Regularity of Projection Operators

Attached to Worm Domains

David E. Barrett, Dariush Ehsani, Marco M. Peloso ^{1}

Received: October 2, 2014 Revised: October 12, 2015

Communicated by Thomas Peternell

Abstract. We construct a projection operator on an unbounded worm domain which maps subspaces of W^{s} to themselves. The subspaces are determined by a Fourier decomposition of W^{s} according to a rotational invariance of the worm domain.

2010 Mathematics Subject Classification: Primary 32W05; Secondary 35B65, 32T20

Introduction

Our work is on the non-smooth unbounded worm domains

D_{β} = \{(z_{1}, z_{2}) ∈ ℂ^{2} : Re (z_{1}e^{−i log z_{2}\overline{z}_{2}}) > 0, | log z_{2}\overline{z}_{2}| < β −π/2\} β > π/2.

On a bounded version of the domains D_{β}, given by

Ω_{c} = \{(z_{1}, z_{2}) : |z_{1} + e^{i log z_{2}\overline{z}_{2}} |^{2} < 1, | log z_{2}\overline{z}_{2}| < β − π/2\} ,

C. Kiselman showed the failure of the Bergman projection to preserve C^{∞}(\overline{Ω}_{c}) [7]. The model domains, D_{β}, were important in [1], where the first author used them to show the Diedrich-Fornæss worm domains (constructed in [5]) provide a counterexample to regularity of the Bergman projection on a smoothly bounded pseudoconvex domain. In a detailed analysis of the Bergman kernel, Krantz and the third author, in [8], studied the L^{p} mapping properties of the Bergman projection on D_{β}, obtaining the exact range of values of p for which the mapping is bounded.

1Work of the first author supported in part by the National Science Foundation under Grant No. 1161735. Work of the second author was (partially) supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant RU 1474/2 within DFG’s Emmy Noether Programme.

In this article we look at regularity in terms of Sobolev spaces. We denote by W^{s}(D_{β}) the space of functions whose derivatives of order ≤ s are in L^{2}(D_{β}), and by W_{𝒟}^{s}(D_{β}) the closure of 𝒟 := C_{0}^{∞}(D_{β}) in W^{s}(D_{β}). The first author’s results on smooth domains relied on the fact (proved in the same paper) that the Bergman projection on the model domain, D_{β}, fails to map W_{𝒟}^{s}(D_{β}) to W^{s}(D_{β}) for large enough s [1]. More precisely, the failure to preserve Sobolev spaces was proved on subspaces (defined as W_{j}^{s}(D_{β}) below). This was instrumental in proving that Condition R, in which for each s ≥ 0 there exists an M ≥ 0 such that the Bergman projection is bounded as a map from W_{𝒟}^{s+M}(Ω) to W^{s}(Ω) ([3]), fails when Ω is the Diederich-Fornæss worm domain [4]. We point out that for a smoothly bounded pseudoconvex domain Condition R is equivalent to the apparently stronger condition in which the larger domain W^{s+M}(Ω) replaces W_{𝒟}^{s+M}(Ω). This equivalence also holds on the domains D_{β}: the first author constructed a composition of first order operators which allow us to consider the Bergman projection acting on functions which vanish to desired order at the boundary, without changing the resulting image of the projection (see Theorem 2.2 and specifically Theorem 3.1 in [2]). The question remained whether there exists another (oblique) projection operator which preserves the level of the Sobolev spaces. We construct such an operator in the present article. We now state our main result. From the rotational invariance of D_{β} with respect to the rotations, ρ_{θ}(z) = (z_{1}, e^{iθ}z_{2}), we can decompose the Bergman space B(D_{β}) = L^{2}(D_{β}) ∩ O(D_{β}) by

B(D_{β}) = ⨁_{j∈ℤ} B_{j}(D_{β}),

where B_{j}(D_{β}) consists of functions f ∈ B(D_{β}) satisfying f ∘ ρ_{θ} ≡ e^{ijθ}f. The space L^{2}(D_{β}) admits a similar decomposition into subspaces L_{j}^{2}(D_{β}), and we can define W_{j}^{s}(D_{β}) = L_{j}^{2}(D_{β}) ∩ W^{s}(D_{β}). Our main theorem is grounded on adjustments to factors which imply the obstruction to regularity of the Bergman projection on worm domains. The Bergman kernel for each space, B_{j}(D_{β}) is explicitly calculated and expressed as an integral in the form:

K_{j}(z, w) =\frac{1}{2π^{2}} z_{2}^{j}\overline{w}_{2}^{j} ∫_{ℝ} \frac{(ξ − \frac{j+1}{2})ξ}{sinh [(2β − π)(ξ − \frac{j+1}{2})] sinh πξ} z_{1}^{iξ−1}\overline{w}_{1}^{−iξ−1}dξ,

where, with an abuse of notation, we write

z_{1}^{α} = (z_{1}e^{−i log z_{2}\overline{z}_{2}})^{α}e^{iα log z_{2}\overline{z}_{2}}.

