmonic functions originate from Olofsson and Wittsten [29]; see [6, 9, 10, 11, 25] for other contributions. A related interesting study is Borichev and Hedenmalm [7]. We mention here also the paper Shimorin [33] concerned with another family of conformally invariant operators. In a discussion of conformal invariance a special role is played by the Bergman kernel function. Let A2 (Ω) be the space of analytic functions f in Ω with finite norm  f 2 =

Ω

|f (z)|2 dA(z).

The space A2 (Ω) is a Hilbert space of analytic functions in Ω known as the Bergman space. The Bergman kernel KΩ is the reproducing kernel function for A2 (Ω). In plain language this means that KΩ : Ω×Ω → C is the function uniquely determined by the properties that KΩ (·, ζ) ∈ A2 (Ω) and f (ζ) = f, KΩ (·, ζ) ,

f ∈ A2 (Ω),

for every ζ ∈ Ω, where ·, · is the scalar product of A2 (Ω). It is well-known that the Bergman kernels have the conformal invariance property that KΩ1 (z, ζ) = ϕ (ζ)ϕ (z)KΩ2 (ϕ(z), ϕ(ζ))

ON A WEIGHTED HARMONIC GREEN FUNCTION

3

for (z, ζ) ∈ Ω21 whenever ϕ : Ω1 → Ω2 is a biholomorphic map. We remind also that KΩ is positive on the diagonal and non-vanishing in Ω × Ω when Ω is simply connected (see for instance Krantz [24, Chapter 1]). The pseudo hyperbolic metric for Ω is the function ρΩ : Ω × Ω → [0, 1) defined by ρΩ (z, ζ) = |ϕ(z)|, where ϕ : Ω → D is a biholomorphic map such that ϕ(ζ) = 0. For α > −1 we consider the Green function GΩ;α for the differential operator ΔΩ;α in Ω. We show that (0.4)

GΩ;α (z, ζ) = KΩ (z, ζ)−α/2 hα (ρΩ (z, ζ)2 )

for (z, ζ) ∈ Ω2 outside the diagonal, where KΩ is the Bergman kernel for Ω, the function ρΩ : Ω × Ω → [0, 1) is the pseudo hyperbolic metric for Ω and  1 (1 − t)α dt hα (x) = − t x for 0 < x < 1 (see Theorem 5.1). The power in (0.4) is defined using a logarithm of KΩ which is real on the diagonal. A special case worthy of particular mention is that of the open unit disc D when formula (0.4) for the Green function Gα = GD;α simplifies to 2   ¯ α hα  ζ − z  (0.5) Gα (z, ζ) = (1 − ζz) ¯ 1 − ζz for (z, ζ) ∈ D2 outside the diagonal (see Corollary 5.1). This latter formula (0.5) for Gα is known and was discovered earlier by Gustav Behm [6]. Another special case worth mention is that of the open upper half-plane (see Corollary 5.2). Notice that the function h0 above is the usual logarithm so that contained in (0.4) is also the standard formula GΩ;0 (z, ζ) = log(ρΩ (z, ζ)2 ) for the Green function for the usual Laplacian ΔΩ;0 = ∂ ∂¯ in Ω. Conformal invariance properties of the Bergman kernel and the pseudo hyperbolic metric leads by (0.4) to the conformal invariance formula GΩ1 ;α (z, ζ) = ϕ (ζ)−α/2 ϕ (z)−α/2 GΩ2 ;α (ϕ(z), ϕ(ζ)) for the Green function, where ϕ : Ω1 → Ω2 is a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) (see Theorem 5.2). Here the power (ϕ )−α/2 is defined in the usual way using some logarithm of ϕ . Powers of the Bergman kernel function are much natural from the point of view of kernel functions for Hilbert spaces of analytic functions. For α > −1 we denote by Aα (Ω) the space of all analytic functions f in Ω with finite norm  2 f Ω;α = (1 + α) |f (z)|2 wΩ;α (z) dA(z). Ω

The space Aα (Ω) is a Hilbert space of analytic functions in Ω in the usual sense of bounded point evaluations. Notice that the space A0 (Ω) = A2 (Ω) is the unweighted Bergman space discussed above. We denote by KΩ;α the reproducing kernel function for the space Aα (Ω). We show that KΩ;α (z, ζ) = KΩ (z, ζ)1+α/2 for (z, ζ) ∈ Ω2 , where KΩ is the Bergman kernel function for Ω (see Theorem 2.2). This latter formula for the kernel function KΩ;α makes a good case that the scale of spaces Aα (Ω) forms a natural analogue for Ω of the standard weighted Bergman

4

ANDERS OLOFSSON

spaces of the open unit disc. We refer to Hedenmalm, Korenblum and Zhu [21] for background on Bergman spaces of the unit disc. We consider also the associated Green potential  GΩ;α [μ](z) = GΩ;α (z, ζ) dμ(ζ) Ω

of a complex Radon measure μ ∈ D0 (Ω) in Ω such that  (0.6) (1 − ρΩ (z1 , ζ)2 )α/2+1 wΩ;α/2 (ζ) d|μ|(ζ) < +∞ Ω

for some z1 ∈ Ω, where |μ| is the total variation measure of μ. The condition (0.6) does not depend on the choice of point z1 ∈ Ω (see Corollary 3.1) and is designed in order to ensure that the above integral GΩ;α [μ](z) exists in the usual Lebesgue sense for quasi every z ∈ Ω (see Section 6). We refer to condition (0.6) on μ saying that the Green potential GΩ;α [μ] exists. For such μ we show that the Green potential u = GΩ;α [μ] satisfies the inhomogeneous Dirichlet problem  ΔΩ;α u = μ in Ω, (0.7) u = 0 on ∂Ω, in a suitable sense (see Theorem 6.4). The first equation in (0.7) is interpreted in a standard distributional sense that ΔΩ;α u = μ in D (Ω), where D (Ω) is the space of distributions in Ω. The boundary condition in (0.7) is interpreted in a certain conformally invariant L1 sense that lim uΩ,a;α;r = 0

r→1

for a ∈ Ω, where the semi-norms ·Ω,a;α;r are as in (3.9). Credit here is again due to Gustav Behm [6] who established (0.7) in the important case of the open unit disc D. Part of our analysis of (0.7) pertains to related uniqueness questions (see Theorems 3.3 and 6.5). In this context we mention also a recent study of α-harmonic functions in the open upper half-plane by Carlsson and Wittsten [9]. The novelty of ours concern the case of a general planar simply connected domain Ω = C. In the final part of the paper we turn to a study of pointwise boundary limits of Green potentials in the case of the open unit disc Ω = D. Let T = ∂D be the unit circle. For the sake of simplicity of notation we write Gα = GD;α for the Green function for D given by (0.5). The Green potential Gα [μ] exists if and only if μ ∈ D0 (D) is a complex Radon measure in D such that  (1 − |ζ|2 )α+1 d|μ|(ζ) < +∞, D

where |μ| is the total variation measure of μ. We introduce a notion of regular ˜ in D and show that almost boundary point eiθ0 ∈ T for a finite complex measure μ every eiθ0 ∈ T is such a regular boundary point (see Section 7). We then prove that (0.8)

lim Gα [μ](reiθ0 ) = 0

r→1

whenever Gα [μ] exists and eiθ0 ∈ T is a regular boundary point for the measure d˜ μ(ζ) = (1 − |ζ|2 )α+1 dμ(ζ),

ζ∈D

(see Theorem 9.2). As a consequence, the limit (0.8) holds for almost every eiθ0 ∈ T. It should be noticed that a Green potential Gα [μ] has no angular boundary limits

ON A WEIGHTED HARMONIC GREEN FUNCTION

5

in general as is the case for α-harmonic functions in D (see [29, Section 6]) or the related transforms Bα [μ] (see Theorem 9.1). Part of our analysis of boundary limits almost everywhere concerns a study of a convolution inequality on T involving the maximal function (see Theorem 8.1). Our approach here is based on a suggestion in the 1968 edition of Katznelson’s classical book [23, Lemma III.2.4]. The study of pointwise limits almost everywhere of Green potentials goes back to a classical result of Littlewood [26] and has attracted considerable attention over the years, see for instance [2, 3, 12, 27, 31] or [13, Chapter XII] in the references. In fact, a starting point of ours was the adaption of Littlewood’s classical work [26] to the current state of affairs of the weighted Green functions Gα . The author thanks the referee for valuable comments. 1. Standard weights and invariance of Laplacians For the sake of convenience of exposition we start with a standard invariance property of automorphisms of the unit disc. We denote by Aut(D) the group of all analytic functions in D mapping D one-to-one onto itself. Lemma 1.1. Let ϕ ∈ Aut(D). Then ϕ (ζ)ϕ (z) (1 −

ϕ(ζ)ϕ(z))2

=

1 ¯ 2 (1 − ζz)

2

for (z, ζ) ∈ D . 

Proof. See for instance [16, Chapter IX]. The function (1.1)

KD (z, ζ) =

1 ¯ 2, (1 − ζz)

(z, ζ) ∈ D2 ,

appearing in Lemma 1.1 is the Bergman kernel for the unit disc relative to normalized area measure dA. More generally, for a simply connected planar domain Ω not equal to the whole plane C, we set (1.2)

KΩ (z, ζ) =

ϕ (ζ)ϕ (z) (1 − ϕ(ζ)ϕ(z))2

,

(z, ζ) ∈ Ω2 ,

where ϕ is a one-to-one analytic function in Ω such that ϕ(Ω) = D. From Lemma 1.1 we have that the function KΩ : Ω2 → C does not depend on the choice of biholomorphic map ϕ : Ω → D. From its definition (1.2) we have the symmetry formula (1.3)

KΩ (z, ζ) = KΩ (ζ, z),

(z, ζ) ∈ Ω2 .

The function KΩ : Ω × Ω → C is known as the Bergman kernel function for Ω relative to normalized area measure dA. Lemma 1.2. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2). Then KΩ1 (z, ζ) = ϕ (ζ)ϕ (z)KΩ2 (ϕ(z), ϕ(ζ)) for (z, ζ) ∈ Ω21 .

6

ANDERS OLOFSSON

Proof. Let ϕ2 be a one-to-one analytic function in Ω2 such that ϕ(Ω2 ) = D. Now the composition ϕ1 = ϕ2 ◦ ϕ is a one-to-one analytic function in Ω1 such that ϕ1 (Ω1 ) = D, and we have that KΩ1 (z, ζ) =

ϕ1 (ζ)ϕ1 (z) (1 − ϕ1 (ζ)ϕ1 (z))2

= ϕ (ζ)ϕ (z)

ϕ2 (ϕ(ζ))ϕ2 (ϕ(z)) (1 − ϕ2 (ϕ(ζ))ϕ2 (ϕ(z)))2



= ϕ (ζ)ϕ (z)KΩ2 (ϕ(z), ϕ(ζ)) for (z, ζ) ∈ Ω21 , where the mid equality follows by the chain rule.



Let Ω = C be a simply connected planar domain. Since the function KΩ (·, ζ) is non-vanishing in Ω it has a logarithm in Ω (see for instance [32, Theorem 13.11]). We choose this logarithm such that log KΩ (ζ, ζ) is real, and define powers KΩ (z, ζ)β in the usual way. Lemma 1.3. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and let β ∈ R. Then KΩ1 (z, ζ)β = ϕ (ζ)β ϕ (z)β KΩ2 (ϕ(z), ϕ(ζ))β for (z, ζ) ∈ Ω21 . As a consequence, the product ϕ (ζ)β ϕ (z)β does not depend on the particular choice of logarithm used to define the power (ϕ )β . Proof. Let be an analytic function in Ω1 such that e(z) = ϕ (z) for z ∈ Ω1 . Fix ζ ∈ Ω1 . By Lemma 1.2 we have that elog KΩ1 (z,ζ) = e(ζ)+(z) elog KΩ2 (ϕ(z),ϕ(ζ)) for z ∈ Ω1 . By this equality it follows that there exists an integer k ∈ Z such that log KΩ1 (z, ζ) = (ζ) + (z) + log KΩ2 (ϕ(z), ϕ(ζ)) + 2πik for z ∈ Ω1 . Setting z = ζ we see that k = 0. By a passage to powers we now have that KΩ1 (z, ζ)β = eβ log KΩ1 (z,ζ) = eβ(ζ) eβ(z) eβ log KΩ2 (ϕ(z),ϕ(ζ)) = ϕ (ζ)β ϕ (z)β KΩ2 (ϕ(z), ϕ(ζ))β for z ∈ Ω1 . This completes the proof of the lemma.



We now define the standard weight functions. Definition 1.1. Let Ω = C be a simply connected planar domain. The standard weight for Ω with weight parameter α ∈ R is the function wΩ;α : Ω → (0, ∞) defined by wΩ;α (z) = KΩ (z, z)−α/2 , z ∈ Ω, where KΩ is the Bergman kernel function for Ω. It is evident from formula (1.1) that wα (z) = wD;α (z) = (1 − |z|2 )α ,

z ∈ D,

for α ∈ R are the usual standard weights for D. Let us calculate also the standard weights for the open upper half-plane (1.4)

H = {z ∈ C : (z) > 0},

where (z) denotes the imaginary part.

ON A WEIGHTED HARMONIC GREEN FUNCTION

7

Proposition 1.1. The standard weights for the open upper half-plane have the form wH;α (z) = 2α ( (z))α , z ∈ H, for α ∈ R. Proof. We calculate the kernel function KH . The M¨obius transformation ϕ(z) = (z − i)/(z + i) maps H one-to-one onto D. From its definition (1.2) we have that KH (z, ζ) =

ϕ (ζ)ϕ (z) (1 −

ϕ(ζ)ϕ(z))2

1 =− ¯ , (ζ − z)2

(z, ζ) ∈ H2 ,

where the last equality is straightforward to check. This yields the conclusion of the proposition.  Standard weights in the upper half-plane have been considered recently by several different authors. See for instance Carlsson and Wittsten [9] or Montes-Rodriguez with collaborators [14]. For the sake of easy reference we record the following invariance property of standard weights. Corollary 1.1. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and α ∈ R. Then |ϕ (z)|α wΩ1 ;α (z) = wΩ2 ;α (ϕ(z)) for z ∈ Ω1 . Proof. The result follows from Lemma 1.3 by passage to diagonal restriction.



Following Garabedian [17] we consider weighted Laplacians of the form ΔΩ;α;z = ∂z wΩ;α (z)−1 ∂¯z ,

z ∈ Ω,

where wΩ;α : Ω → (0, ∞) is a standard weight function and α ∈ R (see Definition 1.1). A novelty here lies in the particular choice of weight function w = wΩ;α . Let ϕ : Ω1 → Ω2 be as in Corollary 1.1 and α ∈ R. We shall consider weighted compositions of the form (1.5)

uϕ,α (z) = ϕ (z)−α/2 (u ◦ ϕ)(z),

z ∈ Ω1 ,

of, say, u ∈ C 2 (Ω2 ), where the power (ϕ )−α/2 is defined using some branch of the logarithm log(ϕ ) in Ω1 . We denote by (1.6)

S(z) = 1/¯ z,

z ∈ C \ {0},

the reflection in the unit circle. Theorem 1.1. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and α ∈ R. Let u ∈ C 2 (Ω2 ) and consider the function uϕ,α ∈ C 2 (Ω1 ) defined as in (1.5). Then ΔΩ1 ;α uϕ,α (z) = S(ϕ (z)−α/2 )|ϕ (z)|2 ((ΔΩ2 ;α u) ◦ ϕ)(z) for z ∈ Ω1 , where S : C \ {0} → C \ {0} is given by (1.6).

8

ANDERS OLOFSSON

Proof. Differentiating using the chain rule we have that  (z) = ϕ (z)−α/2 ϕ (z)−α/2 ¯ ¯ ϕ,α (z) = ϕ (z)−α/2 ∂u(ϕ(z))ϕ ∂u

ϕ (z) ϕ (z)−α/2

¯ ∂u(ϕ(z))

ϕ (z)

= |ϕ (z)|−α

¯ ∂u(ϕ(z)) ϕ (z)−α/2 for z ∈ Ω1 . We now use Corollary 1.1 to conclude that ¯ ϕ,α (z) = ∂u

wΩ1 ;α (z) ϕ (z) ¯ ∂u(ϕ(z)) wΩ2 ;α (ϕ(z)) ϕ (z)−α/2

for z ∈ Ω1 . Thus 1 wΩ1 ;α (z)

¯ ϕ,α (z) = ∂u

ϕ (z)



1

ϕ (z)−α/2 wΩ2 ;α

 ¯ (ϕ(z)), ∂u

and another differentiation gives that ΔΩ1 ;α uϕ,α (z) =

ϕ (z) ϕ (z)−α/2

(ΔΩ2 ;α u)(ϕ(z))ϕ (z)

= S(ϕ (z)−α/2 )|ϕ (z)|2 (ΔΩ2 ;α u)(ϕ(z)) for z ∈ Ω1 . This completes the proof of the theorem.



