Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 55–68

REGULATED DOMAINS AND BERGMAN TYPE PROJECTIONS

Jari Taskinen

University of Joensuu, Department of Mathematics P.O. Box 111, FIN-80101 Joensuu, Finland; Jari.Taskinen@Joensuu.Fi

Abstract. We analyze the relations of the geometry of a regulated complex domain Ω with the existence of Bergman type projections from L_{ω}^{p}(Ω) onto the Bergman space 𝒜_{ω}^{p}(Ω). The main technical device is a Muckenhoupt type weight condition. In particular we find bounded Bergman type projections on 𝒜^{p}(Ω) even in the case Ω has arbitrary inward or outward cusps. As a consequence, 𝒜_{ω}^{p}(Ω) is isomorphic as a Banach space to l_{p}.

1. Introduction

Let Ω ⊂ C be a complex domain bounded by a Jordan curve. We want to find Bergman type projections from L_{ω}^{p}(Ω) onto the corresponding Bergman space. Moreover, we want to analyze the relations of the geometry of Ω with the boundedness of various Bergman projections. Here, the space L_{ω}^{p}(Ω) is with respect to a weighted 2-dimensional Lebesgue measure, where the weight ω is of the simplest possible type: it is a power of the boundary distance,

(1.1)  ω(z) = (dist (z, ∂Ω))^{α}

for some α > −1. By the Bergman space 𝒜_{ω}^{p}(Ω) we mean the space of analytic mappings f: Ω → C endowed with the norm

(1.2)  ∥f∥ := ∥f∥_{p,ω} := (∫_{Ω} |f|^{p}ω dm)^{1/p} < ∞.

We only deal with the case 1 < p < ∞.

As a consequence of the Koebe distortion theorem (see [5, Corollary 1.4]), the weights (1.1) correspond on the open unit disc D to the weights where the weight

(1.3)  ν(z) = |ψ′(z)|^{2+α}(1 − |z|)^{α},

where ψ: D → Ω is a Riemann conformal map. Clearly, ν is in general a nonradial weight on D, although it emerges in a canonical way from a most natural class of weights on Ω.

2000 Mathematics Subject Classification: Primary 46E15; Secondary 46B03, 47B38, 30H05, 30C20.

Academy of Finland project number 38954.

It turns out that certain Muckenhoupt-type conditions will play a central role. Our predecessor is the paper [2, p. 143], where it was shown that the weight |ψ′|^{2−p} on D satisfies the Békollé–Bonami condition B_{p}(0), i.e.,

(1.4)  sup_{S} ∫_{S} |ψ′|^{2−p} dm(∫_{S} |ψ′|^{(2−p)(−p^{∗}/p)} dm)^{p/p^{∗}} ≤ Cm(S)^{p}

where p and p^{∗} are the usual dual indices of each other, if and only if certain geometric conditions for the boundary of Ω are satisfied. In (1.4) the sup is taken over all sets

(1.5)  S := S(θ, ϱ) := {z = re^{it} ∈ D : 1 − ϱ < r < 1 and |θ − t| < 2πϱ},

where 0 ≤ θ ≤ 2π and 0 < ϱ < 1.

Moreover, in [2] it was shown that the condition (1.4) is equivalent to the boundedness of the Bergman projection on the space L^{p}(Ω).

The conditions of [2] for Ω require more than being just a regulated domain in the sense of [5].

In our paper we generalize the above mentioned result of [2] both qualitatively and quantitatively, see Theorem 3.1. On the technical, quantitative side we show that (1.4) still holds if |ψ′| is raised to more arbitrary powers and dm is replaced by a weighted measure

(1.6)  dm_{α} := (1 − |z|)^{α} dm.

On the qualitative side, our results hold for arbitrary regulated domains. In particular we do not need the assumption that the domain should be of bounded boundary rotation type, cf. [2, assumption H2, (ii) on p. 138]. We do not put any restriction for the number of cusps as in H2, (i), on p. 155. On the other hand, we are not able to handle the limiting case (i.e. equality in the condition (3.3)) in this generality. It seems that some additional assumptions on Ω are needed in that case. A version of that result is contained in a different paper [6]; it is not presented here since the proof is technical and based on a completely different method.

Theorem 3.1 implies boundedness results for a large family of Bergman type projections from L_{ω}^{p}(Ω) onto 𝒜_{ω}^{p}(Ω). These are presented in Section 4. It follows from [2] that for example the existence of cusps in Ω may make the standard Bergman projection unbounded on L^{p}(Ω). However, we are able to show that for an arbitrary (regulated) Ω there are more general, but still of the Bergman type, bounded projections, see Corollary 4.6.

It also follows from these results that 𝒜_{ω}^{p}(Ω) is linearly homeomorphic to the Banach space l_{p}; see Corollary 4.7.

Concerning notation, dm denotes the two-dimensional Lebesgue measure and dμ is the one-dimensional Lebesgue measure on the boundary of the unit disc. The notation “dist” and “diam” denotes the euclidean distance of two given points, respectively, the euclidean diameter of a set. We denote by C, C′ or c strictly positive constants which may vary from place to place but not in the same sequence of inequalities. By writing “≅” between two quantities A and B (which may depend on some variables z, θ, …) we mean that there exists a constant C > 0 (independent of z, θ, …) such that C^{−1}B ≤ A ≤ CB. Otherwise we recommend the references [3], [5] and [8].