Such a power of z_{1} is holomorphic on D_{β} as is easy to see, and it is locally constant in |z_{2}|, but not constant if α is not an integer and β > π. In fact, in this case, the fiber over z_{1} is a union of disjoint annuli in z_{2} and the function is constant on each such annulus, but not globally constant. Using the residue calculus, one can compute an asymptotic expansion of the kernel (see [1]). The poles corresponding to non-integer multiples of i of the kernel lead to non-integer powers of z_{1} and w_{1} which ultimately lead to the obstruction of regularity of the operator. We construct a kernel which, when added to the Bergman kernel, eliminates all such poles, and in this way we successfully remove the obstruction to regularity of the Bergman projection on the model domains, D_{β}, and construct new projections which preserve the level of Sobolev spaces:

Main Theorem. Let β > π/2, and D_{β} be defined as above. For all j ∈ ℤ there exists a bounded linear projection

T_{j} : L^{2}(D_{β}) → B_{j}(D_{β})

which satisfies

T_{j} : W_{𝒟}^{s}(D_{β}) → W_{j}^{s}(D_{β})

for every s ≥ 0.

Much of this paper was discussed at collaborative meetings made possible through invitations extended by the University of Michigan and the Università degli Studi di Milano. All authors gratefully acknowledge the support from these institutions. We also thank the referee for a careful reading of the article as well as for many helpful suggestions in addition to pointing out an error in the calculation of the adjoint to the tangential operator, Λ_{t}, in Section 4 which led to our use of the W_{𝒟}^{s}(D_{β}) Sobolev spaces.

1. The Bergman projection on D_{β}

Following [1], we introduce the domains

D′_{β} = \{(z_{1}, z_{2}) ∈ ℂ^{2} : |Im z_{1} − log z_{2}\overline{z}_{2}| < π/2, | log z_{2}\overline{z}_{2}| < β − π/2\}

to aid in our study of the Bergman kernels on D_{β}. D′_{β} is related to D_{β} via the biholomorphic mapping

Ψ : D′_{β} → D_{β} (1.1)

(z_{1}, z_{2}) ↦ (e^{z_{1}}, z_{2}).

Let K_{D_{β}}(z, w) be the Bergman kernel for D_{β}, and K_{j}(z, w) the reproducing kernel for B_{j}(D_{β}); we have the relation

K_{D_{β}}(z, w) = ∑_{j} K_{j}(z, w).

We calculate K_{j}(z, w) using Fourier transforms as in [1]. Let S_{β} denote the strip

S_{β} := {z = x + iy ∈ ℂ : |y| < β},

and let ω_{j}(y) be the continuous bounded function on the interval I_{β} := {y : |y| < β}, given by ω_{j} = π (e^{(j+1)(·)}χ_{β−π/2}) ∗ χ_{π/2}, where for a > 0, χ_{a} := χ_{(−a,a)}, the characteristic function of the interval (−a, a). We denote by ∥ · ∥_{ω_{j}} the L^{2}(S_{β})-norm weighted with the function ω_{j}:

∥f∥_{ω_{j}} := (∫_{S_{β}} |f(x, y)|^{2}ω_{j}(y)dxdy)^{1/2} .

We further define the weighted Bergman spaces on the strip S_{β} by

B_{ω_{j}} = {f holomorphic on S_{β} : ∥f∥_{ω_{j}}^{2} < ∞}.

For f ∈ B_{ω_{j}},

f̂(ξ, y) = ∫_{ℝ} f(x + iy)e^{−ixξ}dx

satisfies

(1.2)  f̂(ξ, y) = e^{−yξ}f̂_{ℝ}(ξ),

where f̂_{ℝ}(ξ) := f̂(ξ, 0). Here and throughout we use the notation for complex variables

z_{1} = x + iy
w_{1} = x′ + iy′.

Define

k′_{j}(ξ, y, w_{1}) = \frac{1}{ω̂_{j}(−2iξ)}e^{iξ(y−\overline{w}_{1})},

where ω̂_{j} refers to the Fourier-Laplace transform of ω_{j}, and satisfies

(1.3)  ω̂_{j}(−2iξ) = π \frac{sinh [(2β − π)(ξ − \frac{j+1}{2})] sinh πξ}{(ξ − \frac{j+1}{2})ξ} .

We note that ω̂_{j} extends to an entire function. We claim that k′_{j} corresponds to the kernel for the orthogonal projection on D′_{β} according to the following lemma:

Lemma 1.1. Let K′_{j}(z, w) denote the reproducing kernel of the space B_{j}(D′_{β}). Then

K′_{j}(z, w) =\frac{1}{2π^{2}} z_{2}^{j}\overline{w}_{2}^{j} ∫_{ℝ} \frac{(ξ − \frac{j+1}{2})ξ}{sinh [(2β − π)(ξ − \frac{j+1}{2})] sinh πξ} e^{i(z_{1}−\overline{w}_{1})ξ}dξ.

Proof. Let Γ : B_{1} → B_{2} be a surjective isometry of two Bergman spaces. Let K_{1}(z, w) be the reproducing kernel of the space B_{1} and K_{2}(z, w) the kernel for B_{2}. Then

(1.4)  K_{2}(z, w) = \overline{Γ_{w}\overline{Γ_{z}K_{1}(z, w)}}.