A locally integrable function u ∈ L1loc (Ω) in Ω is identified with the distribution  u(z)ψ(z) dA(z), ψ ∈ D(Ω), (1.7) u, ψ = Ω

where D(Ω) = Cc∞ (Ω) is the space of C ∞ -smooth test functions with compact support contained in Ω and dA is usual planar Lebesgue area measure normalized so that the open unit disc D has unit area. Let us recall the notion of pull-back of a distribution. Let ϕ : Ω1 → Ω2 be an analytic function with non-vanishing derivative. For a locally integrable function u ∈ L1loc (Ω2 ) in Ω2 we set (u ◦ ϕ)(z) = u(ϕ(z)),

z ∈ Ω1 .

By the change of variables formula this provides us with a map ϕ∗ (u) = u ◦ ϕ from L1loc (Ω2 ) into L1loc (Ω1 ). It is known that this map ϕ∗ extends by continuity to a continuous map ϕ∗ : D (Ω2 ) → D (Ω1 ) acting on distributions. Notably, if ormander limj→∞ uj = u in D (Ω2 ), then limj→∞ ϕ∗ (uj ) = ϕ∗ (u) in D (Ω1 ). See H¨ [22, Section 6.1] for details. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and α ∈ R. For u ∈ D (Ω2 ) we consider the distribution (1.8)

uϕ,α = (ϕ )−α/2 (u ◦ ϕ)

in D (Ω1 ),

where the composition and product are defined in a distributional sense. Notice that (1.8) generalizes (1.5) to a distributional setting. Theorem 1.2. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and α ∈ R. Let u ∈ D (Ω2 ) and consider the distribution uϕ,α ∈ D (Ω1 ) defined by (1.8). Then ΔΩ1 ;α uϕ,α = S((ϕ )−α/2 )|ϕ |2 ((ΔΩ2 ;α u) ◦ ϕ)

in D (Ω1 ),

ON A WEIGHTED HARMONIC GREEN FUNCTION

9

where S : C \ {0} → C \ {0} is given by (1.6). Proof. Choose functions u(j) ∈ C 2 (Ω2 ) for j ≥ 1 such that u = limj→∞ u(j) in D (Ω2 ) (see [22, Theorem 4.1.5]). From Theorem 1.1 we know that  −α/2 ΔΩ1 ;α u(j) )|ϕ |2 ((ΔΩ2 ;α u(j) ) ◦ ϕ) ϕ,α = S((ϕ )

in D (Ω1 ) for j ≥ 1. By continuity properties of distributions we have that  −α/2 ΔΩ1 ;α uϕ,α = lim ΔΩ1 ;α u(j) )|ϕ |2 ((ΔΩ2 ;α u(j) ) ◦ ϕ) ϕ,α = lim S((ϕ ) j→∞

j→∞

 −α/2

= S((ϕ )

 2

)|ϕ | ((ΔΩ2 ;α u) ◦ ϕ)



in D (Ω1 ).



Remark 1.1. We mention that the results of Theorems 1.1 and 1.2 hold locally. Indeed, this is evident by inspection of the proof or a standard cut-off argument. Let ϕ : Ω1 → Ω2 be as in Theorem 1.2 and α ∈ R. For u ∈ D (Ω2 ) we consider the distribution  −α/2 u∨ )|ϕ |2 (u ◦ ϕ) ϕ,α = S((ϕ )

(1.9)

in D (Ω1 ),

where S : C \ {0} → C \ {0} is the reflection map (1.6). In this language the result of Theorem 1.2 says that ΔΩ1 ;α uϕ,α = (ΔΩ2 ;α u)∨ ϕ,α

in D (Ω1 )

for u ∈ D (Ω2 ). Thus Theorem 1.2 restates in the form of a commutative diagram D (Ω2 ) ↓ D (Ω2 )

ΔΩ2 ;α

D (Ω1 ) ↓ D (Ω1 )

→ →

ΔΩ1 ;α

where the upper and lower arrows are the weighted pull-backs defined by (1.8) and (1.9), respectively. We next notice that the operations (1.5) and (1.9) are in a sense dual. Proposition 1.2. Let ϕ : Ω1 → Ω2 be as in Theorem 1.2 and α ∈ R. Let ψj ∈ D(Ωj ) (j = 1, 2) be test functions. Then   ψ1 (z)(ψ2 )ϕ,α (z) dA(z) = (ψ1 )∨ ϕ−1 ,α (z)ψ2 (z) dA(z), Ω1

Ω2



where the logarithms of ϕ and (ϕ 

log ϕ (ϕ

−1 

−1

) are chosen such that

(ζ)) + log(ϕ−1 ) (ζ) = 0

for ζ ∈ Ω2 . Proof. Notice first that   ψ1 (z)(ψ2 )ϕ,α (z) dA(z) = I= Ω1

Ω1

ψ1 (z)ϕ (z)−α/2 ψ2 (ϕ(z)) dA(z).

We now make the change of variables ζ = ϕ(z) to conclude that  I= ϕ (ϕ−1 (ζ))−α/2 |(ϕ−1 ) (ζ)|2 ψ1 (ϕ−1 (ζ))ψ2 (ζ) dA(ζ). Ω2

By the choice of logarithms we have that ϕ (ϕ−1 (ζ))−α/2 = (ϕ−1 ) (ζ)α/2 = S((ϕ−1 ) (ζ)−α/2 )

10

ANDERS OLOFSSON

for ζ ∈ Ω2 . Thus

 I= Ω2

(ψ1 )∨ ϕ−1 ,α (z)ψ2 (z) dA(z), 

which yields the conclusion of the proposition.

Notice that Proposition 1.2 gives the action on test functions of the operations (1.8) and (1.9). 2. Powers of the Bergman kernel Powers of the Bergman kernel arise as kernel functions for natural Hilbert spaces of analytic functions. The purpose of this section is to indicate briefly such a connection from the point of view of Bergman spaces. Let Ω = C be a simply connected planar domain and α > −1. We denote by Aα (Ω) the space of all analytic functions f in Ω with finite norm  f 2Ω;α = (1 + α) |f (z)|2 wΩ;α (z) dA(z), Ω

where wΩ;α is a standard weight function (see Definition 1.1). The space Aα (Ω) is a Hilbert space of analytic functions in Ω in the usual sense of bounded point evaluations. Notice that the space A0 (Ω) is the unweighted Bergman space discussed in the introduction. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2). We shall consider weighted compositions of the form (2.1)

Cϕα f (z) = ϕ (z)1+α/2 f (ϕ(z)),

z ∈ Ω1 ,

of functions f ∈ H(Ω2 ) analytic in Ω2 , where the power is defined using some logarithm of ϕ . Lemma 2.1. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and α > −1. Then Cϕα f 2Ω1 ;α = f 2Ω2 ;α ,

f ∈ Aα (Ω2 ),

where the operation Cϕα is defined by (2.1). Proof. Let f ∈ Aα (Ω2 ). Notice that  α 2 Cϕ f Ω1 ;α = (1 + α) |ϕ (z)|2+α |f (ϕ(z))|2 wΩ1 ;α (z) dA(z). Ω1

By the change of variables ζ = ϕ(z) we have that  Cϕα f 2Ω1 ;α = (1 + α) |ϕ (ϕ−1 (ζ))|2+α |f (ζ)|2 wΩ1 ;α (ϕ−1 (ζ))|(ϕ−1 ) (ζ)|2 dA(ζ). Ω2

We now use Corollary 1.1 to see that  α 2 Cϕ f Ω1 ;α = (1 + α) |ϕ (ϕ−1 (ζ))|2+α |(ϕ−1 ) (ζ)|2+α |f (ζ)|2 wΩ2 ;α (ζ) dA(ζ) Ω2  |f (ζ)|2 wΩ2 ;α (ζ) dA(ζ) = f 2Ω2 ;α , = (1 + α) Ω2

where the mid equality follows by the chain rule.



ON A WEIGHTED HARMONIC GREEN FUNCTION

11

In the language of operator theory Lemma 2.1 says that the operation Cϕα induces an isometry from the space Aα (Ω2 ) into the space Aα (Ω1 ). We next consider products of weighted composition operators. Proposition 2.1. Let ϕ1 : Ω1 → Ω2 and ϕ2 : Ω2 → Ω3 be biholomorphic maps between simply connected planar domains Ωj = C (j = 1, 2, 3) and α ∈ R. Set ϕ = ϕ2 ◦ ϕ1 . Then Cϕα f = Cϕα1 Cϕα2 f, f ∈ H(Ω3 ), provided the logarithms of ϕ , ϕ1 and ϕ2 are chosen such that log ϕ (z) = log ϕ2 (ϕ1 (z)) + log ϕ1 (z) for z ∈ Ω1 . Proof. Let f ∈ H(Ω3 ) and observe that Cϕα1 Cϕα2 f (z) = ϕ1 (z)1+α/2 Cϕα2 f (ϕ1 (z)) = ϕ1 (z)1+α/2 ϕ2 (ϕ1 (z))1+α/2 f (ϕ2 (ϕ1 (z))) for z ∈ Ω1 . By choice of logarithms we have that ϕ1 (z)1+α/2 ϕ2 (ϕ1 (z))1+α/2 = ϕ (z)1+α/2 for z ∈ Ω1 . Thus Cϕα1 Cϕα2 f (z) = ϕ (z)1+α/2 f (ϕ(z)) = Cϕα f (z) for z ∈ Ω1 .



We denote by L(H, K) the space of all bounded linear operators from a Hilbert space H into a Hilbert space K. As a consequence of Proposition 2.1 we have that the isometry Cϕα in L(Aα (Ω2 ), Aα (Ω1 )) from Lemma 2.1 is surjective. The kernel function for the space Aα (Ω) is the function KΩ;α : Ω × Ω → C defined by the properties that KΩ;α (·, ζ) ∈ Aα (Ω) and f (ζ) = f, KΩ;α (·, ζ) Ω;α ,

f ∈ Aα (Ω),

for every ζ ∈ Ω, where ·, · Ω;α is the scalar product of Aα (Ω). It is well-known that kernel functions have the symmetry property that KΩ;α (z, ζ) = KΩ;α (ζ, z) for z, ζ ∈ Ω (see Aronszajn [4, Section I.2]). Proposition 2.2. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and α > −1. Then (Cϕα )∗ KΩ1 ;α (·, ζ) = ϕ (ζ)1+α/2 KΩ2 ;α (·, ϕ(ζ)) for ζ ∈ Ω1 , where (Cϕα )∗ is the adjoint of the operator Cϕα ∈ L(Aα (Ω2 ), Aα (Ω1 )). Proof. Let f = (Cϕα )∗ KΩ1 ;α (·, ζ) ∈ Aα (Ω2 ), where ζ ∈ Ω1 . By the reproducing property of kernel functions we have that f (z) = f, KΩ2 ;α (·, z) Ω2 ;α = KΩ1 ;α (·, ζ), Cϕα KΩ2 ;α (·, z) Ω1 ;α for z ∈ Ω2 . Passing to the complex conjugate we have that f (z) = Cϕα KΩ2 ;α (·, z), KΩ1 ;α (·, ζ) Ω1 ;α = ϕ (ζ)1+α/2 KΩ2 ;α (ϕ(ζ), z) for z ∈ Ω2 , where the last equality follows by the reproducing property of kernel functions. The result now follows by the symmetry property of kernel functions. 

12

ANDERS OLOFSSON

We now turn to conformal invariance of the kernel functions KΩ;α . Theorem 2.1. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected  C (j = 1, 2) and α > −1. Then planar domains Ωj = KΩ1 ;α (z, ζ) = ϕ (ζ)1+α/2 ϕ (z)1+α/2 KΩ2 ;α (ϕ(z), ϕ(ζ)) for z, ζ ∈ Ω1 . Proof. From the reproducing property of kernel functions we have that KΩ1 ;α (z, ζ) = KΩ1 ;α (·, ζ), KΩ1 ;α (·, z) Ω1 ;α for z, ζ ∈ Ω1 . From Lemma 2.1 and Proposition 2.1 we have that the operator (Cϕα )∗ in L(Aα (Ω1 ), Aα (Ω2 )) is an isometry. Thus KΩ1 ;α (z, ζ) = (Cϕα )∗ KΩ1 ;α (·, ζ), (Cϕα )∗ KΩ1 ;α (·, z) Ω2 ;α = ϕ (ζ)1+α/2 KΩ2 ;α (·, ϕ(ζ)), ϕ (z)1+α/2 KΩ2 ;α (·, ϕ(z)) Ω2 ;α for z, ζ ∈ Ω1 , where the last equality follows by Proposition 2.2. We now again use the reproducing property of kernel functions to conclude that KΩ1 ;α (z, ζ) = ϕ (ζ)1+α/2 ϕ (z)1+α/2 KΩ2 ;α (·, ϕ(ζ)), KΩ2 ;α (·, ϕ(z)) Ω2 ;α = ϕ (ζ)1+α/2 ϕ (z)1+α/2 KΩ2 ;α (ϕ(z), ϕ(ζ)) 

for z, ζ ∈ Ω1 . We are now ready for the calculation of KΩ;α .

Theorem 2.2. Let Ω = C be a simply connected planar domain and α > −1. Then KΩ;α (z, ζ) = KΩ (z, ζ)1+α/2 for z, ζ ∈ Ω, where KΩ is the Bergman kernel for Ω. Proof. Let ϕ : Ω → D be a biholomorphic map. From Theorem 2.1 we have that KΩ;α (z, ζ) = ϕ (ζ)1+α/2 ϕ (z)1+α/2 KD;α (ϕ(z), ϕ(ζ)) for z, ζ ∈ Ω. From [21, Section 1.1] we know that KD;α (z, ζ) =

1 ¯ 2+α , (1 − ζz)

z, ζ ∈ D.

We now have that KΩ;α (z, ζ) = ϕ (ζ)1+α/2 ϕ (z)1+α/2

1 (1 − ϕ(ζ)ϕ(z))2+α

= KΩ (z, ζ)1+α/2

for z, ζ ∈ Ω, where the last equality follows by formula (1.2).



In this context of weighted composition operators acting on Bergman spaces the contribution Shimorin [34] should be mentioned; see also Hedenmalm [20]. Recall from complex analysis that ϕ ∈ Aut(D) if and only if it has the form ϕ(z) = cϕζ (z),

z ∈ D,

for some c ∈ T and ζ ∈ D, where (2.2)

¯ ϕζ (z) = (ζ − z)/(1 − ζz)

ON A WEIGHTED HARMONIC GREEN FUNCTION

13

(see for instance Gamelin [16, Section IX.2]). The function ϕζ in (2.2) is selfinvolutive in the sense that ϕζ ◦ ϕζ is the identity, that is, (ϕζ ◦ ϕζ )(z) = z for z ∈ D. We record also the standard formula (1 − |ζ|2 )(1 − |z|2 ) (2.3) 1 − |ϕζ (z)|2 = ¯ 2 |1 − ζz| for ϕζ of the form (2.2). 3. Uniqueness of α-harmonic functions In this section we shall present some criteria for uniqueness of solution for the Dirichlet problem for α-harmonic functions. We start with the case of the open unit disc. For a suitably smooth function u in D we consider functions ur on T defined by ur (eiθ ) = u(reiθ ),

(3.1)

eiθ ∈ T,

for 0 < r < 1. We denote by D (T) the space of distributions on T. An integrable function f ∈ L1 (T) is identified with the distribution  π 1 f, ψ = f (eiθ )ψ(eiθ ) dθ, ψ ∈ C ∞ (T), 2π −π where C ∞ (T) is the space of C ∞ -smooth test functions on T. Let us also denote by φk the exponential monomial φk (eiθ ) = eikθ ,

eiθ ∈ T,

for k ∈ Z. Theorem 3.1. Let α ≤ −1 and let u be α-harmonic in D. Assume that the limit limr→1 ur exists in D (T), where ur is as in (3.1). Then u is analytic in D. Proof. By [29, Theorem 1.2] we know that the function u has the form ∞ ∞  1    c−k tk−1 (1 − t|z|2 )α dt z¯k + ck z k , z ∈ D, (3.2) u(z) = k=1

for some sequence

0

{ck }∞ k=−∞

k=0

of complex numbers such that lim sup|ck |1/|k| ≤ 1. |k|→∞

A calculation using (3.2) and orthogonality of exponential monomials shows that  π  1 1 (3.3) ur , φk = u(reiθ )eikθ dθ = c−k rk tk−1 (1 − tr2 )α dt 2π −π 0 for k ≥ 1 and 0 < r < 1. By monotone convergence we have that  1  1 (3.4) lim tk−1 (1 − tr2 )α dt = tk−1 (1 − t)α dt = +∞ r→1

0

0

for k ≥ 1. By assumption we have that the limits limr→1 ur , φk all exist. In view of (3.3) and (3.4) the existence of this latter limit forces that c−k = 0 for k ≥ 1. From (3.2) we conclude that u is analytic in D.  See Carlsson and Wittsten [9, Theorem 1.5] or Olofsson [28, Theorem 2.3] for other results in a similar vein as Theorem 3.1. We now consider functions with vanishing boundary value.