2. Regulated domains

Before formulating our main result, Theorem 3.1, we recall the properties of regulated domains. Let us start with a simply connected, bounded domain Ω ⊂ C with a locally connected boundary. In this case a Riemann conformal map ψ: D → Ω has a continuous extension to \overline{D} (still denoted by ψ). We can thus define the curve w(t) = ψ(e^{it}), 0 ≤ t ≤ 2π. According to [5, Section 3.5], Ω is called a regulated domain, if each point of ∂Ω is attained only finitely often by ψ, and if

(2.1)  β(t) := lim_{τ→t^{+}} arg(w(τ) − w(t))

exists for all t and defines a regulated function. (Recall that β is regulated, if it can be approximated uniformly by step functions, i.e. for every ε > 0 there exist 0 = t_{0} < t_{1} < ⋯ < t_{n} = 2π and constants γ_{1}, …, γ_{n} such that

(2.2)  |β(t) − γ_{j}| < ε  for t_{j−1} < t < t_{j}, j = 1, …, n.)

Geometrically, β is the direction angle of the forward tangent of ∂Ω at w(t). For more details, see [5, Section 3.5].

Regulated domains can be characterized as follows.

Theorem 2.1. Let Ω ⊂ D be a simply connected domain with locally connected boundary. Then Ω is regulated if and only if, for a Riemann conformal map ψ: D → Ω,

(2.3)  log ψ′(z) = log |ψ′(0)| + \frac{i}{2π} ∫_{0}^{2π} \frac{e^{it} + z}{e^{it} − z} (β(t) − t − \frac{π}{2} ) dt,

where β: [0, 2π] → R is a regulated function.

In the situation of Theorem 2.1 the function β coincides with the direction angle defined above. For a proof, see [5].

Let us denote by BMO(∂D) the BMO-space on the boundary of the unit disc; for a detailed definition, see [8]. Let BMO(D) stand for the space of measurable functions f on the disc with norm

(2.4)  sup_{Q} \frac{1}{m(Q)} ∫_{Q} |f − f_{Q}| dm,

where Q is running over all open Euclidean discs intersecting D and f_{Q} is the mean of f over Q. See [3, p. 282].

We shall need the John–Nirenberg inequality for functions in BMO(∂D) (cf. [3, Theorem VI.2.1]):

Lemma 2.2. There exists a constant C > 0 such that for every positive λ

(2.5)  sup_{I} \frac{μ(\{x ∈ I | |f − f_{I}| > λ\})}{μ(I)} ≤ e^{−Cλ/∥f∥_{BMO(∂D)}},

where I runs over all subarcs of ∂D, μ(I) denotes the 1-dimensional Lebesgue measure of I, f ∈ BMO(∂D), and f_{I} denotes the integral mean of f over I.

Given a function in BMO(∂D) we shall also need to control some twodimensional BMO-like properties of its Poisson extension:

Lemma 2.3. There exists a constant C > 0 as follows. If f ∈ BMO(∂D) with ∥f∥_{BMO} ≤ 1 and F: D → C is the harmonic Poisson extension of f, then

(2.6)  sup_{Q} \frac{m(\{z ∈ Q | |F (z) − F_{Q}| ≥ λ\})}{m(Q)} ≤ e^{−Cλ/∥f∥_{BMO}(∂D)},

where the supremum is taken over all Euclidean discs intersecting D and F_{Q} is the mean value of F on Q.

Proof. It is known that BMO(∂D) is mapped continuously into the harmonic Bloch space ℬ_{h} (the space of harmonic functions on the unit disc with finite Bloch norm) via the Poisson extension. On the other hand, ℬ_{h} is known to have a norm equivalent to that of BMO(D). See e.g. [8, p. 188] and [3, p. 282] for these statements. Hence ∥F ∥_{BMO(D)} ≤ C∥F ∥_{ℬ_{h}} ≤ C′∥f∥_{BMO(∂D)} ≤ C′.

On the other hand, the two-dimensional version of the John–Nirenberg inequality implies

(2.7)  sup_{Q} \frac{m(\{z ∈ Q | |F (z) − F_{Q}| ≥ λ\})}{m(Q)} ≤ e^{−Cλ/∥F ∥_{BMO(D)}}.

The lemma follows.

Finally, we need to control the mean F_{Q} above.

Lemma 2.4. There exists a constant C > 0 as follows. If f, F and Q are as in the previous lemma then

(2.8)  |F_{Q}| ≤ C| log(m(Q))| ∥f∥_{BMO(∂D)} + C|F_{D}|.

Proof. Let Q ⊂ Q_{1} ⊂ Q_{2} ⊂ ⋯ Q_{k} be a sequence (as short as possible) of sets Q_{j} (as in (2.4)) such that m(Q_{j} ∩ D) = 2m(Q_{j−1} ∩ D) for all j (here Q_{0} := Q) and such that m(Q_{k}) ≥ 1. Hence, k ≤ C′|log(m(Q))| for an absolute strictly positive constant C′.