We now apply (1.4) to the spaces B_{1} = B_{ω_{j}} and B_{2} = B_{j}(D′_{β}). From [1]

K_{j}(z_{1}, w_{1}) = \frac{1}{2π} ∫_{ℝ} k′_{j}(ξ, y, w_{1})e^{ixξ}dξ

is the reproducing kernel for B_{ω_{j}}, and

Γ : B_{ω_{j}} → B_{j}(D′_{β})
f(z_{1}) ↦ z_{2}^{j}f(z_{1})

is the isometry between Bergman spaces. Thus, by (1.4)

K′_{j}(z, w) = \frac{1}{2π} z_{2}^{j}\overline{w}_{2}^{j} ∫_{ℝ} k′_{j}(ξ, y, w_{1})e^{ixξ}dξ

from which the lemma follows. □

2. Improving the Bergman projection

Crucial to the proof in [1] of the failure of the Bergman projection to preserve W^{s}(D_{β}) is the existence of poles of k′_{j}(ξ, y, w_{1}) in the ξ variable whose imaginary part is a non-integer multiple of i. We see from (1.3) that such poles of k′_{j}(ξ, y, w_{1}) are due to the zeros of ω̂_{j}(−2iξ) at (j + 1)/2 + ikπ/(2β − π) for k a non-zero integer. In this section we deal with this obstruction by adding a correction term which eliminates such poles. We assume initially that j = −1. To keep the notation that integral operators are defined by integrating functions against conjugates of functions of two variables (the kernel), we will work with terms in the kernel coming from ω̂_{j}(2iξ), observing that \overline{ω̂_{j}(−2iξ)} = ω̂_{j}(2iξ). The goal in this section then is to find a function, denoted by ĥ(ξ, y), defined in ℂ × I_{β} such that ĥ(ξ, y), cancels the poles of the function

(2.1)  \frac{1}{ω̂_{−1}(2iξ)}e^{−ξy}

at ξ = ikν_{β}, for k a non-zero integer, and ν_{β} = π/(2β −π). The function \overline{ĥ(ξ, y)} will have an inverse transform which is orthogonal to B_{ω_{−1}} and satisfy certain L^{2} estimates which will be used in Section 3 to construct an integral operator. To ease notation we set

τ_{k}(ξ) = (−1)^{k} \frac{e^{−k^{2}ν_{β}^{2}}}{(2β − π)π} \frac{ξ^{2}}{sinh(πξ)}e^{−ξ^{2}} k ∈ ℤ.

We define

(2.2)  ĥ_{k}(ξ, y) = \frac{τ_{k}(ξ)e^{(ξ−2ikν_{β})y}}{ξ − ikν_{β}} .

We note that the pole of (2.1) at ξ = ikν_{β}, for k a non-zero integer is the same as the pole of ĥ_{k}. Our aim is to sum ĥ_{k} over k in order to produce a function which will be used to eliminate all such poles of (2.1). The following proposition shows that we can sum over k. To keep track of the poles, we introduce the set P of all poles:

P := {ikν_{β} : k ≠ 0} ∪ {ik : k ≠ 0}.

Proposition 2.1. Let ĥ_{k}(ξ, y) be defined as above. The infinite sum

∑_{k≠0} ĥ_{k}(ξ, ·)

converges in L^{∞}(I_{β}) to a function ĥ(ξ, ·) uniformly in ξ on compact subsets of ℂ \ P. Let B_{r} = ∪B(ikν_{β}; r) denote the union of balls centered at elements of P for some fixed radius r > 0. Let U be any neighborhood of P containing B_{r}. Then on ℂ \ U × I_{β}

(2.3)  |ĥ(ξ, y)| ≲ |ξ|^{2}e^{−Reξ^{2}}e^{(β−π)|Reξ|},

with the constant of inequality depending only on U.

Proof.

∑_{k≠0} ĥ_{k}(ξ, y) = ∑_{k≠0} \frac{τ_{k}(ξ)e^{(ξ−2ikν_{β})y}}{ξ − ikν_{β}}

is a sum of terms of the form

e^{ξy} ∑_{k≠0} a_{k}(ξ)

where

|a_{k}(ξ)| ≲ \frac{1}{k} e^{−k^{2}ν_{β}^{2}}|ξ|^{2}e^{−Reξ^{2}}e^{−π|Reξ|} k ≠ 0.

Inequality (2.3) is then a consequence of

|ĥ(ξ, y)| = |e^{ξy} ∑_{k} a_{k}(ξ)e^{−2ikν_{β}y}|

≲ e^{β|Reξ|} ∑_{k} |a_{k}(ξ)|

≲ |ξ|^{2}e^{−Reξ^{2}}e^{(β−π)|Reξ|}.

□

We note for f ∈ B_{ω_{−1}}:

∫_{ℝ} ∫_{I_{β}} \overline{ĥ_{k}(ξ, y)}f̂(ξ, y)ω_{−1}(y)dydξ = ∫_{ℝ} ∫_{I_{β}} \overline{ĥ_{k}(ξ, y)}e^{−yξ}f̂_{ℝ}(ξ)ω_{−1}(y)dydξ

= ∫_{ℝ} \frac{τ_{k}(ξ)}{ξ + ikν_{β}} f̂_{ℝ}(ξ) [∫_{I_{β}} e^{2ikν_{β}y}ω_{−1}(y)dy] dξ

= 0,

where we use the representation of f in (1.2) in the first step, and the fact that ∫_{I_{β}} e^{2ikν_{β}y}ω_{−1}(y)dy = ω̂_{−1}(−2kν_{β}) = 0 in the last. We collect the essential properties, which follow directly from the above, of the kernel function h(x, y) in the following theorem:

Theorem 2.2. There exists h(x, y) ∈ L_{ω_{−1}}^{2}(S_{β}) with the following properties:

(i) For each y ∈ I_{β}, the poles of

ĥ(ξ, y) + \frac{1}{ω̂_{−1}(2iξ)}e^{−ξy}

with respect to ξ lie at only integer multiples of i. (ii) The kernel given by

H′(z, w) = \frac{1}{2π} \frac{1}{z_{2}\overline{w}_{2}} ∫_{ℝ} \overline{ĥ(ξ, y)}e^{i(x−\overline{w}_{1})ξ}dξ

is orthogonal to the space B_{−1}(D′_{β}) in the sense that H′(·, w) ⊥ B_{−1}(D′_{β}). (iii) Let U be any neighborhood of P containing B_{r} for some r > 0. Then on ℂ \ U × I_{β}

|ĥ(ξ, y)| ≲ |ξ|^{2}e^{−Reξ^{2}}e^{(β−π)|Reξ|},

with the constant of inequality depending only on U.