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ANDERS OLOFSSON

Theorem 3.2. Let α ∈ R and let u be α-harmonic in D. Assume that limr→1 ur = 0 in D (T), where ur is as in (3.1). Then u(z) = 0 for z ∈ D. Proof. By Theorem 3.1 we can assume that α > −1. For α > −1 the result follows by [29, Theorem 5.3].  We shall deduce uniqueness results for α-harmonic functions in simply connected domains Ω = C by applying Theorem 3.2 to weighted pullbacks of the form uϕ,α . First we need some preparation. It is well-known that the quantity  z−ζ    (3.5) ρD (z, ζ) =  ¯ , z, ζ ∈ D, 1 − ζz is symmetric, satisfies the triangle inequality and is invariant under automorphisms of D in the sense that (3.6)

ρD (ϕ(z), ϕ(ζ)) = ρD (z, ζ)

for ϕ ∈ Aut(D) (see [16, Chapter IX]). Consider now a simply connected planar domain Ω = C. We set  ϕ(z) − ϕ(ζ)    (3.7) ρΩ (z, ζ) =  , z, ζ ∈ Ω, 1 − ϕ(ζ)ϕ(z) where ϕ is a one-to-one analytic function in Ω such that ϕ(Ω) = D. From (3.6) we have that the function ρΩ : Ω × Ω → [0, 1) is well-defined in the sense that (3.7) does not depend on the particular choice of biholomorphic map ϕ : Ω → D. The following result is well-known but included here for the sake of convenience. Proposition 3.1. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2). Then ρΩ1 (z, ζ) = ρΩ2 (ϕ(z), ϕ(ζ)) for z, ζ ∈ Ω1 . Proof. Let ϕ2 : Ω2 → D be a biholomorphic map. By (3.7) we have that ρΩ2 (ϕ(z), ϕ(ζ)) = ρD (ϕ2 (ϕ(z)), ϕ2 (ϕ(ζ))) for z, ζ ∈ Ω1 . Now ϕ2 ◦ ϕ : Ω1 → D is a biholomorphic map mapping Ω1 one-to-one onto D and another application of (3.7) gives that ρΩ2 (ϕ(z), ϕ(ζ)) = ρΩ1 (z, ζ) 

for z, ζ ∈ Ω1 .

The above construction provides us with a conformally invariant metric ρΩ in Ω known as the pseudo hyperbolic metric for Ω. The corresponding discs (3.8)

D(ζ, r) = {z ∈ Ω : ρΩ (z, ζ) < r}

are known as pseudo hyperbolic discs in Ω. We set T(ζ, r) = ∂D(ζ, r) for ζ ∈ Ω and 0 < r < 1.

ON A WEIGHTED HARMONIC GREEN FUNCTION

15

For functions u in Ω we shall consider semi-norms of the form  (1 − r2 )α/2+1 (3.9) |u(z)|wΩ;−α/2−1 (z) |dz| uΩ,a;α;r = 2πr T(a,r)  1 (1 − ρΩ (z, a)2 )α/2+1 = |dz| |u(z)| 2πr T(a,r) wΩ;α/2+1 (z) for α ∈ R and 0 < r < 1, where the integration is with respect to arc length. Lemma 3.1. Let Ωj = C be a simply connected planar domain and aj ∈ Ωj (j = 1, 2). Let ϕ : Ω1 → Ω2 be a biholomorphic map such that ϕ(a1 ) = a2 . Let u be a function in Ω2 and consider the function uϕ,α in Ω1 defined as in (1.5). Then uϕ,α Ω1 ,a1 ;α;r = uΩ2 ,a2 ;α;r for α ∈ R and 0 < r < 1. Proof. Notice that ϕ maps T(a1 , r) one-to-one onto T(a2 , r) (see Proposition 3.1). By the change of variables ζ = ϕ(z) we have that  (1 − r2 )α/2+1 |u(ϕ(z))||ϕ (z)|−α/2 wΩ1 ;−α/2−1 (z) |dz| uϕ,α Ω1 ,a1 ;α;r = 2πr T(a1 ,r)  (1 − r2 )α/2+1 = |u(ζ)||ϕ (ϕ−1 (ζ))|−α/2 |(ϕ−1 ) (ζ)|wΩ1 ;−α/2−1 (ϕ−1 (ζ)) |dζ| 2πr T(a2 ,r) for α ∈ R and 0 < r < 1. By Corollary 1.1 we have that (1 − r2 )α/2+1 × 2πr

uϕ,α Ω1 ,a1 ;α;r = 

T(a2 ,r)

=

|u(ζ)||ϕ (ϕ−1 (ζ))|−α/2 |(ϕ−1 ) (ζ)|−α/2 wΩ2 ;−α/2−1 (ζ) |dζ|

(1 − r2 )α/2+1 2πr

 T(a2 ,r)

|u(ζ)|wΩ2 ;−α/2−1 (ζ) |dζ| = uΩ2 ,a2 ;α;r

for α ∈ R and 0 < r < 1, where the mid equality follows by the chain rule.



Of particular interest is the case of the open unit disc. Notice that   1 1 |u(z)| |dz| = |u(reiθ )| dθ (3.10) uD,0;α;r = 2πr |z|=r 2π T for α ∈ R and 0 < r < 1 are the usual L1 (T) integral means. Theorem 3.3. Let Ω = C be a simply connected planar domain and α ∈ R. Let u be α-harmonic in Ω. If lim uΩ,a;α;r = 0 r→1

for some a ∈ Ω, then u(z) = 0 for all z ∈ Ω. Proof. Let ϕ : D → Ω be a biholomorphic map with ϕ(0) = a. We consider the function v = uϕ,α in D, where uϕ,α is defined as in (1.5). By Theorem 1.1 we have that v is α-harmonic in D. By Lemma 3.1 we have that vD,0;α;r = uΩ,a;α;r

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ANDERS OLOFSSON

for 0 < r < 1. In view of formula (3.10) we thus know that  1 |v(reiθ )| dθ = 0. lim r→1 2π T By Theorem 3.2 we conclude that v(z) = 0 for z ∈ D. Thus u(z) = 0 for z ∈ Ω.



In this context should be mentioned also an interesting uniqueness result for α-harmonic functions in the open upper half-plane by Carlsson and Wittsten [9]. Let us show also the strong triangle inequality for the pseudo hyperbolic metric. Theorem 3.4. Let Ω = C be a simply connected planar domain and denote by ρΩ the pseudo hyperbolic metric for Ω. Then ρΩ (z1 , z2 ) + ρΩ (z2 , z3 ) − ρΩ (z1 , z3 ) ≥ ρΩ (z1 , z2 )ρΩ (z2 , z3 )ρΩ (z1 , z3 ) for z1 , z2 , z3 ∈ Ω. Proof. Let z1 , z2 , z3 ∈ Ω. Let ϕ : Ω → D be a biholomorphic map with ϕ(z2 ) = 0. For z ∈ Ω we have that (3.11)

ρΩ (z, z2 ) = ρD (ϕ(z), 0) = |ϕ(z)|,

which follows by Proposition 3.1 and formula (3.5). Again by Proposition 3.1 and (3.5) we have that  ϕ(z ) − ϕ(z ) 2  1 3  1 − ρΩ (z1 , z3 )2 = 1 − ρD (ϕ(z1 ), ϕ(z3 ))2 = 1 −   . 1 − ϕ(z1 )ϕ(z3 ) From formula (2.3) we have that 1 − ρΩ (z1 , z3 )2 =

(1 − |ϕ(z1 )|2 )(1 − |ϕ(z3 )|2 ) |1 − ϕ(z1 )ϕ(z3 )|2

.

We now use the triangle inequality to conclude that 1 − ρΩ (z1 , z3 )2 ≥

(1 − |ϕ(z1 )|2 )(1 − |ϕ(z3 )|2 ) (1 − ρΩ (z1 , z2 )2 )(1 − ρΩ (z3 , z2 )2 ) = , 2 (1 + |ϕ(z1 )||ϕ(z3 )|) (1 + ρΩ (z1 , z2 )ρΩ (z3 , z2 ))2

where the last equality follows by (3.11). Thus ρΩ (z1 , z3 )2 ≤ 1 −

(1 − ρΩ (z1 , z2 )2 )(1 − ρΩ (z3 , z2 )2 )  ρΩ (z1 , z2 ) + ρΩ (z2 , z3 ) 2 = , (1 + ρΩ (z1 , z2 )ρΩ (z3 , z2 ))2 1 + ρΩ (z1 , z2 )ρΩ (z2 , z3 )

where the last equality is straightforward to check. We now conclude that ρΩ (z1 , z3 ) ≤

ρΩ (z1 , z2 ) + ρΩ (z2 , z3 ) . 1 + ρΩ (z1 , z2 )ρΩ (z2 , z3 )

This latter inequality yields the conclusion of the theorem. Observe that the result of Theorem 3.4 is equivalently formulated saying that (3.12)

ρΩ (z1 , z3 ) ≤

ρΩ (z1 , z2 ) + ρΩ (z2 , z3 ) 1 + ρΩ (z1 , z2 )ρΩ (z2 , z3 )

for z1 , z2 , z3 ∈ Ω. Another reformulation of Theorem 3.4 is that |ρΩ (z1 , z2 ) − ρΩ (z2 , z3 )| ≤ ρΩ (z1 , z3 ) 1 − ρΩ (z1 , z2 )ρΩ (z2 , z3 ) for z1 , z2 , z3 ∈ Ω.



ON A WEIGHTED HARMONIC GREEN FUNCTION

17

It should be mentioned that Theorem 3.4 is well-known, see for instance [15, Theorem 1(c)]. Similar inequalities have been studied in the context of uniform algebras, see for instance Bear [5]. It is evident that ρΩ (z1 , z2 ) < 1 for z1 , z2 ∈ Ω. Corollary 3.1. Let Ω = C be a simply connected planar domain. Then 1 − ρΩ (z1 , z3 )2 1 + ρΩ (z1 , z2 ) 1 − ρΩ (z1 , z2 ) ≤ ≤ 1 + ρΩ (z1 , z2 ) 1 − ρΩ (z2 , z3 )2 1 − ρΩ (z1 , z2 ) for z1 , z2 , z3 ∈ Ω, where ρΩ is the pseudo hyperbolic metric for Ω. Proof. Let z1 , z2 , z3 ∈ Ω. From the reformulation (3.12) of Theorem 3.4 we have that  ρ (z , z ) + ρ (z , z ) 2 Ω 1 2 Ω 2 3 1 − ρΩ (z1 , z3 )2 ≥ 1 − 1 + ρΩ (z1 , z2 )ρΩ (z2 , z3 ) (1 − ρΩ (z1 , z2 )2 )(1 − ρΩ (z2 , z3 )2 ) = , (1 + ρΩ (z1 , z2 )ρΩ (z2 , z3 ))2 where the last equality is straightforward to check. Thus 1 − ρΩ (z1 , z3 )2 1 − ρΩ (z1 , z2 )2 ≥ . 2 1 − ρΩ (z2 , z3 ) (1 + ρΩ (z1 , z2 )ρΩ (z2 , z3 ))2 Since 0 ≤ ρΩ (z2 , z3 ) ≤ 1, an easy estimation gives that 1 − ρΩ (z1 , z3 )2 1 − ρΩ (z1 , z2 )2 1 − ρΩ (z1 , z2 ) , ≥ = 2 2 1 − ρΩ (z2 , z3 ) (1 + ρΩ (z1 , z2 )) 1 + ρΩ (z1 , z2 ) which proves the lower bound in the corollary. The upper bound in the corollary follows by symmetry from the lower bound just proved.  4. Calculation of fundamental solutions Motivated by Theorem 1.2 we begin by looking for a fundamental solution at the origin. Rotational symmetry suggests us to consider radial functions in the punctured disc D \ {0}. By a radial function in D \ {0} we understand a function u of the form u(z) = h(|z|2 ),

(4.1)

z ∈ D \ {0},

for some function h on (0, 1). For the sake of simplicity of notation we write Δα = ΔD;α for the weighted Laplacians for D. Lemma 4.1. Let α ∈ R and let u be a function of the form (4.1) for some function h ∈ C 2 (0, 1). Then   Δα u(z) = (1 − |z|2 )−α−1 |z|2 (1 − |z|2 )h (|z|2 ) + (1 − (1 − α)|z|2 )h (|z|2 ) for z ∈ D \ {0}. Proof. Differentiating we have that ¯ ∂u(z) = zh (|z|2 ) for z ∈ D \ {0}. Now ¯ (1 − |z|2 )−α ∂u(z) = z(1 − |z|2 )−α h (|z|2 )

18

ANDERS OLOFSSON

and another differentiation gives that z )h (|z|2 ) Δα u(z) = (1 − |z|2 )−α h (|z|2 ) + z(−α)(1 − |z|2 )−α−1 (−¯ + z(1 − |z|2 )−α h (|z|2 )¯ z for z ∈ D \ {0}. A simplification of terms gives that   Δα u(z) = (1 − |z|2 )−α−1 |z|2 (1 − |z|2 )h (|z|2 ) + (α|z|2 + 1 − |z|2 )h (|z|2 ) for z ∈ D \ {0}, which yields the result.



We now calculate the radial solutions of Δα u = 0 in D \ {0}. Theorem 4.1. Let α ∈ R, 0 < r < 1 and set  x (1 − t)α dt, hα,r (x) = t r2

0 < x < 1.

Let u ∈ C 2 (D \ {0}) be a radial function. Then Δα u = 0 in D \ {0} if and only if u has the form u(z) = a + bhα,r (|z|2 ), z ∈ D \ {0}, for some constants a and b. Proof. Put u of the form (4.1) with h ∈ C 2 (0, 1). By Lemma 4.1 we have that u satisfies the equation Δα u = 0 in D \ {0} if and only if the function h satisfies the ordinary differential equation (4.2)

x(1 − x)h (x) + (1 − (1 − α)x)h (x) = 0

for x ∈ (0, 1). Solving (4.2) using, for instance, an integrating factor we see that the solutions h of (4.2) are precisely the functions of the form h(x) = a + bhα,r (x) for some constants a and b. This yields the conclusion of the theorem.



Remark 4.1. We remark that equation (4.2) above is a hypergeometric equation x(1 − x)y  + (c − (a + b + 1)x)y  − aby = 0 with parameters a = −α, b = 0 and c = 1. We shall need the following lemma from the theory of test functions. For the sake of convenience we include some details of proof. Lemma 4.2. Let f ∈ C ∞ (−1, 1) be an even function. Then there exists a function g ∈ C ∞ [0, 1) such that f (x) = g(x2 ) for all x ∈ (−1, 1). Proof. Since f is even we have that f (2j+1) (0) = 0 for j ≥ 0. By a theorem of Borel there exists φ ∈ C ∞ (−1, 1) such that φ(j) (0)/j! = f (2j) (0)/(2j)! for j ≥ 0 (see H¨ ormander [22, Theorem 1.2.6]). Consider now the function h(x) = f (x) − φ(x2 ), ∞

−1 < x < 1.

Observe that h ∈ C (−1, 1) is even. A look at the Taylor expansion reveals that h(j) (0) = 0 for j ≥ 0. Now set √ h1 (x) = h( x), 0 < x < 1.