We obtain |F_{Q}| ≤ |F_{Q} − F_{Q_{k}}| + |F_{Q_{k}} − F_{D}| + |F_{D}|, and here

(2.9)  |F_{Q} − F_{Q_{k}}| ≤ ∑_{j=0}^{k−1} |F_{Q_{j}} − F_{Q_{j+1}}| ≤ Ck∥F ∥_{BMO(D)},

since

(2.10)  \begin{aligned}|F_{Q_{j}} − F_{Q_{j+1}}| & ≤ \frac{1}{m(Q_{j})} ∫_{Q_{j}} |F − F_{Q_{j+1}}| dm \\ & ≤ \frac{2}{m(Q_{j+1})} ∫_{Q_{j+1}} |F − F_{Q_{j+1}}| dm ≤ 2∥F ∥_{BMO(D)}. \end{aligned}

The term |F_{Q_{k}} − F_{D}| is estimated in the same way, since the sets Q_{k} and D are of comparable 2-dimensional measure. Now, combine (2.9) with the bound for k. The estimate in terms of the BMO-norm of f follows from the inequality above (2.7).

3. Main result

Our main result, Theorem 3.1, gives a connection of a Muckenhoupt type condition (for |ψ′|) with the geometry of Ω. The result holds for arbitrary regulated domains.

Theorem 3.1. Let ψ be a Riemann conformal map from D onto the bounded regulated domain Ω. Let a > 0, 1 < p < ∞, 1/p+1/p^{∗} = 1, and let the numbers α, σ and γ (all > −1) satisfy

(3.1)  α = \frac{σ}{p} + \frac{γ}{p^{∗}} .

Then the (weight) function |ψ′|^{a} satisfies the condition

(3.2)  sup_{S} ∫_{S} |ψ′|^{a} dm_{σ}(∫_{S} |ψ′|^{−ap^{∗}/p} dm_{γ})^{p/p^{∗}} ≤ Cm_{α}(S)^{p},

if

(3.3)  2 + σ > \frac{aδ_{1}}{π}  and  \frac{aδ_{2}}{π} > −\frac{p}{p^{∗}} (γ + 2).

Conversely, if 2 + σ < aδ_{1}/π or aδ_{2}/π < −p(p^{∗})^{−1}(γ + 2), then (3.2) fails.

Remarks. 1^{∘} Here, the sup is as in (1.4) and

(3.4)

δ_{1} := sup_{t} lim_{τ→0^{+}}(β(t + τ) − β(t − τ)) ≥ 0  and
δ_{2} := inf_{t} lim_{τ→0^{+}}(β(t + τ) − β(t − τ)) ≤ 0

which exist e.g. by (2.2), see also [5].

2^{∘} The theorem still holds, if a < 0. In this case (3.3) has to be replaced by “2 + σ > aδ_{2}/π and aδ_{1}/π > −p(γ + 2)/p^{∗} ” and the reverse condition by “2 + σ < aδ_{2}/π or aδ_{1}/π < −p(γ + 2)/p^{∗} ”. This can be deduced from the case a > 0 by replacing a by b := −ap^{∗}/p and p by p^{∗}.

3^{∘} The first (respectively second) condition (3.3) means a restriction for outward (respectively inward) pointing corners and cusps in the boundary of Ω.

The method of proof is basically still the same as in [2]. The main difference is that we explicitly use the notion of regulated domains. This allows us to approximate the direction angle of the boundary curve conveniently just by step functions, and this leads to some simplifications and generalizations in the arguments.

Proof. 1^{∘} Let us assume that (3.3) holds. We first derive in (3.5)–(3.21) a representation for |ψ′|^{a} which reveals the essential factors in (3.2).

Let us fix 0 < ε < 1 such that 2 + σ − aδ_{1}/π > 2 max(1, aδ_{1}/π, |σ|)ε and

γ + 2 + \frac{p^{∗}}{p} \frac{aδ_{2}}{π} > 2 max(1, \frac{p^{∗}}{p} |\frac{aδ_{2}}{π} |, |γ|)ε

and such that

(3.5)  \frac{1}{ε} > 100A max(1, \frac{4π}{a} (1 + \frac{p}{p^{∗}} )),

where A > 0 is the maximum of the constants C occurring in the bound [3, Theorem VI.1.5], and Lemmas 2.2, 2.3 and 2.4.

According to (2.3), the function |ψ′|^{a} has the representation

(3.6)  |ψ′|^{a} = C exp(−\frac{a}{2π} Re ∫_{0}^{2π} \frac{e^{it} + z}{e^{it} − z} β̃(t) dt),

where β̃ is the harmonic conjugate of β − t, β defined by (2.3).

By (2.2), there exist finitely many points 0 = t_{0} < t_{1} < ⋯ < t_{n} = 2π such that |β(t)−t−γ_{j}| < ε^{5} for t_{j−1} < t < t_{j}, for some real constants γ_{j}, j = 1, …, n. We denote by β_{1} and β_{2} the 2π -periodic extensions to R of the functions

(3.7)  β_{1} = ∑_{j=1}^{n} γ_{j}χ_{j},  β_{2} = β − t − β_{1},

where χ_{j}(t) = 1 for t_{j−1} < t < t_{j} and zero elsewhere. Clearly, the modulus of β_{2} is bounded by ε^{5} . By the choice of the points t_{j} we have (with γ_{0} := γ_{n})

(3.8) |δ_{1} − max_{0≤j<n}(γ_{j+1} − γ_{j})| ≤ 2ε^{5}  and  |δ_{2} − min_{0≤j<n}(γ_{j+1} − γ_{j})| ≤ 2ε^{5}.