We also denote the horizontal lines

S_{t} = {ℝ + it}

for t ∈ ℝ. From the Theorem 2.2 iii), we have in particular, on any given S_{t} such that S_{t} ∩ P = ∅, ĥ(ξ, y) satisfies the following estimates uniformly, i.e. with constant of inequality independent of ξ:

(2.4)  ∫_{I_{β}} |ĥ(ξ, y)|^{2} dy ≲ |ξ|^{4}e^{−2Reξ^{2}}e^{2(β−π)|Reξ|}.

3. Mapping properties

We begin this section with some integral estimates for our constructed correction term. We let H′(z, w) be as in Theorem 2.2. Due to the \overline{z}_{2}^{−1} factor in \overline{H′(z, w)}, the operator determined by the kernel, H′(z, w), will have its action restricted to the L_{−1}^{2}(D′_{β}) component of a given function in L^{2}(D′_{β}) We use the equivalence between Bergman spaces given in Lemma 1.1 in the proof of the next proposition: for G ∈ B_{−1}(D′_{β}), G is of the form G = g(z_{1})z_{2}^{−1}, where g ∈ B_{ω_{−1}}, and ∥G∥_{B_{−1}(D′_{β})} = ∥g∥_{B_{ω_{−1}}} .

Proposition 3.1. Let β > π/2, and H′ be the integral operator

H′f(w) = ∫_{D′_{β}} f(z)\overline{H′(z, w)}dV (z),

where

H′(z, w) = \frac{1}{2π} \frac{1}{z_{2}\overline{w}_{2}} ∫_{ℝ} \overline{ĥ(ξ, y)}e^{i(x−\overline{w}_{1})ξ}dξ.

Then

H′ : L^{2}(D′_{β}) → B_{−1}(D′_{β}),

and

∥H′f∥_{B_{−1}(D′_{β})} ≲ ∥f∥_{L_{−1}^{2}(D′_{β})}.

1214 D. E. Barrett, D. Ehsani, M. M. Peloso Proof. We write D′_{β} = ℝ × d′_{β}, where

d′_{β} = \{(y, z_{2}) ∈ ℝ × ℂ : |y − log z_{2}\overline{z}_{2}| < π/2, | log z_{2}\overline{z}_{2}| < β − π/2\}. Then,

H′f(w) = \frac{1}{2π} \frac{1}{w_{2}} ∫_{D′_{β}} \frac{1}{\overline{z}_{2}} ∫_{ℝ} ĥ(ξ, y)e^{−i(x−w_{1})ξ}dξf(z)dV (z)

= \frac{1}{2π} \frac{1}{w_{2}} ∫_{d′_{β}} \frac{1}{\overline{z}_{2}} ∫_{ℝ} ∫_{ℝ} ĥ(ξ, y)e^{−i(x−w_{1})ξ}f(x, y, z_{2})dxdξdydV (z_{2})

= \frac{1}{2π} \frac{1}{w_{2}} ∫_{d′_{β}} \frac{1}{\overline{z}_{2}} ∫_{ℝ} ĥ(ξ, y)e^{iw_{1}ξ}f̂(ξ, y, z_{2})dξdydV (z_{2}).

We use a decomposition of f according to

f(z) = ∑_{j} f_{j}(z), f_{j}(z) ∈ L_{j}^{2}(D′_{β}).

Using the orthogonality of powers of z_{2} (over circular regions) we can isolate any f_{j} by integrating through \overline{z}_{2}^{j}. This is used in the third step below where after integrating over z_{2} only f_{−1}(z) terms remain:

∥H′f∥_{B_{−1}(D′_{β})}^{2} = \frac{1}{4π^{2}} ∥∫_{d′_{β}} \frac{1}{\overline{z}_{2}} ∫_{ℝ} ĥ(ξ, y)e^{i(·)ξ}f̂(ξ, y, z_{2})dξdydV (z_{2})∥_{B_{ω_{−1}}}^{2}

= \frac{1}{4π^{2}} ∫_{I_{β}} ∫_{ℝ} |∫_{d′_{β}} \frac{1}{\overline{z}_{2}} ∫_{ℝ} ĥ(ξ, y)e^{−y′ξ}e^{ix′ξ}f̂(ξ, y, z_{2})dξdydV (z_{2})|^{2} dx′ω_{−1}(y′)dy′

= \frac{1}{4π^{2}} ∫_{I_{β}} ∫_{ℝ} |∫_{d′_{β}} \frac{1}{\overline{z}_{2}} ĥ(ζ, y)e^{−y′ζ} f̂_{−1}(ζ, y, z_{2})dydV (z_{2})|^{2} dζω_{−1}(y′)dy′

≲ ∫_{I_{β}} ∫_{ℝ} [ (∫_{I_{β}} |ĥ(ζ, y)|^{2} ω_{−1}(y)dy) ×

(∫_{d′_{β}} |f̂_{−1}(ζ, y, z_{2})|^{2} dydV (z_{2})) e^{−2y′ζ}]dζω_{−1}(y′)dy′.