ON A WEIGHTED HARMONIC GREEN FUNCTION

19 (j)

Clearly h1 ∈ C ∞ (0, 1). From Taylor’s formula we have that limx→0 h1 (x) = 0 for j ≥ 0, which gives that h1 ∈ C ∞ [0, 1). From construction we have that f (x) = g(x2 ) for x ∈ (−1, 1) with 0 ≤ x < 1.

g(x) = φ(x) + h1 (x), Clearly g ∈ C ∞ [0, 1).



We mention that Lemma 4.2 appears as Exercise 1.1 in H¨ ormander [22]. A function u in D is said to be homogeneous of degree k ∈ Z with respect to rotations if it has the property that u(eiθ z) = eikθ u(z),

z ∈ D,

for all eiθ ∈ T. Notice that in this terminology a radial function is a function of homogeneity 0 with respect to rotations. Theorem 4.2. Let u ∈ C ∞ (D) be homogeneous of degree k ≥ 0 with respect to rotations. Then there exists a function f ∈ C ∞ [0, 1) such that u(z) = z k f (|z|2 ),

z ∈ D.

Proof. By differentiation we have ∂ n ∂¯m u(0)ei(n−m)θ = eikθ ∂ n ∂¯m u(0) for eiθ ∈ T, which gives that ∂ n ∂¯m u(0) = 0

for n − m = k.

Thus, the function u vanishes at the origin of order k. Consider the function v(z) = z¯k u(z),

z ∈ D.

∞

The function v ∈ C (D) is radial and vanishes of order 2k at the origin. In particular, the restriction of v to the interval (−1, 1) is even. By Lemma 4.2 there exists a function g ∈ C ∞ [0, 1) such that v(x) = g(x2 ) for x ∈ (−1, 1). Now z¯k u(z) = v(z) = v(|z|) = g(|z|2 ) for z ∈ D. Also the function g vanishes at the origin of order k since v vanishes at the origin of order 2k. By Taylor’s formula this gives that g(x) = xk f (x),

x ∈ [0, 1),

for some function f ∈ C ∞ [0, 1). Now z¯k u(z) = |z|2k f (|z|2 ) for z ∈ D and a  cancellation of a factor z¯k gives the result. Remark 4.2. If u ∈ C ∞ (D) is homogeneous of degree k < 0 with respect to rotations, then (4.3)

u(z) = z¯|k| f (|z|2 ),

z ∈ D,

for some function f ∈ C ∞ [0, 1). Indeed, if u ∈ C ∞ (D) has homogeneity k < 0, then its complex conjugate u ¯ has homogeneity −k = |k| ≥ 0 and an application of Theorem 4.2 yields (4.3).

20

ANDERS OLOFSSON

We shall consider rotation operators acting on functions u in D by Reiθ u(z) = u(eiθ z),

z ∈ D,

iθ

for e ∈ T. As a byproduct of Theorem 1.1 we have that the differential operator Δα commutes with the action of rotations, that is, (4.4)

Δα Reiθ = Reiθ Δα

for eiθ ∈ T. Notice also that u ∈ C ∞ (D) is homogeneous of degree k ∈ Z with respect to rotations if and only if Reiθ u = eikθ u for eiθ ∈ T. From the commutativity relation (4.4) we have that if u is homogeneous of degree k with respect to rotations, then so is the function Δα u (compare [28, Lemma 2.2]). It is known from harmonic analysis that every function u ∈ C ∞ (D) has a canonical expansion of the form (4.5)

∞ 

u(z) =

z ∈ D,

uk (z),

k=−∞

convergent in C ∞ (D), where uk ∈ C ∞ (D) is homogeneous of degree k with respect to rotations. Moreover, the k-th term uk in (4.5) is uniquely determined by (4.5) to the extent that  1 (4.6) uk (z) = u(eiθ z)e−ikθ dθ, z ∈ D, 2π T for k ∈ Z. We shall refer to the expansion (4.5)-(4.6) as the homogeneous expansion of u ∈ C ∞ (D). We now turn to distributional derivatives. Recall the distributional pairing (1.7). The symbol δζ denotes a unit Dirac mass at ζ ∈ C. Theorem 4.3. Let α ∈ R, 0 < r < 1 and let hα,r be as in Theorem 4.1. Set Eα,r (z) = hα,r (|z|2 ),

z ∈ D \ {0}.

Then Eα,r ∈ L1loc (D) and Δα Eα,r = δ0 in D (D). Proof. It is straightforward to check that the function hα,r has a logarithmic singularity at the origin in the sense that hα,r (x) = log(x) + O(1)

as x → 0

(compare Lemma 5.1 below). Therefore Eα,r ∈ L1loc (D). We now show that Δα Eα,r = δ0 in D (D). Let ψ ∈ D(D) be a compactly supported test function. Notice that Δα Eα,r , ψ = Eα,r , Δ∗α ψ , where Δ∗α,z = ∂¯z (1 − |z|2 )−α ∂z ,

z ∈ D.

Consider next the  homogeneous expansion (4.5)-(4.6) of ψ. Since ψ ∈ D(D) the expansion ψ = ψk converges in D(D). We now have that Δα Eα,r , ψ =

∞  k=−∞

Eα,r , Δ∗α ψk = Eα,r , Δ∗α ψ0 ,

ON A WEIGHTED HARMONIC GREEN FUNCTION

21

where the rightmost equality follows by cancellation since Eα,r is radial and Δ∗α ψk is homogeneous of degree k with respect to rotations (see the discussion following formula (4.4) above). We now write ψ0 (z) = f0 (|z|2 ),

z ∈ D,

with f0 ∈ C ∞ [0, 1) having compact support which is possible by Theorem 4.2. By Lemma 4.1 we have that   hα,r (|z|2 )(1 − |z|2 )−α−1 |z|2 (1 − |z|2 )f0 (|z|2 ) Δα Eα,r , ψ = D  + (1 − (1 − α)|z|2 )f0 (|z|2 ) dA(z)  1   rhα,r (r2 )(1 − r2 )−α−1 r2 (1 − r2 )f0 (r2 ) + (1 − (1 − α)r2 )f0 (r2 ) dr, =2 0

where the last equality follows by change to polar coordinates. By the change of variables t = r2 we see that (4.7)



Δα Eα,r , ψ = 

1

= 0

1 0

  hα,r (t)(1 − t)−α−1 t(1 − t)f0 (t) + (1 − (1 − α)t)f0 (t) dt

hα,r (t)t(1 − t)−α f0 (t) dt +



1 0

hα,r (t)(1 − t)−α−1 (1 − (1 − α)t)f0 (t) dt.

An integration by parts in the leftmost integral on the right hand side in (4.7) gives that Δα Eα,r , ψ  1  hα,r (t)t(1 − t)−α + hα,r (t)(1 − t)−α + hα,r (t)tα(1 − t)−α−1 f0 (t) dt =− 0 1

 + 0

hα,r (t)(1 − t)−α−1 (1 − (1 − α)t)f0 (t) dt



=−

1 0

hα,r (t)t(1 − t)−α f0 (t) dt,

where the last equality is straightforward to check. By construction we have that hα,r (t)t(1 − t)−α = 1 for 0 < t < 1. Thus  1 f0 (t) dt = f0 (0) = ψ0 (0) = ψ(0). Δα Eα,r , ψ = − 0

Varying ψ ∈ D(D) we conclude that Δα Eα,r = δ0 in D (D).



We now proceed to calculate fundamental solutions for general points. Theorem 4.4. Let α ∈ R, 0 < r < 1 and let hα,r be as in Theorem 4.1. Let Ω = C be a simply connected planar domain and let ϕ : Ω → D be a biholomorphic map with ϕ(ζ) = 0, where ζ ∈ Ω. Set Eα,r (z) = ϕ (ζ)−α/2 ϕ (z)−α/2 hα,r (|ϕ(z)|2 ), Then Eα,r ∈ L1loc (Ω) and ΔΩ;α Eα,r = δζ in D (Ω).

z ∈ Ω \ {ζ}.

22

ANDERS OLOFSSON

Proof. Clearly Eα,r ∈ L1loc (Ω) since the function hα,r has a logarithmic singularity at the origin. By Theorems 1.2 and 4.3 we have that ΔΩ;α Eα,r = ϕ (ζ)−α/2 S((ϕ )−α/2 )|ϕ |2 (δ0 ◦ ϕ) = ϕ (ζ)−α/2 (δ0 )∨ ϕ,α in D (Ω), where S given by (1.6) is the reflection in the unit circle and the distribution (δ0 )∨ ϕ,α is defined according to (1.9). By Proposition 1.2 and an approximation argument we have −1 ) (0)−α/2 ψ(ζ) (δ0 )∨ ϕ,α , ψ = (ϕ for ψ ∈ D(Ω), which gives that −1 ) (0)−α/2 δ (δ0 )∨ ζ ϕ,α = (ϕ

in D (Ω). We now have that ΔΩ;α Eα,r = ϕ (ζ)−α/2 (ϕ−1 ) (0)−α/2 δζ = δζ in D (Ω), where the last equality follows from compatibility of logarithms (see Proposition 1.2). This completes the proof of the theorem.  For α > −1 we consider the function  1 (1 − t)α dt, (4.8) hα (x) = − t x

0 < x ≤ 1.

Notice that hα (1) = 0. The function hα has some relationship to the family of Beta functions. In fact, hα (x) = −B(1 − x; α + 1, 0) for 0 < x ≤ 1, where  x ta−1 (1 − t)b−1 dt B(x; a, b) = 0

is the incomplete Beta function (see [1, Section 6.6] or [19, Chapter 1]). Observe that h0 is the usual logarithm function. The function hα differs from the function hα,r by an additive constant, where hα,r is as in Theorem 4.1. It is straightforward to check that the results of Theorems 4.1, 4.3 and 4.4 all hold true with hα,r replaced by hα when α > −1. 5. The Green function Let Ω = C be a simply connected planar domain and α > −1. By a Green function for the differential operator ΔΩ;α we mean a distribution-valued function Ω  ζ → GΩ;α (·, ζ) ∈ D (Ω) such that (5.1)



ΔΩ;α GΩ;α (·, ζ) = δζ GΩ;α (·, ζ) = 0

in Ω, on ∂Ω,

in a suitable sense for every ζ ∈ Ω. We interpret the first condition in (5.1) in a distributional sense that ΔΩ;α GΩ;α (·, ζ) = δζ in D (Ω), where δζ is the unit Dirac mass at ζ ∈ Ω. By standard regularity theory (Weil’s lemma) this latter condition implies that GΩ;α (·, ζ) ∈ C ∞ (Ω \ {ζ}). We then interpret the boundary condition in (5.1) that (5.2)

lim GΩ;α (·, ζ)Ω,a;α;r = 0

r→1

ON A WEIGHTED HARMONIC GREEN FUNCTION

23

for every a ∈ Ω, where the semi-norms ·Ω,a;α;r are defined as in (3.9). By Theorem 3.3 there exists at most one Green function for ΔΩ;α . Theorem 5.1. Let α > −1 and let Ω = C be a simply connected planar domain. Then the Green function GΩ;α for ΔΩ;α exists. Furthermore, GΩ;α (z, ζ) = KΩ (z, ζ)−α/2 hα (ρΩ (z, ζ)2 )

(5.3)

for (z, ζ) ∈ Ω2 outside the diagonal, where KΩ is the Bergman kernel for Ω, ρΩ is the pseudo hyperbolic metric for Ω and the function hα is as in (4.8). Proof. Let ζ ∈ Ω and set u(z) = KΩ (z, ζ)−α/2 hα (ρΩ (z, ζ)2 ),

z ∈ Ω \ {ζ}.

L1loc (Ω)

We proceed to check that u ∈ and ΔΩ;α u = δζ in D (Ω). For this purpose let ϕ : Ω → D be a biholomorphic map such that ϕ(ζ) = 0. From (3.7) we have that ρΩ (z, ζ) = |ϕ(z)| for z ∈ Ω. By Lemma 1.3 and formula (1.1) we have that KΩ (z, ζ)−α/2 = ϕ (ζ)−α/2 ϕ (z)−α/2 for z ∈ Ω. Thus u(z) = ϕ (ζ)−α/2 ϕ (z)−α/2 hα (|ϕ(z)|2 ),

z ∈ Ω \ {ζ}.

Recall also that the function hα differs from the function hα,r by an additive constant, where hα,r is as in Theorem 4.1. By Theorem 4.4 we conclude that u ∈ L1loc (Ω) and ΔΩ;α u = δζ in D (Ω). Let a ∈ Ω. We proceed to check that limr→1 uΩ,a;α;r = 0. For this purpose let ϕ : D → Ω be a biholomorphic map such that ϕ(0) = a. By Lemma 3.1 we have that  π 1 |uϕ,α (reiθ )| dθ (5.4) uΩ,a;α;r = uϕ,α D,0;α;r = 2π −π for 0 < r < 1, where the last equality follows by (3.10) and uϕ,α is as in (1.5). We now calculate uϕ,α . Let ζ1 = ϕ−1 (ζ). Clearly, (5.5)

uϕ,α (z) = ϕ (z)−α/2 KΩ (ϕ(z), ζ)−α/2 hα (ρΩ (ϕ(z), ζ)2 ),

z ∈ D \ {ζ1 }.

By Proposition 3.1 we have that

 z−ζ   1  ρΩ (ϕ(z), ζ) = ρD (z, ζ1 ) =   1 − ζ¯1 z

for z ∈ D, where the last equality follows by (3.5). By Lemma 1.3 we have that ϕ (z)−α/2 KΩ (ϕ(z), ζ)−α/2 = S(ϕ (ζ1 )−α/2 )KD (z, ζ1 )−α/2 = S(ϕ (ζ1 )−α/2 )(1 − ζ¯1 z)α for z ∈ D, where S is the reflection map given by (1.6) and the last equality follows by (1.1). Thus, from (5.5) we now have that  z − ζ 2   1  uϕ,α (z) = S(ϕ (ζ1 )−α/2 )(1 − ζ¯1 z)α hα   1 − ζ¯1 z for z ∈ D. As a consequence, we have that uϕ,α (z) → 0 as |z| → 1. In view of (5.4) this makes evident that limr→1 uΩ,a;α;r = 0. This completes the proof of the theorem. 

24

ANDERS OLOFSSON

A formal interpretation of Theorem 5.1 is that the distribution GΩ;α (·, ζ) in D (Ω) is induced by a function in L1loc (Ω) determined by formula (5.3). Recall that we use the distributional pairing (1.7). Observe from (4.8) that the function hα is real-valued. From this observation, formula (1.3) and symmetry of the pseudo hyperbolic metric, we have the symmetry formula (5.6)

GΩ;α (z, ζ) = GΩ;α (ζ, z),

(z, ζ) ∈ Ω2 , z = ζ,

for the Green function. The following result is due to Gustav Behm [6]. Corollary 5.1 (Behm). Let α > −1. Then the Green function GD;α for ΔD;α has the form 2   ¯ α hα  ζ − z  GD;α (z, ζ) = (1 − ζz) ¯ 1 − ζz 2 for (z, ζ) ∈ D outside the diagonal, where the function hα is as in (4.8). Proof. Recall formulas (1.1) and (3.5) for the Bergman kernel and pseudo hyperbolic metric for D, respectively. The result now follows by Theorem 5.1.  Another domain of interest is the open upper half-plane H given in (1.4). By the principal branch of the logarithm we mean the logarithm function defined in the slit plane C \ (−∞, 0] which attains real values on the positive real axis. Corollary 5.2. Let α > −1. Then the Green function GH;α for ΔH;α has the form  ζ − z 2    GH;α (z, ζ) = eiπα/2 (ζ¯ − z)α hα  ¯  ζ −z for (z, ζ) ∈ H2 outside the diagonal, where the function hα is as in (4.8) and the power is defined using the principal branch of the logarithm. Proof. From the proof of Proposition 1.1 we know that the Bergman kernel for H has the form 1 , (z, ζ) ∈ H2 . KH (z, ζ) = − ¯ (ζ − z)2 It is straightforward to check that log KH (z, ζ) = −2 log(ζ¯ − z) − iπ for z, ζ ∈ H, where the principal branch of the logarithm is used on the right hand side. We now see that KH (z, ζ)−α/2 = eiπα/2 (ζ¯ − z)α for z, ζ ∈ H. We now calculate the pseudo hyperbolic metric for H. The M¨obius transformation ϕ(z) = (z − i)/(z + i) maps H one-to-one onto D. From Proposition 3.1 and formula (3.5) we have that  ϕ(ζ) − ϕ(z)    ρH (z, ζ) = ρD (ϕ(z), ϕ(ζ)) =   1 − ϕ(ζ)ϕ(z) for z, ζ ∈ H. It is now straightforward to check that ζ − z    ρH (z, ζ) =  ¯ , z, ζ ∈ H. ζ −z The result now follows by Theorem 5.1. 