Let us define for j = 1, 2

(3.9)

ν_{j}(z) := exp(−\frac{a}{2π} Re ∫_{0}^{2π} \frac{e^{it} + z}{e^{it} − z} β̃_{j}(t) dt)

= exp(−\frac{a}{2π} Im ∫_{0}^{2π} \frac{e^{it} + z}{e^{it} − z} β_{j}(t) dt);

we thus have |ψ′|^{a} = e^{C}ν_{1}ν_{2} .

Concerning ν_{2} , we want to show that, for ϱ = 1 + ε^{−1} (which may be a large number),

(3.10)  ∫_{S} ν_{2}^{ϱ} dm(∫_{S} ν_{2}^{−ϱp^{∗}/p} dm)^{p/p^{∗}} ≤ Cm(S)^{p}.

This follows from [2, Théorème 1.2], as soon as we show that (notation as in (2.5))

(3.11)  (\frac{1}{μ(I)} ∫_{I} ν_{2}^{ϱ/2} dμ)(\frac{1}{μ(I)} ∫_{I} ν_{2}^{−ϱp^{∗}/(2p)} dμ)^{p/p^{∗}} ≤ C.

Notice that by the theorem for conjugate functions, [3, Theorem VI.1.5],

(3.12)  ∥β̃_{2}∥_{BMO(∂D)} ≤ A∥β_{2}∥_{L^{∞}(∂D)} = A ess sup_{t∈[0,2π]} |β_{2}(t)| < ε^{4},

see the choice of ε above. By [3, Lemma VI.6.5], (3.11) is satisfied, if

(3.13)

sup_{I} \frac{1}{μ(I)} ∫_{I} e^{ϱa(β̃_{2}−β̃_{2,I})/4π} dμ < C and

sup_{I} \frac{1}{μ(I)} ∫_{I} e^{−ϱap(β̃_{2}−β̃_{2,I})/4πp^{∗}} dμ < C.

The proof of these is standard: let I be an interval and denote, for the positive real numbers s < s′, I[s, s′] := {t ∈ I | s ≤ |β̃_{2}(t) − β̃_{2,I}| ≤ s′}. Then (3.14)

∫_{I} e^{ϱa(β̃_{2}−β̃_{2,I})/4π} dμ = ( ∫_{I[0,4π/(ϱa)]} + ∫_{I[4π/(ϱa),1]} + ∑_{k=1}^{∞} ∫_{I[k,k+1]} )e^{ϱa(β̃_{2}−β̃_{2,I})/4π} dμ.

Here the integrand is bounded by a constant on I[0, 4π/(ϱa)]. On I[4π/(ϱa), 1], the integrand is bounded by e^{ϱa/4π}, but the measure of the set I[4π/(ϱa), 1] is at most μ(I) times

(3.15)  e^{−C4π/ϱa∥β̃_{2}∥_{BMO}} ≤ e^{−C4π/ϱaε^{4}} ≤ e^{−2ϱa/4π},

see the John–Nirenberg inequality (2.5), (3.5), and (3.12), and notice also 1/ε ≤ ϱ ≤ 2/ε. This integral thus is less than \frac{1}{2} . In the same way one finds that the integral over I[k, k + 1] is bounded by 2^{−k}. We thus obtain the first inequality of (3.13). The proof of the second one is similar.

For ν_{1} we want to prove

(3.16)  ∫_{S} ν_{1}^{ϱ^{∗}} dm_{σϱ^{∗}}(∫_{S} ν_{1}^{−ϱ^{∗}p^{∗}/p}dm_{γϱ^{∗}})^{p/p^{∗}} ≤ Cm_{αϱ^{∗}}(S)^{p},

where 1/ϱ + 1/ϱ^{∗} = 1, hence, 1 < ϱ^{∗} = 1 + ε. It is enough to prove this for so small sets S that dist (S, e^{it_{j}}) ≤ 2m(S) < 1 for at most one j. First, ν_{1} has a representation

(3.17)  ν_{1}(z) = ν_{1}(re^{iθ}) = ν̂_{1}(z) ∏_{j=0}^{n−1}((1 − r)^{2} + r(θ − t_{j})^{2})^{−a(γ_{j+1}−γ_{j})/(2π)},

where ν̂_{1} is a bounded function on D which is also bounded away from zero. One obtains (3.17) easily by taking the convolution of (3.7) times −a/(2π) with the conjugate Poisson kernel −Q_{z}(t), see [3, p. 102]; the product stems from the principal part 2r(θ − t)/((1 − r)^{2} + r(θ − t)^{2}) of −Q_{z}(t). Now

(3.18) (1 − r)^{2} + r(θ − t_{j})^{2} ≅ (1 − r)^{2} + (θ − t_{j})^{2} ≅ dist (e^{iθ}, re^{it_{j}})^{2} ≅ dist (re^{iθ}, e^{it_{j}})^{2}.

Keeping this in mind, we want to estimate ∫_{S} ν_{1}^{ϱ^{∗}} dm_{σϱ^{∗}}.