From Theorem 2.2 (iii) and (2.4) we have that

∫_{I_{β}} |ĥ(ζ, y)|^{2} ω_{−1}(y)dy ≲ |ζ|^{4}e^{−2Reζ^{2}}e^{2(β−π)|Reζ|}.

We continue with our estimate of ∥H′f∥_{B_{−1}(D′_{β})}:

∥H′f∥_{B_{−1}(D′_{β})}^{2} ≲∫_{ℝ} ∫_{d′_{β}} |f̂_{−1}(ζ, y, z_{2})|^{2} dydV (z_{2})|ζ|^{4}e^{−2ζ^{2}}e^{2(β−π)|ζ|}ω̂_{−1}(−2iζ)dζ
≲ ∥f_{−1}∥_{L^{2}(D′_{β})}^{2},

where the last estimate follows by the fact that the term |ζ|^{4}e^{−2ζ^{2}}e^{2(β−π)|ζ|}ω̂_{−1}(−2iζ) is bounded with respect to ζ. □

Documenta Mathematica 20 (2015) 1207–1225

We recall the biholomorphic mapping Ψ : D′_{β} → D_{β} from (1.1). Through a change of variables Ψ^{−1}, H′ induces an integral operator on L^{2}(D_{β}): g ↦ (g ∘ Ψ)det(Ψ^{−1})′, (Ψ^{−1})′ being the complex Jacobian of (Ψ^{−1})′, is an isometry between L^{2}(D_{β}) and L^{2}(D′_{β}), and in fact since Ψ is biholomorphic, between Bergman spaces (see also (1.4)). In this regard, we define the kernel

(3.1)  H(z, w) = \frac{1}{z_{1}\overline{w}_{1}} H′(Ψ^{−1}z, Ψ^{−1}w),

using the fact that det(Ψ^{−1}(z))′ = \frac{1}{z_{1}} . Let H be the integral operator

Hf(w) = ∫_{D_{β}} f(z)\overline{H(z, w)}dV (z),

where H(z, w) is given by (3.1). Then as a result of Proposition 3.1, we have the following

Corollary 3.2. We have that

H : L^{2}(D_{β}) → B_{−1}(D_{β}),

and

∥Hf∥_{B_{−1}(D_{β})} ≲ ∥f∥_{L_{−1}^{2}(D_{β})}.

We now define the projection operator T_{−1} as

T_{−1} = P_{−1} + H,

where P_{−1} : L^{2}(D_{β}) → B_{−1}(D_{β}) is the orthogonal projection operator.

4. Properties of the projection T_{−1}

Theorem 4.1. Let β > π/2 and T_{−1} = P_{−1} + H. Then

T_{−1} : L^{2}(D_{β}) → B_{−1}(D_{β}).

Furthermore, T_{−1} is a projection, and has the regularity property

(4.1)  T_{−1} : W_{𝒟}^{k}(D_{β}) → W_{−1}^{k}(D_{β})  ∀k,

and

∥T_{−1}f∥_{W_{−1}^{k}(D_{β})} ≲ ∥f∥_{W^{k}(D_{β})}

for f ∈ W_{𝒟}^{k}(D_{β}).

Proof. The mapping from L^{2}(D_{β}) to B_{−1}(D_{β}) follows from the corresponding properties of P_{−1} and H (see Corollary 3.2). That T_{−1} is a projection follows from P_{−1} being a projection and from the restriction of H to B_{−1}(D_{β}) being equivalently 0 (from Theorem 2.2 ii.):

T_{−1}^{2} = P_{−1}^{2} + P_{−1}H + HP_{−1} + H^{2}
= P_{−1} + H
= T_{−1}.

Since T_{−1}f is holomorphic, to prove (4.1) we estimate the L^{2} norm of holomorphic derivatives of T_{−1}f. Also, T_{−1}f is of the form g(w_{1}, |w_{2}|)w_{2}^{−1}, where the function g(w_{1}, |w_{2}|) is holomorphic and locally constant in w_{2}, so its derivatives in w_{2} are zero and we only need to estimate the derivatives with respect to the first variable. To prove the theorem we thus show

(4.2)  ∥\frac{∂^{k}}{∂w_{1}^{k}} T_{−1}f∥_{L^{2}(D_{β})} ≲ ∥f_{−1}∥_{W^{k}(D_{β})},

for f ∈ W_{𝒟}^{k}(D_{β}). The domain D′_{β} is related to D_{β} via the biholomorphic mapping Ψ. We can then read off the kernels attached to the domain D_{β} from the transformation formula applied to the corresponding kernels on D′_{β}, as in (3.1). We have the relations

K_{−1}(z, w) = \frac{1}{z_{1}\overline{w}_{1}} K′_{−1}(Ψ^{−1}z, Ψ^{−1}w)

H(z, w) = \frac{1}{z_{1}\overline{w}_{1}} H′(Ψ^{−1}z, Ψ^{−1}w)