ON A WEIGHTED HARMONIC GREEN FUNCTION

25

See Proposition 1.1 for the exact form of the standard weights wH;α for H. Inherent in Theorem 5.1 is a conformal invariance property of the Green function. Theorem 5.2. Let α > −1. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2). Then GΩ1 ;α (z, ζ) = ϕ (ζ)−α/2 ϕ (z)−α/2 GΩ2 ;α (ϕ(z), ϕ(ζ)) for (z, ζ) ∈ Ω21 outside the diagonal, where the power (ϕ )−α/2 is defined using some logarithm of ϕ . Proof. Recall Lemma 1.3 and Proposition 3.1. By Theorem 5.1 we have that ϕ (ζ)−α/2 ϕ (z)−α/2 GΩ2 ;α (ϕ(z), ϕ(ζ)) = ϕ (ζ)−α/2 ϕ (z)−α/2 KΩ2 (ϕ(z), ϕ(ζ))−α/2 hα (ρΩ2 (ϕ(z), ϕ(ζ))2 ) = KΩ1 (z, ζ)−α/2 hα (ρΩ1 (z, ζ)2 ) = GΩ1 ;α (z, ζ) for (z, ζ) ∈ Ω21 outside the diagonal.



We remark in passing that the invariance property of Green functions in Theorem 5.2 equivalently means that the function GΩ;α has the form GΩ;α (z, ζ) = KΩ (z, ζ)−α/2 h(ρΩ (z, ζ)2 ) for (z, ζ) ∈ Ω2 outside the diagonal, where h : (0, 1) → C is some function. The symmetry property (5.6) for GΩ;α thus corresponds to the function hα in (5.3) being real-valued. We omit the details. We now turn to estimates of Green functions. We first consider the function hα defined in (4.8). Lemma 5.1. Let α > −1 and let the function hα be as in (4.8). Then   1  1  max 1, (1 − x)α log(x) ≤ hα (x) ≤ min 1, (1 − x)α log(x) α+1 α+1 for 0 < x < 1. Proof. An integration by parts shows that hα (x) = (1 − x)α log(x) − α



1 x

log(t)(1 − t)α−1 dt

for 0 < x < 1. We shall next estimate the integral  1 log(t)(1 − t)α−1 dt I(x) = x

appearing above. Since the logarithm is negative on the interval (0, 1), we have that I(x) ≤ 0 for 0 < x < 1. By concavity of the logarithm we have that 1−t log(x) ≤ log(t) 1−x for 0 < x < t < 1. An integration now shows that  1 log(x) 1 (1 − x)α log(x) (1 − t)α dt = I(x) ≥ 1−x x α+1 for 0 < x < 1. These upper and lower bounds for I(x) lead to the conclusion of the lemma. 

26

ANDERS OLOFSSON

Remark 5.1. From the above proof we have that the function (0, 1)  x → log(x)/(1 − x) is increasing. We now return to the Green function. Theorem 5.3. Let α > −1 and let Ω = C be a simply connected planar domain. Set   1 HΩ;α (z, ζ) = wΩ;α/2 (z)wΩ;α/2 (ζ)(1 − ρΩ (z, ζ)2 )α/2 log 2 ρΩ (z, ζ) for (z, ζ) ∈ Ω2 outside the diagonal, where ρΩ is the pseudo hyperbolic metric for Ω. Then   1  1  HΩ;α (z, ζ) ≤ |GΩ;α (z, ζ)| ≤ max 1, HΩ;α (z, ζ) (5.7) min 1, α+1 α+1 for (z, ζ) ∈ Ω2 outside the diagonal. Proof. Let ϕ : Ω → D be a biholomorphic map. From Theorem 5.2 we have that |GΩ;α (z, ζ)| = |ϕ (ζ)|−α/2 |ϕ (z)|−α/2 |Gα (ϕ(z), ϕ(ζ))| for (z, ζ) ∈ Ω2 outside the diagonal, where Gα = GD;α is the Green function for D. We now use Corollary 1.1 to conclude that   |Gα (ϕ(z), ϕ(ζ))| (5.8) |GΩ;α (z, ζ)| = wΩ;α/2 (ζ)wΩ;α/2 (z) 2 α/2 2 α/2 (1 − |ϕ(z)| ) (1 − |ϕ(ζ)| ) for (z, ζ) ∈ Ω2 outside the diagonal. We now consider the rightmost factor in (5.8). By Corollary 5.1 we have that  (1 − |ζ|2 )(1 − |z|2 ) −α/2  ζ − z 2  |Gα (z, ζ)|   = − hα  ¯ 2 ¯  (1 − |z|2 )α/2 (1 − |ζ|2 )α/2 |1 − ζz| 1 − ζz = −(1 − |ϕζ (z)|2 )−α/2 hα (|ϕζ (z)|2 ) for (z, ζ) ∈ D2 outside the diagonal, where ϕζ is as in (2.2) and the last equality follows by (2.3). Recall the pseudo hyperbolic metric ρD given by (3.5). By Lemma 5.1 we have that |Gα (z, ζ)| ≤ Mα (1 − ρD (z, ζ)2 )α/2 log(1/ρD (z, ζ)2 ) (1 − |z|2 )α/2 (1 − |ζ|2 )α/2 for (z, ζ) ∈ D2 outside the diagonal, where Mα = max(1, 1/(α + 1)). By conformal invariance we have that |Gα (ϕ(z), ϕ(ζ))| ≤ Mα (1 − ρΩ (z, ζ)2 )α/2 log(1/ρΩ (z, ζ)2 ) (1 − |ϕ(z)|2 )α/2 (1 − |ϕ(ζ)|2 )α/2 for (z, ζ) ∈ Ω2 outside the diagonal, where we have used Proposition 3.1. The upper bound in (5.7) now follows by (5.8). The lower bound in (5.7) is proved similarly.  For the sake of simplicity of notation we write Hα = HD;α , where HD;α is as in Theorem 5.3. A calculation using formula (2.3) shows that   (1 − |z|2 )α (1 − |ζ|2 )α 1 (5.9) Hα (z, ζ) = log 2 α ¯ |ϕζ (z)| |1 − ζz| for (z, ζ) ∈ D2 outside the diagonal, where ϕζ is as in (2.2).

ON A WEIGHTED HARMONIC GREEN FUNCTION

27

Lemma 5.2. Let 0 ≤ λ ≤ 1. Then (1 − |z|2 )λ (1 − |ζ|2 )1−λ ≤ 22|λ−1/2| ¯ |1 − ζz| for z, ζ ∈ D. Proof. Set

(1 − |z|2 )λ (1 − |ζ|2 )1−λ ¯ |1 − ζz| z,ζ∈D for 0 ≤ λ ≤ 1. From the triangle inequality we have that M (0) = M (1) = 2. From formula (2.3) we have that M (1/2) = 1. It is straightforward to check that the function M : [0, 1] → (0, ∞) is logarithmically convex. These facts lead to the conclusion of the lemma.  M (λ) = sup

We now return to the function Hα . Proposition 5.1. Let α > −1 and let Hα be as in (5.9). Then   (1 − |z|2 )α+1 (1 − |ζ|2 )α+1  1 1 + log Hα (z, ζ) ≤ 2 α+2 ¯ |ϕζ (z)| |1 − ζz| for (z, ζ) ∈ D2 outside the diagonal, where ϕζ is as in (2.2). As a consequence   (1 − |z|2 )α+1 (1 − |ζ|2 )α+1 1 Hα (z, ζ) ≤ + 2|α| (1 − |ζ|2 )α log 2 α+2 ¯ |ϕζ (z)| |1 − ζz| for (z, ζ) ∈ D2 outside the diagonal. Proof. We first show that log x ≥ −1 + log x 1−x

(5.10) for 0 < x < 1. Set

h(x) =

x log x. 1−x

A differentiation shows that

 1  log x +1 ≤0 1−x 1−x for 0 < x < 1, where the last inequality follows from the monotonicity property of the logarithm pointed out in Remark 5.1. Thus the function h is decreasing on (0, 1), which leads to the inequality that h(x) ≥ −1 for 0 < x < 1. This latter inequality yields (5.10). We now estimate the quantity Hα (z, ζ). By (5.10) we have that h (x) =

(1 − |z|2 )α (1 − |ζ|2 )α (1 − |ϕζ (z)|2 )(−1 + log(|ϕζ (z)|2 )) ¯ α |1 − ζz|  (1 − |z|2 )α+1 (1 − |ζ|2 )α+1  1 + log(1/|ϕζ (z)|2 ) = α+2 ¯ |1 − ζz|

Hα (z, ζ) ≤ −

for (z, ζ) ∈ D2 outside the diagonal, where the in the last equality we have used (2.3). This proves the first estimation in the proposition. By Lemma 5.2 we have that (1 − |z|2 )α+1 (1 − |ζ|2 ) ≤ 2|α| ¯ α+2 |1 − ζz|

28

ANDERS OLOFSSON

for (z, ζ) ∈ D2 . This latter estimation shows that the last inequality in the proposition is a consequence of the first.  We mention that Theorem 5.3 and Proposition 5.1 generalize Behm [6, Lemma 3]. 6. The Green potential In this section we define the α-harmonic Green potential and discuss some of its basic properties. A Radon measure is a distribution of order 0. The space of such distributions in Ω is denoted by D0 (Ω). We denote by B(z0 , r) = {z ∈ C : |z − z0 | < r} the Euclidean disc with center z0 ∈ C and radius r > 0. Proposition 6.1. Let α > −1 and let Ω = C be a simply connected planar domain. Denote by ρΩ the pseudo hyperbolic metric for Ω. Let z ∈ Ω and let μ ∈ D0 (Ω) be a positive Radon measure in Ω. Then GΩ;α (z, ·) ∈ L1 (Ω, dμ) if and only if  (1 − ρΩ (z, ζ)2 )α/2+1 wΩ;α/2 (ζ) dμ(ζ) < +∞ (6.1) Ω and B(z,ε) log|z − ζ| dμ(ζ) > −∞ for some 0 < ε < dist(z, C \ Ω). The condition (6.1) does not depend on the point z ∈ Ω. The function GΩ;α (z, ·) belongs to L1 (Ω, dμ) for quasi every z ∈ Ω if (6.1) holds. Proof. Let 0 <  < 1 and let D(z, ) be the pseudo hyperbolic disc in Ω with center z and radius  as in (3.8). We write  |GΩ;α (z, ζ)| dμ(ζ) = I1, (z) + I2, (z), Ω

where I1, (z) =



 D(z,)

|GΩ;α (z, ζ)| dμ(ζ)

and

I2, (z) =

Ω\D(z,)

|GΩ;α (z, ζ)| dμ(ζ).

Recall also Theorem 5.3. We consider first the integral I1, (z). For ζ ∈ D(z, ) we have that the quantity to the quantity log(1/|z − ζ|). This gives that |GΩ;α (z, ζ)| is boundedly equivalent I1, (z) < +∞ if and only if B(z,ε) log|z − ζ| dμ(ζ) > −∞ for some 0 < ε < dist(z, C \ Ω). We next consider the integral I2, (z). For ζ ∈ Ω \ D(z, ) we have that the quantity |GΩ;α (z, ζ)| is boundedly equivalent to wΩ;α/2 (ζ)(1−ρΩ (z, ζ)2 )α/2+1 . This gives that I2, (z) < +∞ is equivalent to (6.1). From Corollary 3.1 we have that condition (6.1) does not depend on the choice of point z ∈ Ω. It is well-known that the logarithmic potential of a compactly supported measure converges q.e., that is, outside of a Gδ subset of (outer) logarithmic capacity 0 (see for instance [30, Section 3.5]). This well-known fact gives that the function  GΩ;α (z, ·) belongs to L1 (Ω, dμ) for q.e. z ∈ Ω if (6.1) holds. Let α > −1 and let Ω = C be a simply connected planar domain. Let μ ∈ D0 (Ω) be a complex Radon measure in Ω such that  (1 − ρΩ (z1 , ζ)2 )α/2+1 wΩ;α/2 (ζ) d|μ|(ζ) < +∞ (6.2) Ω

ON A WEIGHTED HARMONIC GREEN FUNCTION

29

for some z1 ∈ Ω, where |μ| is the total variation measure of μ. The α-harmonic Green potential of μ in Ω is the function GΩ;α [μ] defined by  GΩ;α (z, ζ) dμ(ζ), z ∈ Ω, (6.3) GΩ;α [μ](z) = Ω

whenever the integral exists. By Proposition 6.1 we have that the integral in (6.3) exists for quasi every z ∈ Ω. We say that the Green potential GΩ;α [μ] exists if μ ∈ D0 (Ω) satisfies (6.2). Notice that condition (6.2) for μ ∈ D0 (Ω) does not depend on the particular choice of point z1 ∈ Ω, which follows by Corollary 3.1. We denote by Cc (Ω) the space of all continuous functions in Ω with compact support contained in Ω. Lemma 6.1. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2) and α ∈ R. Let μ ∈ D0 (Ω2 ) and let μ∨ ϕ,α be as in (1.9). Then   ψ(z) d|μ∨ |(z) = ψ(ϕ−1 (z))|(ϕ−1 ) (z)|−α/2 d|μ|(z) ϕ,α Ω1

Ω2

for ψ ∈ Cc (Ω1 ). Proof. By Proposition 1.2 and an approximation argument we have that   ∨ ψ1 (z) dμϕ,α (z) = (ϕ−1 ) (z)−α/2 ψ1 (ϕ−1 (z)) dμ(z) Ω1

Ω2

for ψ1 ∈ Cc (Ω1 ). Letting ψ1 above approximate a characteristic function we see that  ∨ (ϕ−1 ) (z)−α/2 dμ(z) μϕ,α (E1 ) = ϕ(E1 )

for every Borel subset E1 of Ω1 . Passing to the total variation measure using a countable measurable partition we have that  ∨ |μϕ,α |(E1 ) ≤ |(ϕ−1 ) (z)|−α/2 d|μ|(z) ϕ(E1 )

for every Borel subset E1 of Ω1 . In terms of test functions this latter inequality says that   (6.4) ψ1 (z) d|μ∨ |(z) ≤ ψ1 (ϕ−1 (z))|(ϕ−1 ) (z)|−α/2 d|μ|(z) ϕ,α Ω1

Ω2

for 0 ≤ ψ1 ∈ Cc (Ω1 ). It remains to show that equality holds in (6.4). Applying (6.4) with the function 0 ϕ−1 : Ω2 → Ω1 playing the role of ϕ and μ∨ ϕ,α ∈ D (Ω1 ) playing the role of μ in (6.4) we have that   ∨ ψ2 (z) d|(μ∨ ) |(z) ≤ ψ2 (ϕ(z))|ϕ (z)|−α/2 d|μ∨ (6.5) ϕ,α ϕ−1 ,α α,ϕ |(z) Ω2

Ω1

∨ 0 for 0 ≤ ψ2 ∈ Cc (Ω2 ). Notice that (μ∨ ϕ,α )ϕ−1 ,α = μ in D (Ω2 ) by compatibility of logarithms (see Proposition 1.2). We now apply inequality (6.5) with ψ2 =

30

ANDERS OLOFSSON

ψ1 ◦ ϕ−1 |(ϕ−1 ) |−α/2 to see that  ψ1 (ϕ−1 (z))|(ϕ−1 ) (z)|−α/2 d|μ|(z) Ω2   ψ1 (z)|(ϕ−1 ) (ϕ(z))|−α/2 |ϕ (z)|−α/2 d|μ∨ |(z) = ≤ ϕ,α Ω1

Ω1

ψ1 (z) d|μ∨ ϕ,α |(z)

for 0 ≤ ψ1 ∈ Cc (Ω1 ), where the last equality follows by the chain rule. This proves that equality holds in (6.4).  We now prove conformal invariance of the Green potential. Theorem 6.1. Let α > −1. Let ϕ : Ω1 → Ω2 be a biholomorphic map between simply connected planar domains Ωj = C (j = 1, 2). Let μ ∈ D0 (Ω2 ) be a Radon ∨ measure such that u = GΩ2 ;α [μ] exists. Then GΩ1 ;α [μ∨ ϕ,α ] exists, where μϕ,α is as in (1.9). Furthermore, uϕ,α (z) = GΩ1 ;α [μ∨ ϕ,α ](z),

z ∈ Ω1 ,

whenever the integrals exist, where uϕ,α is as in (1.5). Proof. We first show that GΩ1 ;α [μ∨ ϕ,α ] exists. Let z1 ∈ Ω1 and set  wΩ1 ;α/2 (ζ)(1 − ρΩ1 (z1 , ζ)2 )α/2+1 d|μ∨ I= ϕ,α |(ζ). Ω1

By Lemma 6.1 we have that  wΩ1 ;α/2 (ϕ−1 (ζ))(1 − ρΩ1 (z1 , ϕ−1 (ζ))2 )α/2+1 |(ϕ−1 ) (ζ)|−α/2 d|μ|(ζ). I= Ω2

We now use Corollary 1.1 and Proposition 3.1 to conclude that  wΩ2 ;α/2 (ζ)(1 − ρΩ2 (ϕ(z1 ), ζ)2 )α/2+1 d|μ|(ζ) < +∞ I= Ω2

since GΩ2 ;α [μ] exists. We consider now the Green potential GΩ1 ;α [μ∨ ϕ,α ]. By Proposition 1.2 and an approximation argument we have that  ](z) = GΩ1 ;α (z, ζ) dμ∨ GΩ1 ;α [μ∨ ϕ,α ϕ,α (ζ) Ω1  (ϕ−1 ) (ζ)−α/2 GΩ1 ;α (z, ϕ−1 (ζ)) dμ(ζ) = Ω2

for q.e. z ∈ Ω1 . From Theorem 5.2 we have that (ϕ−1 ) (ζ)−α/2 GΩ1 ;α (z, ϕ−1 (ζ)) =

1 GΩ2 ;α (ϕ(z), ζ) (ϕ−1 ) (ϕ(z))−α/2

= ϕ (z)−α/2 GΩ2 ;α (ϕ(z), ζ) for (z, ζ) ∈ Ω1 × Ω2 such that ϕ(z) = ζ, where the last equality follows by compatibility of logarithms (see Proposition 1.2). Thus   −α/2 ](z) = ϕ (z) GΩ2 ;α (ϕ(z), ζ) dμ(ζ) = uϕ,α (z) GΩ1 ;α [μ∨ ϕ,α Ω2

for q.e. z ∈ Ω1 .