Because of (3.17) and (3.18), we can replace ν_{1} by

(3.19)  dist (re^{iθ}, e^{it_{j}})^{−a(γ_{j+1}−γ_{j})/π}

for some j; let us also denote diam(S) by h, hence m(S) ≅ h^{2} . Let us fix a point Re^{iT} ∈ S, where 1 − h < R < 1, 0 ≤ T ≤ 2π. Applying the linear change of variables L: z → z − Re^{iT} we find that L(S) is contained in the set {w ∈ C | |w| ≤ 10h}. Hence, using polar coordinates,

(3.20)  \begin{aligned}∫_{S} ν_{1}^{ϱ^{∗}} dm_{σϱ^{∗}} & ≅ ∫_{S} dist (re^{iθ}, e^{it_{j}})^{−aϱ^{∗}(γ_{j+1}−γ_{j})/π} dm_{σϱ^{∗}} \\ & ≤ ∫_{0}^{10h} ∫_{0}^{2π} r^{−ϱ^{∗}a(γ_{j+1}−γ_{j})/π}r^{σϱ^{∗}}(cos θ)^{σϱ^{∗}}r dr dθ. \end{aligned}

Since (3.3) and (3.8) hold, the total exponent of r in (3.20) becomes larger than −1 (we have ϱ^{∗} = 1 + ε, see also the choice of ε). Hence, (3.20) converges and is bounded by a constant times

(3.21)  h^{−ϱ^{∗}ai(γ_{j+1}−γ_{j})/π+ϱ^{∗}σ+2}.

In the same way one proves that

(3.22)  ∫_{S} ν_{1}^{−ϱ^{∗}p^{∗}/p} dm_{γϱ^{∗}} ≤ Ch^{ϱ^{∗}p^{∗}a(γ_{j+1}−γ_{j})pπ+ϱ^{∗}γ+2}.

As a consequence of (3.21) and (3.22) we obtain (3.16): (3.23)

\begin{matrix}∫_{S} ν_{1}^{ϱ^{∗}} dm_{σϱ^{∗}}(∫_{S} ν_{1}^{−ϱ^{∗}p^{∗}/p} dm_{γϱ^{∗}})^{p/p^{∗}} \\ ≤ Ch^{ϱ^{∗}((a/π)−(a/π))(γ_{j+1}−γ_{j})+ϱ^{∗}(σ+γp/p^{∗})+2(1+p/p^{∗})} = Ch^{ϱ^{∗}pα+2p} ≅ m_{ϱ^{∗}α}(S)^{p} \end{matrix}

For the last step one uses m_{c}(S(θ, h)) ≅ h^{2+c}.

Moreover, to see (3.2) we combine (3.10) and (3.16) and use Hölder:

∫S|ψ′|^{a}dm_{σ}(∫_{S} |ψ′|^{−ap^{∗}/p}dm_{γ})^{p/p^{∗}}

≤ (∫_{S} ν_{1}^{ϱ^{∗}} dm_{σϱ^{∗}})^{1/ϱ^{∗}}(∫_{S} ν_{2}^{ϱ} dm)^{1/ϱ}

× (∫_{S} ν_{1}^{−ϱ^{∗}p^{∗}/p} dm_{γϱ^{∗}})^{p/(ϱ^{∗}p^{∗})}(∫_{S} ν_{2}^{−ϱp^{∗}/p}dm)^{p/(ϱp^{∗})}
≤ Cm(S)^{p/ϱ}m_{ϱ^{∗}α}(S)^{p/ϱ^{∗}} ≤ C′m_{α}(S)^{p}.

2^{∘} We consider the reverse direction. Assume for example 2 + σ < aδ_{1}/π. This time we choose ε > 0 such that −(2 + σ) + aδ_{1}/π > 2 max(1, aδ_{1}/π, |σ|)ε and such that (3.5) also holds. We again define the functions β_{j} and ν_{j} as in (3.6)–(3.9).

Using the methods of part 1^{∘} we find an essential lower bound for ν_{1} at least in a large enough subset of D, as follows. We again use the representation (3.17) and Lemma 2.2. Let us fix j such that γ_{j+1} − γ_{j} > δ_{1} − 2ε^{5} ; consider the sets S(t_{j}, ϱ), where notation is as in (1.5) and 0 < ϱ ≤ \frac{1}{2} is so small that ϱ ≤ min_{ι}(|e^{it_{ι+1}} − e^{it_{ι}}|)/100. As in (3.19) we see that

(3.24)  ν_{1}(z) ≥ C dist (z, e^{it_{j}})^{−a(γ_{j+1}−γ_{j})/π} ≥ C dist (z, e^{it_{j}})^{−a(δ_{1}−2ε^{5})/π}

for z ∈ S(t_{j}, ϱ). Applying again the linear change of variables z → e^{i(π−t_{j})}(z−e^{it_{j}}) we thus obtain the bound (z = re^{iθ})

∫_{S} |ψ′|^{a} dm_{σ} = C̃ ∫_{S} ν_{1}ν_{2} dm_{σ} ≥ C ∫_{S} dist (re^{iθ}, e^{it_{j}})^{−a(δ_{1}−2ε^{5})/π}ν_{2}(re^{iθ}) dm_{σ}

≥ \frac{C}{4} ∫_{0}^{h} ∫_{−π/4}^{π/4} r^{−a(δ_{1}−2ε^{5})/π}r^{σ}(cos θ)^{σ}rν_{2}(re^{iθ} − e^{it_{j}}) dr dθ. (3.25)

This is bounded from below by a constant times

(3.26)  ∫_{0}^{h} ∫_{−π/4}^{π/4} r^{a′}ν_{2}(re^{iθ} − e^{it_{j}}) dr dθ,

where a′ = −a(δ_{1} − 2ε^{5})/π + σ + 1 < −1 − ε.