T_{−1}(z, w) = \frac{1}{z_{1}\overline{w}_{1}} T′_{−1}(Ψ^{−1}z, Ψ^{−1}w),

where K_{−1}, H, T_{−1} (resp. K′_{−1}, H′, T′_{−1}) are the kernels for, respectively, P_{−1}, H, T_{−1} (resp. P′_{−1}, H′, T′_{−1}). Using integration by parts, we relate \frac{∂^{k}}{∂w_{1}^{k}} T_{−1}f to k^{th} order derivatives falling on f. From above, we have

T_{−1}f(w) = ∫_{D_{β}} \overline{T_{−1}(z, w)}f(z)dV (z),

where

\overline{T_{−1}(z, w)} = \frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ}(\frac{1}{ω̂_{−1}(2iξ)}\overline{z}_{1}^{−iξ−1}w_{1}^{iξ−1}

+ ĥ(ξ, (log z_{1} − log \overline{z}_{1})/2i)z_{1}^{−iξ/2−1}\overline{z}_{1}^{−iξ/2}w_{1}^{iξ−1})dξ.

By virtue of the factor \overline{z}_{2}^{−1} in \overline{T_{−1}(z, w)}, all action is isolated on f_{−1}(z). Thus,

T_{−1}f(w) = ∫_{D_{β}} \overline{T_{−1}(z, w)}f(z)dV (z)
= ∫_{D_{β}} \overline{T_{−1}(z, w)}f_{−1}(z)dV (z).

Furthermore,

(4.3)  \frac{∂^{k}}{∂w_{1}^{k}} T_{−1}f = ∫_{D_{β}} \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)}f_{−1}(z)dV (z),

and

\frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)} =

\frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ}(iξ − 1)(iξ − 2) ⋯ (iξ − k)(\frac{1}{ω̂_{−1}(2iξ)}\overline{z}_{1}^{−iξ−1}w_{1}^{iξ−k−1}

+ ĥ(ξ, (log z_{1} − log \overline{z}_{1})/2i)z_{1}^{−iξ/2−1}\overline{z}_{1}^{−iξ/2}w_{1}^{iξ−k−1})dξ. (4.4)

Our strategy is roughly as follows: we use shifts of contours of integration to write the integrands of (4.4) using derivatives with respect to z_{1}; we make sure Fubini’s theorem applies with respect to the z and ξ integrals and then we take the z_{1} derivatives outside the ξ integrals; finally we can then perform an integration by parts in the z_{1} variable in (4.3). When shifting the contour of integration, in order to verify that Fubini’s theorem applies, we work with the two cases, each of which determines a different direction of shift:

i) |w_{1}| < |z_{1}|

ii) |z_{1}| < |w_{1}|.

To illustrate the cases, we consider integrals of the form

ϕ_{t}(w_{1}) = ∫_{U} \frac{1}{\overline{z}_{2}} ∫_{Im(ξ)=t} σ_{w_{1}}(ξ, z_{1}, \overline{z}_{1})f_{−1}(z)dξdV (z),

where σ_{w_{1}} will be either

(iξ − 1)(iξ − 2) ⋯ (iξ − k)\frac{1}{ω̂_{−1}(2iξ)} \frac{1}{z_{1}w_{1}^{k+1}} (\frac{w_{1}}{\overline{z}_{1}} )^{iξ}

or

(iξ − 1)(iξ − 2) ⋯ (iξ − k)ĥ(ξ, (log z_{1} − log \overline{z}_{1})/2i)\frac{1}{z_{1}w_{1}^{k+1}} (\frac{w_{1}}{|z_{1}|})^{iξ} ,

and the domain of integration U will be either D_{β} ⋂{|w_{1}| < |z_{1}|} or D_{β} ⋂{|z_{1}| < |w_{1}|}. Using the estimates for ω̂_{−1}(2iξ) and the estimate in (2.3) for ĥ, we have

(4.5)  |ϕ_{t}(w_{1})| ≲ ∫_{U} |\frac{1}{\overline{z}_{2}} \frac{1}{z_{1}w_{1}^{k+1}} | (\frac{|z_{1}|}{|w_{1}|})^{t} |f_{−1}(z)|dV (z).

We see Fubini’s theorem applies in case i) when t < 0 and in case ii) when t > 0. The signs of t correspond to shifts in the lower- and upper half planes, respectively. We now proceed to the write an expression for the kernel \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)} in terms of derivatives with respect to the z variable, corresponding to the two cases. It will be shown in both cases we are lead to the same expression. Case i). By construction of the term h in Section 2 the integrand exhibits poles only at integer multiples of i, of which those at −i, −2i, …, −ik are cancelled. We therefore deform the contour of integration in (4.4) to ℝ − ik. The contribution of the sides of the contour are null due to the exponential decay in ξ of the integrand. We now work with the contour of integration in (4.4) deformed to ℝ − ik. We first consider

\frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ−ik}(iξ − 1)(iξ − 2) ⋯ (iξ − k)\frac{1}{ω̂_{−1}(2iξ)}\overline{z}_{1}^{−iζ−1}w_{1}^{iξ−k−1}dξ =

\frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ}(iζ + k − 1)(iζ + k − 2) ⋯ (iζ)\frac{1}{ω̂_{−1}(2i(ζ − ik))}\overline{z}_{1}^{−iζ−k−1}w_{1}^{iζ−1}dζ.