ON A WEIGHTED HARMONIC GREEN FUNCTION

31

We now turn our attention to the case of the open unit disc D. Let α > −1. For simplicity of notation we write Gα = GD;α for the Green function and  Gα (z, ζ) dμ(ζ), z ∈ D, Gα [μ](z) = D

for the corresponding Green potential. Notice that Gα [μ] exists if and only if μ ∈ D0 (D) is a complex Radon measure in D such that  (6.6) (1 − |ζ|2 )α+1 d|μ|(ζ) < +∞, D

which should be compared with (6.2). We equip the space L1 (T) with the usual norm  1 f L1 (T) = |f (eiθ )| dθ 2π T for f ∈ L1 (T). The following result is due to Gustav Behm [6]. Theorem 6.2 (Behm). Let α > −1 and let μ ∈ D0 (D) be a complex Radon measure in D such that (6.6) holds. Let u = Gα [μ] and consider the functions ur as in (3.1). Then ur ∈ L1 (T) for 0 < r < 1 and limr→1 ur = 0 in L1 (T). Proof. By the triangle inequality we have that    1 |Gα (reiθ , ζ)| d|μ|(ζ) dθ ur L1 (T) ≤ 2π   T D  1 |Gα (reiθ , ζ)| dθ d|μ|(ζ) = D 2π T for 0 < r < 1, where the last equality follows by a change of order of integration. We consider now the functions  1 |Gα (reiθ , ζ)| dθ, ζ ∈ D, vr (ζ) = 2π T for 0 < r < 1. Notice that (6.7)

 ur L1 (T) ≤

D

vr (ζ) d|μ|(ζ)

for 0 < r < 1 by the previous consideration. By the defining property (5.2) of the Green function we have that vr (ζ) → 0 as r → 1 for every fixed ζ ∈ D. We proceed to estimate the functions vr . By Theorem 5.3 and Proposition 5.1 we have that   1  (1 − |ζ|2 )α+1 (1 − r2 )α+1 vr (ζ) ≤ max 1, ¯ iθ α+2 dθ α+1 2π T |1 − ζre |   2|α| (1 − |ζ|2 )α G0 (reiθ , ζ) dθ − 2π T ¯ By rotation for ζ ∈ D and 0 < r < 1, where G0 is the Green function for Δ0 = ∂ ∂. invariance and monotonicity we have that    1 1 1 (1 − r2 )α+1 (1 − r2 )α+1 (1 − (|ζ|r)2 )α+1 dθ = dθ ≤ dθ. iθ α+2 iθ α+2 ¯ 2π T |1 − ζre | 2π T |1 − |ζ|re | 2π T |1 − |ζ|reiθ |α+2

32

ANDERS OLOFSSON

We can now apply [28, Theorem 3.1] or [29, Corollary 2.8] to conclude that  Γ(α + 1) (1 − r2 )α+1 1 ¯ iθ |α+2 dθ ≤ Γ(α/2 + 1)2 2π T |1 − ζre for ζ ∈ D and 0 < r < 1, where Γ is the usual Gamma function. Also, by a straightforward argument we have that  1 G0 (reiθ , ζ) dθ = max(log(r2 ), log(|ζ|2 )) 2π T for ζ ∈ D and 0 < r < 1. Thus  1  Γ(α + 1) vr (ζ) ≤ max 1, (1 − |ζ|2 )α+1 α + 1 Γ(α/2 + 1)2  + 2|α| (1 − |ζ|2 )α min(log(1/r2 ), log(1/|ζ|2 )) for ζ ∈ D and 0 < r < 1. This latter inequality allows us to apply dominated convergence in (6.7) to conclude that limr→1 ur L1 (T) = 0. This completes the proof of the theorem.  Let us consider also the associated transforms Hα [μ] defined by  Hα [μ](z) = Hα (z, ζ) dμ(ζ), z ∈ D, D

whenever the integral exists, where Hα is as in (5.9) and μ ∈ D0 (D) satisfies (6.6). Corollary 6.1. Let α > −1 and let μ ∈ D0 (D) be a complex Radon measure in D such that (6.6) holds. Let u = Hα [μ] and consider the functions ur as in (3.1). Then ur ∈ L1 (T) for 0 < r < 1 and limr→1 ur = 0 in L1 (T). Proof. The result follows from the proof of Theorem 6.2.



Notice that Corollary 6.1 yields nontrivial information about integrability of the function Gα . The following result is due to Gustav Behm [6]. Theorem 6.3 (Behm). Let α > −1 and let μ ∈ D0 (D) be a complex Radon measure in D such that (6.6) holds. Let u = Gα [μ]. Then Δα u = μ in D (D). Proof. Let ϕ ∈ D(D). Recall Theorem 5.3. The result of Corollary 6.1 allows us to apply Fubini’s theorem to conclude that    Gα (z, ζ) dμ(ζ) Δ∗α ϕ(z) dA(z) Δα u, ϕ = u, Δ∗α ϕ = D D    ∗ Gα (z, ζ)Δα ϕ(z) dA(z) dμ(ζ), = D

D

where

Δ∗α,z = ∂¯z (1 − |z|2 )−α ∂z , z ∈ D. Since Δα Gα (·, ζ) = δζ in D (D) we have that  Gα (z, ζ)Δ∗α ϕ(z) dA(z) = ϕ(ζ) D

for ζ ∈ D. Thus

 Δα u, ϕ =

ϕ(ζ) dμ(ζ). D

ON A WEIGHTED HARMONIC GREEN FUNCTION

Since ϕ ∈ D(D) is arbitrary, this yields the conclusion of the proposition.

33



We now return to a simply connected planar domain Ω = C. Theorem 6.4. Let α > −1 and let Ω = C be a simply connected planar domain. Let μ ∈ D0 (Ω) be a complex Radon measure in Ω satisfying (6.2). Let u = GΩ;α [μ] be the Green potential. Then ΔΩ;α u = μ in D (Ω). Furthermore, lim uΩ,a;α;r = 0

(6.8)

r→1

for every a ∈ Ω, where the semi-norms ·Ω,a;α;r are as in (3.9). Proof. Let a ∈ Ω and let ϕ : D → Ω be a biholomorphic map with ϕ(0) = a. We shall make use of the operations (1.5) and (1.9). From Theorem 6.1 we have that ∨ uϕ,α (z) = Gα [μ∨ ϕ,α ](z) for q.e. z ∈ D. By Theorem 6.3 we have that Δα uϕ,α = μϕ,α  in D (D). An application of Theorem 1.2 now gives that ∨ ΔΩ;α (uϕ,α )ϕ−1 ;α = (μ∨ ϕ,α )ϕ−1 ,α

in D (Ω).

∨ Since (uϕ,α )ϕ−1 ;α = u and (μ∨ ϕ,α )ϕ−1 ,α = μ by compatibility of logarithms, we conclude that ΔΩ;α u = μ in D (Ω). We now consider the boundary limit (6.8). Since uϕ,α = Gα [μ∨ ϕ,α ] we have from Theorem 6.2 that  π 1 lim |uϕ,α (reiθ )| dθ = 0. r→1 2π −π

From Lemma 3.1 we have that uΩ,a;α;r = uϕ,α D,0;α;r =

1 2π



π

−π

|uϕ,α (reiθ )| dθ

for 0 < r < 1, where the last equality follows by (3.10). The limit (6.8) now follows.  We record also the following uniqueness property of Green potentials. Theorem 6.5. Let α > −1 and let Ω = C be a simply connected planar domain. Let u ∈ D (Ω) be such that ΔΩ;α u = μ in D (Ω), where μ ∈ D0 (Ω) is a complex Radon measure in Ω satisfying (6.2). Assume that lim uΩ,a;α;r = 0

r→1

for some a ∈ Ω, where the semi-norms ·Ω,a;α;r are as in (3.9). Then u = GΩ;α [μ] in D (Ω). Proof. We consider the distribution v = u − GΩ;α [μ] in D (Ω). By Theorem 6.4 and assumption we have that ΔΩ;α v = μ − μ = 0 in D (Ω). By Weil’s lemma we conclude that v is α-harmonic in Ω. Again by Theorem 6.4 and assumption we have that limr→1 vΩ,a;α;r = 0. By Theorem 3.3 we conclude that v(z) = 0 for  z ∈ Ω. Thus u = GΩ;α [μ] in D (Ω).

34

ANDERS OLOFSSON

7. Regular boundary points The purpose of this section is to set up some basic machinery of regular boundary points needed for calculation of pointwise boundary limits of Green potentials almost everywhere. For eiθ ∈ T and h > 0 we denote by A(eiθ , h) the open arc A(eiθ , h) = {eiτ ∈ T : |τ − θ| < h}.

(7.1)

We denote by M (T) the space of all complex regular Borel measures on T equipped with the norm of finite total variation ν = |ν|(T) for ν ∈ M (T), where |ν| is the total variation measure of ν. The maximal function of a complex measure ν ∈ M (T) is the function M ν : T → [0, ∞] defined by π (7.2) M ν(eiθ ) = sup |ν|(A(eiθ , h)), eiθ ∈ T. h>0 h The factor π above stems from the fact that we use normalized Lebesgue measure on T, see for instance Theorem 8.1 below. It is well-known that the function M ν : T → [0, ∞] is lower semi-continuous in the usual sense that the set {eiθ ∈ T : M ν(eiθ ) > λ} is open for every real number λ. A deeper fact is that there is a weak type estimate of the form (7.3)

|{eiθ ∈ T : M ν(eiθ ) > λ}| ≤ Cν/λ

for λ > 0, where C is a finite positive constant. See [32, Chapter 7] for details. We now turn to measures carried by the open disc D. Definition 7.1. Let μ ∈ M (D) be a complex regular Borel measure on D. For 0 < ρ < 1 we define a positive measure νρ ∈ M (T) by the requirement that   ϕ(eiθ )dνρ (eiθ ) = ϕ(ζ/|ζ|)d|μ|(ζ) (7.4) T

¯ D\ρD

for ϕ ∈ C(T). By a regular boundary point for μ we understand a point eiθ ∈ T such that limρ→1 M νρ (eiθ ) = 0, where M is the maximal function on T. Let us comment on the situation in Definition 7.1. Remark 7.1. By a standard approximation procedure we have that formula (7.4) holds true whenever ϕ is a nonnegative measurable function on T. (See for instance [32, Chapter 2] for details.) Remark 7.2. For 0 < ρ1 < ρ2 < 1 we have νρ1 ≥ νρ2 ≥ 0 in M (T), and therefore M νρ1 (eiθ ) ≥ M νρ2 (eiθ ) ≥ 0 for eiθ ∈ T. As a consequence the limit limρ→1 M νρ (eiθ ) exists in [0, ∞] for every eiθ ∈ T. For eiθ ∈ T, 0 < ρ < 1 and h > 0 we denote by S(eiθ , ρ, h) the box S(eiθ , ρ, h) = {reiτ ∈ D : ρ < r < 1 and |τ − θ| < h}. We next describe the regular boundary points in more detail.

ON A WEIGHTED HARMONIC GREEN FUNCTION

35

Theorem 7.1. Let μ ∈ M (D) be a complex regular Borel measure on D. Then eiθ ∈ T is a regular boundary point for μ if and only if   1 (7.5) lim lim sup |μ|(S(eiθ , ρ, h)) = 0, ρ→1 h→0 h where the box S(eiθ , ρ, h) is as above. Proof. Let the sequence {νρ }0<ρ<1 in M (T) be associated to μ as in Definition 7.1. We observe first that (7.6)

νρ (A(eiθ , h)) = |μ|(S(eiθ , ρ, h))

for h > 0 and 0 < ρ < 1. Assume now that eiθ is a regular boundary point for μ. From (7.6) we have that 1 1 1 lim sup |μ|(S(eiθ , ρ, h)) ≤ sup νρ (A(eiθ , h)) = M νρ (eiθ ) → 0 h h π h→0 h>0 as ρ → 1, which yields (7.5). Assume next that (7.5) holds. Let ε > 0 be given. By (7.5) there exists ρ0 ∈ (0, 1) such that 1 lim sup |μ|(S(eiθ , ρ, h)) < ε/2 h h→0 for ρ0 ≤ ρ < 1. From here we see, in particular, that there exists h0 > 0 such that 1 |μ|(S(eiθ , ρ0 , h)) < ε h for 0 < h < h0 . Since the function (0, 1)  ρ → |μ|(S(eiθ , ρ, h)) is decreasing for h > 0 fixed, we conclude that 1 |μ|(S(eiθ , ρ, h)) < ε (7.7) h for ρ0 ≤ ρ < 1 and 0 < h < h0 . We shall now estimate the function value M νρ (eiθ ). Observe that for h ≥ h0 we have 1 1 νρ (A(eiθ , h)) ≤ νρ (T) h h0 for all 0 < ρ < 1. For 0 < h < h0 and ρ0 ≤ ρ < 1 we have from (7.6) that 1 1 νρ (A(eiθ , h)) = |μ|(S(eiθ , ρ, h)) < ε, h h where the last inequality follows by (7.7). Passing to the maximal function we have that M νρ (eiθ ) ≤ π max(ε, νρ (T)/h0 ) for ρ0 ≤ ρ < 1. Observe that νρ (T) → 0 as ρ → 1. Passing to the limit we have that lim M νρ (eiθ ) ≤ πε. ρ→1

Since ε > 0 is arbitrary we conclude that eiθ is a regular boundary point for μ.



We now prove that almost every boundary point is regular. Theorem 7.2. Let μ ∈ M (D) be a complex regular Borel measure on D. Then almost every eiθ ∈ T is a regular boundary point for μ.