Our aim is now to obtain the lower bound ∞ for (3.26). The main difficulty is to prove that ν_{2} is large enough on a large enough set. We prove this by taking advantage of the BMO-property of β̃_{2} . It might be dangerous if |β̃_{2}| were large; but the set where this happens is very small. One of the technical difficulties is to extend the estimates on the boundary to the inside of the disc. We claim that

(3.27)  m({z | |ν_{2}(z)| ≤ k^{−ε}}) ≤ Ck^{−ε^{−1}}

for every k ∈ N. Let us denote by B: D → C the Poisson extension of β̃_{2} . The mean value (with respect to z) over the whole disc of the conjugate Poisson kernel (e^{it} + z)/(e^{it} − z) is a bounded function with respect to t, with a bound, say, 10. Hence, since the sup-norm of β_{2} is smaller than ε^{4} , we find that the mean value B_{D} of B over the unit disc satisfies |B_{D}| ≤ ε^{3} . Let us define a decomposition of D into subsets Q′_{n,j} as follows. Denote

(3.28) Q′_{n,j} := {z | 1−2^{−n} ≤ |z| < 1−2^{−n−1}, 2πj2^{−n} ≤ arg(z) ≤ 2π(j +1)2^{−n}}

for all n ∈ N, 1 ≤ j ≤ 2^{n}. Let Q_{n,j} be a Euclidean disc containing Q′_{n,j} such that m(Q_{n,j}) ≤ 2m(Q′_{n,j}). By Lemma 2.4,

(∗)  |B_{Q_{n,j}}| ≤ Cn∥β̃_{2}∥_{BMO(∂D)} + CB_{D} ≤ C′nε^{3}.

Moreover, by Lemma 2.3,

(3.29)

m(\{z ∈ Q_{n,j} | |B(z) − B_{Q_{n,j}}| ≥ \frac{1}{2}ε log k\})
≤ C′m(Q_{n,j})e^{−Cε log k/∥β̃_{2}∥_{BMO(∂D)}} ≤ C′′m(Q_{n,j})k^{−ε^{−2}}.

Given k, denote now by N(k) the set of pairs (n, j), n ∈ N, 1 ≤ j ≤ 2^{n}, such that |B_{Q_{n,j}}| ≥ \frac{1}{2}ε log k for (n, j) ∈ N(k). Clearly, (∗) implies n ≥ Cε^{−2} log k. Hence, by (3.28),

(3.30)  m( ⋃_{(n,j)∈N(k)} Q_{n,j}) ≤ C2^{−Cε^{−2} log k} ≤ C′k^{−cε^{−2}}.

Moreover, for (n, j) ∉ N(k) we have |B_{Q_{n,j}}| ≤ ε log k/2. Hence, for such (n, j) and for z ∈ Q_{n,j}, if |B(z)| ≥ ε log k, then |B(z) − B_{Q_{n,j}}| ≥ ε log k/2. This combined with (3.29) yields m({z ∈ Q_{n,j} | |B(z)| ≥ ε log k}) ≤ Cm(Q_{n,j})k^{−ε^{−2}} .

This and (3.30) together imply m({z ∈ D | |B(z)| ≥ ε log k}) ≤ Ck^{−cε^{−2}} . Hence, (3.27) follows.

We denote by Λ_{k} the set consisting of z with the properties (k +1)^{−1} ≤ |z| ≤ k^{−1} and |ν_{2}(z)| ≥ k^{−ε}. Now (3.27) clearly implies m(Λ_{k}) ≥ C′k^{−2} . Hence, by (3.26),

(3.31)  \begin{aligned}∫_{S} |ψ′|^{a} dm_{σ} & = C̃ ∫_{S} ν_{1}ν_{2} dm_{σ} ≥ C ∑_{k=0}^{∞} ∫_{Λ_{k}} k^{−a′}k^{−ε} \\ & = C ∑_{k=0}^{∞} m(Λ_{k})k^{−a′}k^{−ε} ≥ C′ ∑_{k=0}^{∞} k^{−2−a′−ε} = ∞, \end{aligned}

since −2 − a′ − ε > −1. Hence, (3.2) cannot hold.

4. Applications: Bergman type projections

In this section we assume that ψ is a conformal map onto a bounded regulated domain Ω, φ := ψ^{−1}: Ω → D, and β, δ_{1} and δ_{2} are as in Theorem 3.1.