We use

\frac{1}{ω̂_{−1}(2i(ζ − ik))} = (−1)^{k} \frac{1}{π} \frac{(ζ − ik)^{2}}{sinh[(2β − π)(ζ − ik)] sinh(πζ)},

hence

\frac{(iζ + k − 1)(iζ + k − 2) ⋯ (iζ)}{ω̂_{j}(2i(ζ − ik))}\overline{z}_{1}^{−iζ−k−1}w_{1}^{iζ−1}

= (−1)^{k} \frac{1}{π} \frac{(ζ − ik)(ζ − ik)}{sinh[(2β − π)(ζ − ik)] sinh(πζ)}×

(iζ + k − 1)(iζ + k − 2) ⋯ (iζ)\overline{z}_{1}^{−iζ−k−1}w_{1}^{iζ−1}

= (−1)^{k} \frac{1}{π} \frac{(ζ − ik)(iζ)}{sinh[(2β − π)(ζ − ik)] sinh(πζ)}×

(ζ − ik)(iζ + k − 1) ⋯ (iζ + 1)\overline{z}_{1}^{−iζ−k−1}w_{1}^{iζ−1}

= \frac{1}{π} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)} \frac{∂^{k}}{∂\overline{z}_{1}^{k}} \overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}.

For the integral above we thus have

\frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ−ik}(iξ − 1)(iξ − 2) ⋯ (iξ − k)\frac{1}{ω̂_{−1}(2iξ)}\overline{z}_{1}^{−iζ−1}w_{1}^{iξ−k−1}dξ =

\frac{1}{2π^{2}} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)} \frac{∂^{k}}{∂\overline{z}_{1}^{k}} \overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ.

Similarly, we work with

(4.6) \frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ−ik}(iξ − 1)(iξ − 2) ⋯ (iξ − k)ĥ(ξ, (log z_{1} − log \overline{z}_{1})/2i)×
z_{1}^{−iξ/2−1}\overline{z}_{1}^{−iξ/2}w_{1}^{iξ−k−1}dξ.

Let us write

ĥ(ξ, (log z_{1} − log \overline{z}_{1})/2i) = \frac{ξ}{sinh(πξ)}g(ξ, z_{1}),

and note that g(ξ, z_{1}) has the property

Λ_{t}g(ξ, z_{1}) = 0,

where

Λ_{t} := ((\frac{z_{1}}{\overline{z}_{1}} )^{1/2} \frac{∂}{∂z_{1}} + (\frac{\overline{z}_{1}}{z_{1}} )^{1/2} \frac{∂}{∂\overline{z}_{1}} ) .

The integrand in (4.6) can thus be written according to

(iζ + k − 1)(iζ + k − 2) ⋯ (iζ)ĥ(ζ − ik, (log z_{1} − log \overline{z}_{1})/2i)×
z_{1}^{(−iζ−k)/2−1}\overline{z}_{1}^{(−iζ−k)/2}w_{1}^{iζ−1}

= (−1)^{k} \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})×

(iζ + k)(iζ + k − 1) ⋯ (iζ + 1)z_{1}^{(−iζ−k)/2−1}\overline{z}_{1}^{(−iζ−k)/2}w_{1}^{iζ−1}

= \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})×

((\frac{z_{1}}{\overline{z}_{1}} )^{1/2} \frac{∂}{∂z_{1}} + (\frac{\overline{z}_{1}}{z_{1}} )^{1/2} \frac{∂}{∂\overline{z}_{1}} )^{k} z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}

= \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})(Λ_{t})^{k}z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}.

Therefore,

∫ _{|w_{1}|<|z_{1}|} \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)}f(z)dV (z) = −\frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ×

∫_{Rew_{1}<Rez_{1}} [\frac{1}{π} ∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)} \frac{∂^{k}}{∂\overline{z}_{1}^{k}} \overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ+

∫_{ℝ} \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})(Λ_{t})^{k}z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}dζ]f_{−1}(z)dV (z).

We remark that, as outlined above, the ζ and z integrations can be switched (just consider the integral ϕ_{k} in (4.5)). Case ii). We begin by writing (4.4) in the form:

\frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)} = \frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ×

∫ℝ [\frac{(−1)^{k}}{π} \frac{(ξ + ik)(ξ)}{sinh [(2β − π)(ξ)] sinh(πξ)} \frac{∂^{k}}{∂\overline{z}_{1}^{k}} \overline{z}_{1}^{−iξ+(k−1)}w_{1}^{iξ−1−k}+

(−1)^{k} \frac{ξ + ik}{sinh(πξ)}g(ξ, z_{1}) (Λ_{t})^{k} z_{1}^{−iξ/2+k/2−1}\overline{z}_{1}^{−iξ/2+k/2}w_{1}^{iξ−k−1}]dξ,

which is also obtained by deforming the contour of integration to ℝ + ik (using that the sides of the contour give no contributions in the same manner as that of case i)) of the following integral

− \frac{1}{2π} ∫_{ℝ} [\frac{1}{π} \frac{ζ(ζ − ik)}{sinh [(2β − π)(ζ − ik)] sinh(πζ)} \frac{∂^{k}}{∂\overline{z}_{1}^{k}} \overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}

+ \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1}) (Λ_{t})^{k} z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}]dζ,

noting that the contribution from the poles at integer multiples of i are cancelled due to the differential operators. Combining the results in cases i) and ii), we have

∫D_{β} \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)}f(z)dV (z) =

∫_{|w_{1}|<|z_{1}|} \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)}f(z)dV (z) + ∫_{|w_{1}|>|z_{1}|} \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)}f(z)dV (z),

where

∫|w_{1}|<|z_{1}| \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)}f(z)dV (z) =
−\frac{1}{2π} ∫_{|w_{1}|<|z_{1}|} \frac{1}{\overline{z}_{2}w_{2}} ×

[\frac{1}{π} ∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)} \frac{∂^{k}}{∂\overline{z}_{1}^{k}} \overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ+

∫_{ℝ} \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})(Λ_{t})^{k}z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}dζ]f_{−1}(z)dV (z)

and

∫|w_{1}|>|z_{1}| \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)}f(z)dV (z) =
−\frac{1}{2π} ∫_{|w_{1}|>|z_{1}|} \frac{1}{\overline{z}_{2}w_{2}} ×

[\frac{1}{π} ∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)} \frac{∂^{k}}{∂\overline{z}_{1}^{k}} \overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ+

∫_{ℝ} \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})(Λ_{t})^{k}z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}dζ]f_{−1}(z)dV (z).