36

ANDERS OLOFSSON

Proof. Let the sequence {νρ }0<ρ<1 in M (T) be associated to μ as in Definition 7.1. Recall from Remark 7.2 that the limit limρ→1 M νρ (eiθ ) exists in [0, ∞] for every eiθ ∈ T. We shall employ the weak type inequality (7.3) to show that this latter limit vanishes a.e. Observe that E = {eiθ ∈ T : lim M νρ (eiθ ) > 0} = {eiθ ∈ T : M νρk (eiθ ) > 1/2j }, ρ→1

j≥1 k≥1

where {ρk } is an increasing sequence such that ρk → 1 as k → ∞. Since νρ  → 0 as ρ → 1 we have from the weak type inequality (7.3) that M νρ → 0 in measure as ρ → 1, that is, |{eiθ ∈ T : M νρ (eiθ ) > λ}| → 0 as ρ → 1 for every λ > 0. This gives that the set E above is a null set.



8. A convolution inequality In this section we study a classical estimate involving the maximal function giving pointwise control of certain convolutions, see Theorem 8.1 below. Recall the maximal function (7.2). We denote by E the closure of a set E. Lemma 8.1. Let ν ∈ M (T) be a complex regular Borel measure on T. Then M ν(eiθ ) = sup

0
for e

iθ

|ν|(A(eiθ , h)) (h/π)

∈ T.

Proof. Observe first that (8.1)

sup

0
|ν|(A(eiθ , h)) |ν|(A(eiθ , h)) = sup ≥ M ν(eiθ ) (h/π) (h/π) h>0

since A(eiθ , h) ⊃ A(eiθ , h). We proceed to prove the reverse inequality. Let 0 < h0 ≤ π and h > h0 . By monotonicity we have that |ν|(A(eiθ , h0 )) h |ν|(A(eiθ , h)) h ≤ ≤ M ν(eiθ ). (h0 /π) h0 (h/π) h0 Letting h → h0 we obtain that |ν|(A(eiθ , h0 ))/(h0 /π) ≤ M ν(eiθ ). Thus sup

0
|ν|(A(eiθ , h)) ≤ M ν(eiθ ), (h/π) 

which together with (8.1) yields the conclusion of the lemma. 1

Let us digress on translation operators. The translation of a function f ∈ L (T) by eiθ ∈ T is the function Teiθ f = feiθ defined by feiθ (eiτ ) = f (ei(τ −θ) ),

eiτ ∈ T.

More generally, the translation of a measure ν ∈ M (T) by eiθ ∈ T is the measure νeiθ ∈ M (T) defined by   ϕ(eiτ ) dνeiθ (eiτ ) = ϕ(ei(τ +θ) ) dν(eiτ ) T

T

for ϕ ∈ C(T). By an approximation argument we see that νeiθ (E) = ν(e−iθ E) for every Borel subset E of T, where eiθ E is defined in the obvious way. For the total

ON A WEIGHTED HARMONIC GREEN FUNCTION

37

variation measure we have that |νeiθ | = |ν|eiθ for eiθ ∈ T. As a consequence, we have the translation invariance formula for the maximal function: M (νeiθ )(eiτ ) = M ν(ei(τ −θ) ),

eiτ ∈ T,

for eiθ ∈ T. We denote by χE the characteristic function for a set E. Proposition 8.1. Let ϕ be a function of the form  1 ϕ(eiτ ) = χA(1,h) (eiτ ) dσ(h), (0,π] h

eiτ ∈ T,

for some finite positive measure σ on (0, π]. Let ν ∈ M (T). Then   1   ϕ(ei(θ−τ ) ) d|ν|(eiτ ) ≤ (M ν)(eiθ ) ϕ(eiτ ) dτ 2π T T for eiθ ∈ T. Proof. We first prove that   1   (8.2) ϕ(eiτ ) d|ν|(eiτ ) ≤ (M ν)(1) ϕ(eiτ ) dτ 2π T T for functions ϕ as above. Set 1 χ (eiτ ), eiτ ∈ T, h A(1,h) for 0 < h ≤ π. From Lemma 8.1 we have that  1 (8.3) ϕh (eiτ ) d|ν|(eiτ ) ≤ (M ν)(1) π T ϕh (eiτ ) =

for 0 < h ≤ π. Notice that iτ



ϕ(e ) = (0,π]

ϕh (eiτ ) dσ(h),

eiτ ∈ T,

where ϕ is as above. An integration using Fubini’s theorem shows that (8.4)     1 ϕ(eiτ ) d|ν|(eiτ ) = ϕh (eiτ ) d|ν|(eiτ ) dσ(h) ≤ (M ν)(1)σ((0, π]), π T (0,π] T where the last inequality follows by (8.3). Similarly, we have that     1  1 1 ϕ(eiτ ) dτ = ϕh (eiτ ) dτ dσ(h) = σ((0, π]). 2π T 2π π (0,π] T This latter equality together with (8.4) proves (8.2). Observe that ϕ(eiτ ) = ϕ(e−iτ ) for eiτ ∈ T. Using translation invariance of the maximal function we have from (8.2) that    ϕ(ei(θ−τ ) ) d|ν|(eiτ ) = ϕ(ei(τ −θ) ) d|ν|(eiτ ) = ϕ(eiτ ) d|νe−iθ |(eiτ ) T T T  1    1   iτ iθ ϕ(e ) dτ = (M ν)(e ) ϕ(eiτ ) dτ ≤ (M νe−iθ )(1) 2π T 2π T for eiθ ∈ T. This completes the proof of the proposition. We now calculate the function ϕ in Proposition 8.1.



38

ANDERS OLOFSSON

Proposition 8.2. Let ϕ and σ be as in Proposition 8.1. Then ϕ(eiτ ) = ϕ0 (|τ |) for eiτ ∈ T with |τ | ≤ π, where  1 ϕ0 (t) = dσ(h), t ∈ (0, π], h [t,π] and ϕ0 (0) = (0,π] h1 dσ(h). Proof. For 0 < |τ | ≤ π we have ϕ(eiτ ) =

 [|τ |,π]

1 dσ(h) = ϕ0 (|τ |), h 

and similarly when τ = 0. We shall next study functions ϕ0 of the form  1 (8.5) ϕ0 (t) = dσ(h), t ∈ (0, π], h [t,π]

for some positive Radon measure σ on (0, π]. Observe that ϕ0 (0+) = (0,π] h1 dσ(h) by monotone convergence. It is evident that the function ϕ0 : (0, π] → R given by (8.5) is decreasing, nonnegative and left continuous. It is straightforward to check that σ(t) = −tϕ0 (t)

for t ∈ (0, π)

in the distributional sense and that ϕ0 (π) =

1 π σ({π}).

Proposition 8.3. Let ϕ0 : (0, π] → R be a decreasing, nonnegative and left continuous function. Then ϕ0 has the form (8.5) for some positive Radon measure σ on (0, π]. Furthermore  π ϕ0 (t) dt = σ((0, π]), 0

where σ is as in (8.5). Proof. Let μ = ϕ0 in D (0, π). Let 0 < a < π. From distribution theory we know that μ is a negative Radon measure on (0, π) (see [22, Theorem 4.1.6]). Furthermore,  (8.6) C − ϕ0 (x) = dμ(t) [x,a)

for a.e. 0 < x < a, where C is a constant (see [22, Theorem 3.1.4]). Since both sides of (8.6) are left continuous, we conclude that (8.6) holds for all x ≤ a. Choosing x = a in (8.6) we see that C = ϕ0 (a). Thus  (8.7) ϕ0 (a) − ϕ0 (x) = dμ(t) [x,a)

for 0 < x ≤ a. Letting a → π in (8.7) we obtain that  dμ(t) ϕ0 (π) − ϕ0 (x) = [x,π)

for 0 < x ≤ π. In particular, μ([1, π)) > −∞. Set dσ(t) = −tdμ(t) + πϕ0 (π)dδπ (t)

for t ∈ (0, π]

ON A WEIGHTED HARMONIC GREEN FUNCTION

39

in the sense of distribution theory. It is straightforward to check that σ is a positive Radon measure on (0, π] such that (8.5) holds. Finally, for ϕ of the form (8.5) an application of Fubini’s theorem shows that  π   1  π ϕ0 (t) dt = χ[t,π] (h) dt dσ(h) = σ((0, π]). (0,π] h 0 0 This completes the proof of the proposition.



We now return to the convolution estimate from Proposition 8.1. Theorem 8.1. Let ϕ : T → R ∪ {∞} be a function of the form ϕ(eiτ ) = ϕ0 (|τ |) for eiτ ∈ T with 0 < |τ | ≤ π, where π ϕ0 : (0, π] → R is decreasing on (0, π], nonnegative on (0, π) and such that 0 < 0 ϕ0 (t) dt < +∞. Let ν ∈ M (T). Then   1   ϕ(ei(θ−τ ) ) d|ν|(eiτ ) ≤ (M ν)(eiθ ) ϕ(eiτ ) dτ (8.8) 2π T T for eiθ ∈ T. Proof. In proving (8.8) we can without loss of generality assume that eiθ ∈ T is such that (M ν)(eiθ ) < +∞. For such eiθ we have |ν|({eiθ }) = 0, so that the convolution integral  ϕ ∗ |ν|(eiθ ) =

ϕ(ei(θ−τ ) ) d|ν|(eiτ )

T

does not depend on how the individual function value ϕ(1) is defined. Let ϕ(e ˜ iτ ) = ϕ˜0 (|τ |) for eiτ ∈ T with 0 < |τ | ≤ π, where ϕ˜0 (t) = ϕ0 (t−) for 0 < t ≤ π. Observe that ϕ ≤ ϕ˜ on T\{1} and π ϕ˜0 and ϕ0 coincide πthat the functions at all but countably many points, so that 0 ϕ˜0 (t) dt = 0 ϕ0 (t) dt. Proposition 8.3 applies to show that the function ϕ˜0 has the form  1 ϕ˜0 (t) = dσ(h), t ∈ (0, π], h [t,π] for some finite positive measure σ on (0, π]. By Proposition 8.2 the function ϕ˜ is now admissible for Proposition 8.1. By Proposition 8.1 we now have that    1   i(θ−τ ) iτ i(θ−τ ) iτ iθ ϕ(e ) d|ν|(e ) ≤ ϕ(e ˜ ) d|ν|(e ) ≤ (M ν)(e ) ϕ(e ˜ iτ ) dτ 2π T T T  1   = (M ν)(eiθ ) ϕ(eiτ ) dτ 2π T for eiθ ∈ T, which yields (8.8).



We mention that maximal function estimates of the above form (8.8) using an even decreasing majorant goes back at least to Calderon and Zygmund [8, Chapter II]; see also Stein [35, Theorem III.2(a)] or Grafakos [18, Section 2.1.2]. Our presentation above is inspired by the 1968 edition of Katznelson [23, Lemma III.2.4]. 9. Littlewood’s theorem The purpose of this section is to extend Littlewood’s theorem [26] about the classical Green potential for the unit disc to weighted Green potentials of the form Gα [μ]. Along the way we shall establish also vanishing non-tangential boundary limits a.e. on T for some related transforms Bα [μ] defined below.

40

ANDERS OLOFSSON

We shall first discuss transforms of the form  (1 − |z|2 )α+1 (9.1) Bα [μ](z) = ¯ α+2 dμ(ζ), D |1 − ζz|

z ∈ D,

where α > −1 and μ ∈ M (D). Notice that in (9.1) the measure μ is carried by the open unit disc D. Observe also that if μ has compact support contained in D, then Bα [μ](z) → 0 as |z| → 1. For the purpose of estimation of Bα [μ] we shall use the following elementary lemma. Lemma 9.1. Let z ∈ D. Then |1 − ρz| ≥ ((1 + ρ)/2)|1 − z| ≥ |1 − z|/2 for 0 < ρ < 1. Proof. Notice that

 z  1 − ρz = (1 − z) + (1 − ρ)z = (1 − z) 1 + (1 − ρ) . 1−z

The M¨obius transformation z → w = z/(1 − z) maps D one-to-one onto the halfplane (w) > −1/2. As a consequence we have that |1 − ρz| ≥ |1 − z|(1 − (1 − ρ)/2) = ((1 + ρ)/2)|1 − z| ≥ |1 − z|/2, 

which yields the conclusion of the lemma.

We denote by Γβ (eiθ0 ) a standard non-tangential approach region of the form Γβ (eiθ0 ) = {reiθ ∈ D : |θ − θ0 | < β(1 − r), r ≥ 0, θ ∈ R}, where eiθ0 ∈ T and β > 0 is a positive constant. Theorem 9.1. Let α > −1 and μ ∈ M (D). Let eiθ0 ∈ T be a regular boundary point for μ. Then (9.2)

lim

Γβ (eiθ0 ) z→eiθ0

Bα [μ](z) = 0

for every β > 0. As a consequence (9.2) holds for almost every eiθ0 ∈ T. Proof. Recall from Theorem 7.2 that a.e. eiθ0 ∈ T is a regular boundary point for μ. We proceed to prove (9.2) for such a point eiθ0 . Let 0 < ρ < 1. We write Bα [μ](z) = I1,ρ (z) + I2,ρ (z), where   (1 − |z|2 )α+1 (1 − |z|2 )α+1 dμ(ζ) and I (z) = I1,ρ (z) = 2,ρ ¯ α+2 ¯ α+2 dμ(ζ). ¯ |1 − ζz| ¯ |1 − ζz| ρD D\ρD Observe that I1,ρ (z) → 0 as |z| → 1 for every 0 < ρ < 1. We shall next estimate the integral I2,ρ (z). By Lemma 9.1 we have that   (1 − |z|2 )α+1 (1 − |z|2 )α+1 α+2 d|μ|(ζ) = 2 dνρ (eiτ ), |I2,ρ (z)| ≤ 2α+2 −iτ z|α+2 α+2 ¯ ¯ |1 − (ζ/|ζ|)z| D\ρD T |1 − e where νρ is associated to μ as in Definition 7.1. This latter integral is a Poisson integral of the type considered in [28].

ON A WEIGHTED HARMONIC GREEN FUNCTION

41

Let z = reiθ ∈ Γβ (eiθ0 ) with r ≥ 0. Now e−iτ z ∈ Γβ (ei(θ0 −τ )) and by [29, Lemma 6.1] we have that  (1 − r2 )α+1 (9.3) dνρ (eiτ ) |I2,ρ (z)| ≤ 2α+2 (1 + β)α+2 i(θ0 −τ ) |α+2 T |1 − re = (2(1 + β))α+2 (Kα,r ∗ νρ )(eiθ0 ), where

(1 − r2 )α+1 , eiτ ∈ T. |1 − reiτ |α+2 From [29, Corollary 2.8] or [28, Theorem 3.1] we know that Kα,r (eiτ ) =

Kα,r L1 (T) ≤ Γ(α + 1)/Γ(α/2 + 1)2 . An application of Theorem 8.1 to the convolution in (9.3) now gives that |I2,ρ (z)| ≤ Cα,β M νρ (eiθ0 )

(9.4) for z ∈ Γβ (eiθ0 ), where

Cα,β = (2(1 + β))α+2 Γ(α + 1)/Γ(α/2 + 1)2 . Let ε > 0 be given. Since eiθ0 ∈ T is a regular boundary point for μ we can choose 0 < ρ < 1 above such that M νρ (eiθ0 ) < ε/(2Cα,β + 1) (see Definition 7.1). Next choose 0 < rε < 1 so that |I1,ρ (z)| < ε/(2Cα,β + 1) if rε < |z| < 1. For z ∈ Γβ (eiθ0 ) with |z| > rε we now have that |Bα [μ](z)| ≤ |I1,ρ (z)| + |I2,ρ (z)| < ε/(2Cα,β + 1) + Cα,β ε/(2Cα,β + 1) < ε, where in the mid inequality we have used (9.4). Since ε > 0 is arbitrary this proves (9.2).  Recall from Section 3 the notion of pseudo hyperbolic disc. In our context of the unit disc Ω = D, the pseudo hyperbolic disc with center z ∈ D and radius 0 <  < 1 is defined by D(z, ) = {ζ ∈ D : |ϕz (ζ)| < }, where ϕz (ζ) = (z − ζ)/(1 − z¯ζ). Observe that 0 ∈ D(z, ) if and only if |z| < . We denote by B(z, r) the Euclidean disc with center z ∈ C and radius r > 0, see the paragraph preceding Proposition 6.1. Lemma 9.2. Let z ∈ D and 0 <  < 1. Then the pseudo hyperbolic disc D(z, ) is an Euclidean disc B(P (z, ), R(z, )) with center P (z, ) =

1 − 2 z 1 − 2 |z|2

and radius R(z, ) = 

1 − |z|2 . 1 − 2 |z|2

Proof. Notice that ζ ∈ D(z, ) if and only if |ζ − z|2 < 2 |1 − z¯ζ|2 . A completion of squares in the ζ-variable shows that this latter inequality is equivalent to  1 − 2 2 (1 − |z|2 )2  z  < 2 . ζ − 2 2 1 −  |z| (1 − 2 |z|2 )2 This yields the conclusion of the lemma. 