1^{∘} The standard Bergman projection. Using Theorem 3.1 and [2, Lemma 1.1] we obtain

Theorem 4.1. The Bergman projection

(4.1)  P_{Ω}f(z) := \frac{1}{π} ∫_{Ω} \frac{φ′(z)\overline{φ′(ζ)}}{(1 − φ(z)\overline{φ(ζ)})^{2}} f(ζ) dm(ζ),

is a bounded operator from L^{p}(Ω) onto 𝒜^{p}(Ω), if

(4.2) (2 − p)\frac{δ_{2}}{π} > −\frac{2p}{p^{∗}} (in case p ≤ 2), or, 2 > (2 − p)\frac{δ_{2}}{π} (in case p ≥ 2).

Conversely, if

(4.3) (2 − p)\frac{δ_{2}}{π} < −\frac{2p}{p^{∗}} (in case p ≤ 2), or, 2 < (2 − p)\frac{δ_{2}}{π} (in case p ≥ 2),

then P_{Ω} is unbounded in the given space.

Proof. Take α = γ = σ = 0 and a = 2 − p in Theorem 3.1. Use (3.3) or Remark 2 after Theorem 3.1 depending on whether p ≤ 2 or p ≥ 2. Notice that the first condition (3.3) in Theorem 3.1, 2 > (2 − p)δ_{1}/π, always holds, since 0 ≤ δ_{1} ≤ π. The same is true for the analogous condition in Remark 2.

See the case 3^{∘} for the relation of P_{Ω} and the standard Bergman projection on the disc. The conditions (4.2) and (4.3) are analogous to (i), p. 143 of [2]. Theorem 4.1 is more general than Theorems 2.1 or 3.3 of [2] in the sense that we do not need a ‘bounded boundary rotation’-type assumption, cf. H2, (ii), on p. 138 of [2]. Moreover, we do not put any restriction for the number of cusps or for the local geometry around cusps as in H2, (i), on p. 155.

Corollary 4.2. If \frac{4}{3} < p < 4, then the Bergman projection (4.1) is bounded on L^{p}(Ω) for every bounded regulated domain Ω.

2^{∘} The conjugated Bergman projection. It is clear that the composition operator C_{ψ}: f ↦ f ∘ ψ is a linear homeomorphism from L^{p}(Ω) onto L_{ν}^{p}(D) (this space endowed with the norm ∥ · ∥_{p,ν}, see (1.2)) and from 𝒜^{p}(Ω) onto 𝒜_{ν}^{p}(D), where the weight ν on D is defined by ν(z) = |ψ′(z)|^{2} . So, the operator P_{ψ} := (C_{ψ})^{−1}P_{D}C_{ψ}, where P_{D}f = ∫_{D} f(ζ)/(1 − zζ̄)^{2} dm(ζ) is the standard Bergman projection on the disc, is again algebraically a projection operator on L^{p}(Ω).

It is known by [1] that P_{D} is bounded on L_{ν}^{p}(D) if and only if

(4.4)  sup_{S} ∫_{S} |ν| dm(∫_{S} |ν|^{−p^{∗}/p} dm)^{p/p^{∗}} ≤ Cm(S)^{p},

where S is as before. From this and Theorem 3.1 we obtain

Proposition 4.3. The conjugated Bergman projection P_{ψ} is bounded on L^{p}(Ω), if

(4.5)  \frac{δ_{1}}{π} < 1  and  \frac{δ_{2}}{π} > −\frac{p}{p^{∗}} .

The converse statement is analogous to Theorem 4.1.

Here the first condition (4.5) always excludes outward cusps. The second condition is vacuous if p > 2. So in this case arbitrary inward cusps are allowed.

3^{∘} A general family of projections. Finally we want to show that given an arbitrary regulated domain Ω, an arbitrary 1 < p < ∞ and an arbitrary weight

(4.6)  ω(z) = (dist (z, ∂Ω))^{a},

a > −1, one can find bounded Bergman type projections from L_{ω}^{p}(Ω) onto 𝒜_{ω}^{p}(Ω). These will be picked out of a family which depends on the indices n ∈ Z and α > −1. We consider the space L_{ω}^{p}(Ω) and the operator

(4.7) \begin{aligned}P f(z) & := P_{φ,α,n}f(z) \\ & := (α + 1) ∫_{Ω} \frac{φ′(z)^{2−n}φ′(ζ)^{n−1}\overline{φ′(ζ)}(1 − |φ(ζ)|^{2})^{α}}{(1 − φ(z)\overline{φ(ζ)})^{2+α}} f(ζ) dm(ζ). \end{aligned}

Notice that the standard Bergman projection is the case α = 0, n = 1 and the conjugated Bergman projection is the case α = 0, n = 2.

Formally, P reproduces analytic functions, since

(4.8)  P f(z) = (α + 1)φ′(z)^{1−n} ∫_{Ω} K(z, ζ)φ′(ζ)^{n−1}f(ζ) dm(ζ),

and here the kernel

(4.9)  K(z, ζ) := \frac{φ′(z)\overline{φ′(ζ)}(1 − |φ(ζ)|^{2})^{α}}{(1 − φ(z)\overline{φ(ζ)})^{2+α}}

is reproducing. To see this, use (i) the fact that

(4.10)  P_{D}^{(α)}f(z) = (α + 1) ∫_{D} \frac{(1 − |ζ|^{2})^{α}}{(1 − zζ̄)^{2+α}} f(ζ) dm(ζ)

is the orthogonal projection from L_{(1−|z|^{2})^{α}}^{2}(D) onto 𝒜_{(1−|z|^{2})^{α}}^{2}(D), and

(ii) the Hilbert space isomorphism f ↦ (f ∘ ψ)ψ′ from L_{(1−|φ(z)|^{2})^{α}}^{2}(Ω) onto L_{(1−|z|^{2})^{α}}^{2}(D), and from 𝒜_{(1−|φ(z)|^{2})^{α}}^{2}(Ω) onto 𝒜_{(1−|z|^{2})^{α}}^{2}(D).