42

ANDERS OLOFSSON

We shall need some size estimates of pseudo hyperbolic discs. Lemma 9.3. Let z ∈ D and 0 <  < 1. Then |z| +  |z| −  ≤ |ζ| ≤ 1 − |z| 1 + |z| for ζ ∈ D(z, ). Proof. The lemma is a consequence of Theorem 3.4. We omit the details.



Notice that the lower bound in Lemma 9.3 is interesting only when |z| > . Recall the pseudo hyperbolic metric ρD for D defined in (3.5). Lemma 9.4. Let z, ζ ∈ D and 0 <  < 1. Then 1 − |ζ|2 1+ 1− ≤ ≤ 1+ 1 − |z|2 1− if ρD (z, ζ) < . Proof. The lemma is a consequence of Corollary 3.1. We omit the details.



We now estimate D(z, ) in the angular direction. Lemma 9.5. Let z ∈ D and 0 <  < |z|. Then  (1 − |z|2 )  |arg(ζ z¯)| < arcsin |z|(1 − 2 ) for ζ ∈ D(z, ). Proof. In view of Lemma 9.2 it is geometrically clear that the disc D(z, ) is contained in a sector centered around the point z with opening angle 2θ, where sin(θ) = R(z, )/|P (z, )| =

(1 − |z|2 ) |z|(1 − 2 )

and 0 < θ < π/2. This yields the conclusion of the lemma.



We shall need some estimates of Green functions. Lemma 9.6. Let α > −1. Let z ∈ D and 0 <  < 1. Then |Gα (z, ζ)| ≤ C1 (α, ) for ζ ∈ D \ D(z, ), where

(1 − |z|2 )α+1 (1 − |ζ|2 )α+1 ¯ α+2 |1 − ζz|

 C1 (α, ) = max 1,

1  log(1/2 ) . α+1 1 − 2

Proof. Let the quantity Hα (z, ζ) be as in (5.9). By a monotonicity property of the logarithm we have that log(2 ) log(|ϕz (ζ)|2 ) ≥ 1 − |ϕz (ζ)|2 1 − 2 for ζ ∈ D \ D(z, ) (see Remark 5.1). Thus (1 − |z|2 )α (1 − |ζ|2 )α log(2 ) (1 − |ϕz (ζ)|2 ) ¯ α 1 − 2 |1 − ζz| log(1/2 ) (1 − |z|2 )α+1 (1 − |ζ|2 )α+1 = , ¯ α+2 1 − 2 |1 − ζz|

Hα (z, ζ) ≤ −

ON A WEIGHTED HARMONIC GREEN FUNCTION

43

where the last equality follows by formula (2.3). The conclusion of the lemma now follows from Theorem 5.3.  Recall that (z) denotes the imaginary part of a complex number z ∈ C. Lemma 9.7. For z, ζ ∈ C with ζ = 0 we have that 2 ¯ . |z − ζ|2 = (|z| − |ζ|)2 + ( ((ζ/|ζ|)z)) Proof. Observe first that ¯ − z¯ζ + |ζ|2 . |z − ζ|2 = |z|2 − ζz ¯ Set λ = ((ζ/|ζ|)z). A completion of squares shows that |z − ζ|2 = |ζ|2 − 2|ζ|λ + |z|2 = (|ζ| − λ)2 − λ2 + |z|2 . 2 ¯ . This yields the conclusion of the lemma. Observe that |z|2 − λ2 = ( ((ζ/|ζ|)z))  For easy reference we record the formula 1 (1 − |ζ|2 )(1 − |z|2 ) = 1 + , |ϕζ (z)|2 |ζ − z|2

(9.5)

where ϕζ is as in (2.2). Formula (9.5) is straightforward to check. Lemma 9.8. Let α > −1. Let z ∈ D and 0 <  < |z|. Then  (1 − |ζ|2 )α+1 1 +  (1 − |z|2 )2  |Gα (z, ζ)| ≤ C2 (α, ) log 1 + 2 ¯ 1 − |z|2 1 −  ( ((ζ/|ζ|)z)) for ζ ∈ D(z, ), where

 C2 (α, ) = max 1,

1  2α+2 . α + 1 1 − 2

Proof. Recall Theorem 5.3. We shall estimate the quantity Hα (z, ζ) in (5.9). Let us first consider the quantity 1/|ϕz (ζ)|2 . Recall formula (9.5). By Lemmas 9.4 and 9.7 we have that 1 1 +  (1 − |z|2 )2 ≤ 1 + 2 ¯ |ϕz (ζ)|2 1 −  ( ((ζ/|ζ|)z)) for ζ ∈ D(z, ). Since the logarithm is increasing we have that  1 +  (1 − |z|2 )2  (9.6) log(1/|ϕz (ζ)|2 ) ≤ log 1 + 2 ¯ 1 −  ( ((ζ/|ζ|)z)) for ζ ∈ D(z, ). It remains to estimate the factor F (z, ζ) =

(1 − |z|2 )α (1 − |ζ|2 )α ¯ α |1 − ζz|

in (5.9). By formula (2.3) we have that (1 − |z|2 )(1 − |ζ|2 ) = 1 − |ϕz (ζ)|2 ≥ 1 − 2 ¯ 2 |1 − ζz| for ζ ∈ D(z, ). Thus F (z, ζ) ≤

1 (1 − |z|2 )α+1 (1 − |ζ|2 )α+1 2α+2 (1 − |ζ|2 )α+1 ≤ 2 α+2 1− |1 − ζz| 1 − 2 1 − |z|2

44

ANDERS OLOFSSON

for ζ ∈ D(z, ), where the last inequality is straightforward to check. By Theorem 5.3 this latter estimation together with (9.6) yield the result of the lemma.  Notice that the assumption 0 <  < |z| in Lemma 9.8 means that 0 ∈ D(z, ). Theorem 9.2. Let α > −1. Let μ ∈ D0 (D) be a complex Radon measure in D such that  (1 − |ζ|2 )α+1 d|μ|(ζ) < +∞ D

and set d˜ μ(ζ) = (1 − |ζ|2 )α+1 d|μ|(ζ),

ζ ∈ D.

˜. Then Let eiθ0 ∈ T be a regular boundary point for μ lim Gα [μ](reiθ0 ) = 0.

(9.7)

r→1

As a consequence (9.7) holds for almost every eiθ0 ∈ T. Proof. Recall from Theorem 7.2 that a.e. eiθ0 ∈ T is a regular boundary point for μ ˜. We proceed to prove (9.7) for such a point eiθ0 . Let 0 <  < 1. Let z = reiθ0 with  < r < 1. We write Gα [μ](z) = I1, (z) + I2, (z), where   I1, (z) = Gα (z, ζ) dμ(ζ) and I2, (z) = Gα (z, ζ) dμ(ζ). D\D(z,)

D(z,)

We first consider the integral I1, (z). By Lemma 9.6 we have that  (1 − |z|2 )α+1 (1 − |ζ|2 )α+1 d|μ|(ζ) |I1, (z)| ≤ C1 (α, ) ¯ α+2 |1 − ζz| D\D(z,) μ](z), ≤ C1 (α, )Bα [˜ where the transform Bα [˜ μ] is defined as in (9.1). By Theorem 9.1 we conclude that lim I1, (reiθ0 ) = 0

(9.8)

r→1

for every 0 <  < 1. We now consider the integral I2, (z). By Lemma 9.8 we have that   C2 (α, ) 1 +  (1 − |z|2 )2  d˜ μ(ζ), log 1 + (9.9) |I2, (z)| ≤ 2 ¯ 1 − |z|2 D(z,) 1 −  ( ((ζ/|ζ|)z)) where the constant C2 (α, ) is as in Lemma 9.8. From Lemmas 9.3 and 9.5 we have that the disc D(z, ) is contained in the box   ¯ iθ0 )| < θ(r) and ρ(r) < |ζ| < 1 , S(eiθ0 , ρ(r), θ(r)) = ζ ∈ D : |arg(ζe where θ(r) = arcsin

 (1 − r2 )  r(1 −

2 )

and

ρ(r) =

r− . 1 − r

Observe that ρ(r) → 1 as r → 1. We shall use also the circular arc A(eiθ0 , θ(r)) defined as in (7.1). We introduce measures νρ in M (T) for 0 < ρ < 1 corresponding

ON A WEIGHTED HARMONIC GREEN FUNCTION

45

to the positive measure μ ˜ in M (D) as in Definition 7.1. From (9.9) we now have that   C2 (α, ) 1 +  (1 − |z|2 )2  log 1 + (9.10) |I2, (z)| ≤ d˜ μ(ζ) 2 2 ¯ 1 − |z| S(eiθ0 ,ρ(r),θ(r)) 1 −  ( ((ζ/|ζ|)z))   1 +  (1 − r2 )2  C2 (α, ) log 1 + = dνρ(r) (eiτ ) 2 1−r 1 −  r2 sin2 (θ0 − τ ) A(eiθ0 ,θ(r)) = C2 (α, )(kr, ∗ νρ(r) )(eiθ0 ), where kr, (eiτ ) =

 1 1 +  (1 − r2 )2  log 1 + 2 1−r 1 −  r2 sin2 (τ )

for eiτ ∈ A(1, θ(r)) and kr, (eiτ ) = 0 for eiτ ∈ T \ A(1, θ(r)). We shall next estimate the integral of kr, . We have that  θ(r)  1 1 +  (1 − r2 )2  kr, L1 (T) = log 1 + dτ 2π(1 − r2 ) −θ(r) 1 −  r2 sin2 (τ )  θ(r)  1 +  (1 − r2 )2  1 dτ. log 1 + = π(1 − r2 ) 0 1 −  r2 sin2 (τ ) By the change of variables t = sin τ we have that  sin θ(r)  1 1 +  (1 − r2 )2  1 √ 1 kr, L (T) = log 1 + dt 2 π(1 − r ) 0 1 −  r 2 t2 1 − t2  sin θ(r)  r(1 − 2 ) 1 1 +  (1 − r2 )2   ≤ log 1 + dt, 2 π(1 − r ) (r2 − 2 )(1 − 2 r2 ) 0 1 −  r 2 t2 where the last inequality is straightforward to check. Notice that this latter integral B has the form 0 log(1 + 1/(At)2 ) dt with  1 −  1/2 r (1 − r2 ) A= . and B = 2 1+ 1−r r(1 − 2 ) By the change of variables t → At we have that  B      1 AB 1 ∞ π 1  1 1 dt = dt ≤ dt = , log 1 + log 1 + log 1 + 2 2 2 (At) A t A t A 0 0 0 where the last equality follows by integration by parts. In our case we obtain an estimate  1 +  1/2 (1 − 2 ) kr, L1 (T) ≤  (r2 − 2 )(1 − 2 r2 ) 1 −  for the function kr, . We now return to the estimation of I2, (z). Notice that the functions kr, satisfy the assumptions of Theorem 8.1. An application of that result to the convolution in (9.10) gives that |I2, (reiθ0 )| ≤ C2 (α, )kr, L1 (T) M νρ(r) (eiθ0 )  1 +  1/2 (1 − 2 ) M νρ(r) (eiθ0 ). ≤ C2 (α, )  (r2 − 2 )(1 − 2 r2 ) 1 − 

46

ANDERS OLOFSSON

Passing to the limit we conclude that lim I2, (reiθ0 ) = 0

r→1

˜ (see Definition 7.1). for every 0 <  < 1 since eiθ0 is a regular boundary point for μ Combined with (9.8) this yields (9.7). This completes the proof of the theorem.  We mention that Proposition 5.1 can be used to reduce Theorem 9.2 to Theorem 9.1 and the special case α = 0 of Theorem 9.2. We omit the details. References [1] Abramowitz M, Stegun I, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, 1964. [2] Aikawa H, Thin sets at the boundary, Proc. London Math. Soc. (3) 65 (1992) 357-382. [3] Ancona A, Sur la th´ eorie du potentiel dans les domaines de John, Publ. Mat. 51 (2007) 345-396. [4] Aronszajn N, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950) 337-404. [5] Bear HS, Lectures on Gleason parts, Lecture Notes in Mathematics 121, Springer-Verlag, 1970. [6] Behm G, Solving Poisson’s equation for the standard weighted Laplacian in the unit disc, arXiv:1306.2199v2 [math.AP] 16 Apr 2014. [7] Borichev A, Hedenmalm H, Weighted integrability of polyharmonic functions, Adv. Math. 264 (2014) 464-505. [8] Calderon AP, Zygmund A, On the existence of certain singular integrals, Acta Math. 88 (1952) 85-139. [9] Carlsson M, Wittsten J, The Dirichlet problem for standard weighted Laplacians in the upper half plane, J. Math. Anal. Appl. 436 (2016) 868-889. [10] Chen S, Vuorinen M, Some properties of a class of elliptic partial differential operators, J. Math. Anal. Appl. 431 (2015) 1124-1137. [11] Chen X, Kalaj D, A representation theorem for standard weighted harmonic mappings with an integer exponent and its applications, J. Math. Anal. Appl. 444 (2016) 1233-1241. [12] Dahlberg B, On the existence of radial boundary value for functions subharmonic in a Lipschitz domain, Indiana Univ. Math. J. 27 (1978) 515-526. [13] Doob JL, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften 262, Springer-Verlag, New York, 1984. [14] Duren P, Gallardo-Guti´ errez EA, Montes-Rodriguez A, A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces, Bull. Lond. Math. Soc. 39 (2007) 459-466. [15] Duren P, Weir R, The pseudohyperbolic metric and Bergman spaces in the ball, Trans. Amer. Math. Soc. 359 (2007) 63-76. [16] Gamelin TW, Complex analysis, Springer-Verlag, New York, 2001. [17] Garabedian PR, A partial differential equation arising in conformal mapping, Pacific J. Math. 1 (1951) 485-524. [18] Grafakos L, Classical Fourier analysis, third edition, Graduate Texts in Mathematics 249, Springer, New York, 2014. [19] Gupta A, Nadarajah S, Handbook of Beta distribution and its applications, Dekker, 2004. [20] Hedenmalm H, The dual of a Bergman space on simply connected domains, J. Anal. Math. 88 (2002) 311-335. [21] Hedenmalm H, Korenblum B, Zhu K, Theory of Bergman spaces, Springer-Verlag, 2000. [22] H¨ ormander L, The analysis of linear partial differential operators I, Springer, 1990. [23] Katznelson Y, An introduction to harmonic analysis, third edition, Cambridge, 2002. [24] Krantz SG, Geometric function theory. Explorations in complex analysis, Birkh¨ auser Boston, 2006. [25] Krantz SG, W´ ojcicki P, The weighted Bergman kernel and the Green’s function, Complex Anal. Oper. Theory 11 (2017) 217-225. [26] Littlewood JE, On functions subharmonic in a circle (II), Proc. London Math. Soc., Ser. 2, 28 (1928) 383-394.

ON A WEIGHTED HARMONIC GREEN FUNCTION

47

[27] Na¨ım L, Sur le rˆ ole de la frontiere de R. S. Martin dans la th´eorie du potentiel, Ann. Inst. Fourier (Grenoble) 7 (1957) 183-281. [28] Olofsson A, Differential operators for a scale of Poisson type kernels in the unit disc, J. Anal. Math. 123 (2014) 227-249. [29] Olofsson A, Wittsten J, Poisson integrals for standard weighted Laplacians in the unit disc, J. Math. Soc. Japan 65 (2013) 447-486. [30] Ransford T, Potential theory in the complex plane, London Mathematical Society Student Texts 28, Cambridge University Press, Cambridge, 1995. [31] Rippon PJ, On the boundary behaviour of Green potentials, Proc. London Math. Soc. (3) 38 (1979) 461-480. [32] Rudin W, Real and complex analysis, third edition, McGraw-Hill, 1987. [33] Shimorin S, On a family of conformally invariant operators, Algebra i Analiz 7 (1995) 133-158; translation in St. Petersburg Math. J. 7 (1996) 287-306. [34] Shimorin S, Weighted composition operators associated with conformal mappings, Quadrature domains and their applications, 217-237, Oper. Theory Adv. Appl. 156, Birkh¨ auser, Basel, 2005. [35] Stein EM, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970. Mathematics, Faculty of Science, Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden E-mail address: [email protected]