We have the following

Lemma 4.4. The operator P, (4.7), can be extended as a bounded projection from L_{ω}^{p}(Ω) onto 𝒜_{ω}^{p}(Ω), if and only if the projection P_{D}^{(α)} is bounded from L_{ν}^{p}(D) onto 𝒜_{ν}^{p}(D), where

(4.11)  ν(z) := |ψ′(z)|^{2−(2−n)p+a}(1 − |z|)^{a}.

Proof. Since P reproduces an analytic function, we only need to worry about its boundedness. But the proof for this is straightforward. Use e.g. the Koebe distortion theorem (see [5, Corollary 1.4]), to see that the weight ω ∘ ψ(z) on D is equivalent to the weight (1 − |z|)^{a}|ψ′(z)|^{a}. We omit the details.

It is very useful for us that the boundedness of P_{D}^{(α)} in the situation of Lemma 4.4 can be characterized in terms of a condition like (3.2), see [1]: P_{D}^{(α)} is bounded on L_{ν}^{p}(D), if and only if

(4.12)  sup_{S} ∫_{S} |ν| dm_{α}(∫_{S} |ν|^{−p^{∗}/p} dm_{α})^{p/p^{∗}} ≤ Cm_{α}(S)^{p}.

This immediately gives us

Theorem 4.5. Let Ω be an arbitrary regulated domain, let, for some a > −1, ω(z) = (dist (z, ∂Ω))^{a} and 1 < p < ∞. Let P be as in (4.7) and assume 2 − (2 − n)p + a > 0. Then P is a bounded projection from L_{ω}^{p}(Ω) onto 𝒜_{ω}^{p}(Ω), if

(4.13)  \begin{aligned}2 + α + a & > (2 − (2 − n)p + a)\frac{δ_{1}}{π}  and \\ (2 − (2 − n)p + a)\frac{δ_{2}}{π} & > −\frac{p}{p^{∗}} (α + 2) + a. \end{aligned}

Here δ_{1} and δ_{2} are as in Theorem 3.1. The converse statement is analogous to Theorem 4.1. Also the formulation of the case 2 − (2 − n)p + a < 0 is left to the reader.

Proof. The boundedness of P is equivalent to (4.12), by Lemma 4.4. This is equivalent, in view of (4.11), to (4.14)

sup_{S} ∫_{S} |ψ′|^{2−(2−n)p+a} dm_{α+a}(∫_{S} |ψ′|^{− \frac{p^{∗}}{p} (2−(2−n)p+a)} dm_{α−ap^{∗}/p})^{p/p^{∗}}≤ Cm_{α}(S)^{p},

which by Theorem 3.1 is satisfied, if (4.13) holds.

Taking a large enough α in Theorem 4.5 one obtains

Corollary 4.6. Let Ω be an arbitrary regulated domain, let, for some a > −1, ω(z) = (dist (z, ∂Ω))^{a} and 1 < p < ∞. There exists a bounded projection from L_{ω}^{p}(Ω) onto 𝒜_{ω}^{p}(Ω). The space 𝒜_{ω}^{p}(Ω) is isomorphic to l_{p}.

Proof. The operator f ↦ (f ∘ ψ)ψ′ is a linear homeomorphism from L_{ω}^{p}(Ω) onto L_{ν}^{p}(D) and from 𝒜_{ω}^{p}(Ω) onto 𝒜_{ν}^{p}(D), where ν(z) = |ψ′(z)|^{2−p+a}(1 − |z|)^{a}. Taking n = 1 and α large enough, Lemma 4.4 and Theorem 4.5 show that P_{D}^{(α)} is a bounded projection L_{ν}^{p}(D) onto 𝒜_{ν}^{p}(D). But in view of [7, proof of Theorem III.A.11], this suffices to establish that 𝒜_{ν}^{p}(D) is isomorphic to l_{p}.

References

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[2] Békollé, D.: Projections sur des espaces de fonctions holomorphes dans des domains plans. - Canad. J. Math. XXXVIII, 1986, 127–157.

[3] Garnett, J.: Bounded Analytic Functions. - Academic Press, New York, 1981.

[4] Luecking, D.: Representation and duality in weighted spaces of analytic functions. - Indiana Univ. Math. J. 34, 1985, 319–336. Erratum: Indiana Univ. Math. J. 35, 1986, 927–928.

[5] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. - Grundlehren der mathematischen Wissenschaften 299, Springer-Verlag, 1992.

[6] Taskinen, J.: Muckenhoupt-type condition for regulated domains. - Manuscript, 2001.

[7] Wojtaszczyk, P.: Banach Spaces for Analysts. - Cambridge University Press, 1991.

[8] Zhu, K.: Operator Theory in Function Spaces. - Marcel Decker, New York, 1995.

Received 16 May 2001