We now use Fubini’s theorem in both case i) and ii) to take the derivatives outside of the ζ integrals, and then combine the results above. Before doing so, we note

Λ_{t} = ((\frac{z_{1}}{\overline{z}_{1}} )^{1/2} \frac{∂}{∂z_{1}} + (\frac{\overline{z}_{1}}{z_{1}} )^{1/2} \frac{∂}{∂\overline{z}_{1}} )
=∂_{r_{1}},

where r_{1} = |z_{1}|, is a tangential differential operator. To calculate the adjoint of Λ_{t} we note for fixed z_{2}, z_{1} can be written with coordinates t_{1} and d_{1}, d_{1} representing the distance to the boundary Re z_{1}e^{−i log z_{2}\overline{z}_{2}} = 0, via

(4.7)  z_{1} = (t_{1} + id_{1})e^{iα}

where α = log |z_{2}|^{2} − π/2. In these coordinates we calculate

Λ_{t} = \frac{t_{1}}{\sqrt{t_{1}^{2} + d_{1}^{2}}} \frac{∂}{∂t_{1}} + \frac{d_{1}}{\sqrt{t_{1}^{2} + d_{1}^{2}}} \frac{∂}{∂d_{1}} .

Then,

(Λ_{t})^{∗} = − Λ_{t} − \frac{∂}{∂t_{1}} (\frac{t_{1}}{\sqrt{t_{1}^{2} + d_{1}^{2}}} ) − \frac{∂}{∂d_{1}} (\frac{d_{1}}{\sqrt{t_{1}^{2} + d_{1}^{2}}} )
= − Λ_{t} − \frac{1}{\sqrt{t_{1}^{2} + d_{1}^{2}}}
(4.8)  = − Λ_{t} − \frac{1}{|z_{1}|}.

Furthermore, from the relation (4.7), we can write

\frac{∂}{∂\overline{z}_{1}} = α_{1} \frac{∂}{∂z_{1}} + α_{2} \frac{∂}{∂t_{1}} ,

where α_{1}(|z_{2}|) and α_{2}(|z_{2}|) are bounded away from 0 and depend smoothly on |z_{2}|. We recall that that g(ξ, z_{1}) has the property Λ_{t}g(ξ, z_{1}) = 0, and so

g(ζ − ik, z_{1})(Λ_{t})^{k}z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2} =(Λ_{t})^{k} [z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}g(ζ − ik, z_{1})] .

We thus have, after commuting the z derivatives with the ζ integrals,

∫D_{β} \frac{∂^{k}}{∂w_{1}^{k}} \overline{T_{−1}(z, w)}f(z)dV (z) =
−\frac{1}{2π} ∫_{D_{β}} \frac{1}{\overline{z}_{2}w_{2}} ×

[\frac{1}{π} \frac{∂^{k}}{∂\overline{z}_{1}^{k}} ∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)}\overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ+

(Λ_{t})^{k} ∫_{ℝ} \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}dζ]f_{−1}(z)dV (z). (4.9)

Documenta Mathematica 20 (2015) 1207–1225

Integrating by parts in the first integral on the right in (4.9) gives

− \frac{1}{2π^{2}} ∫_{D_{β}} \frac{1}{\overline{z}_{2}w_{2}} ×

[\frac{∂^{k}}{∂\overline{z}_{1}^{k}} ∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)}\overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ] f_{−1}(z)dV (z)

= −\frac{1}{2π^{2}} ∫_{D_{β}} \frac{1}{\overline{z}_{2}w_{2}} ×

[(α_{2}∂_{t_{1}})^{k} ∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)}\overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ] f_{−1}(z)dV (z)

= (−1)^{k+1} \frac{1}{2π^{2}} ∫_{D_{β}} \frac{1}{\overline{z}_{2}w_{2}} ×

[∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)}\overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ] (α_{2}∂_{t_{1}})^{k}f_{−1}(z)dV (z). (4.10)

Similarly, we perform an integration by parts in the second integral in (4.9), using (4.8).

−\frac{1}{2π}∫_{D_{β}}\frac{1}{\overline{z}_{2}w_{2}} (Λ_{t})^{k}∫_{ℝ} \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}dζf_{−1}(z)dV (z)

= (−1)^{k+1} \frac{1}{2π} ∫_{D_{β}} \frac{1}{\overline{z}_{2}w_{2}} [ ∫_{ℝ} \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}dζ]×

(4.11)  (Λ_{t} + |z_{1}|^{−1})^{k} f_{−1}(z)dV (z).

To finish the proof we note that the proof of Proposition 3.1 , with ĥ(ξ, y)e^{−ixξ} replaced with

\frac{(ξ − ik)ξ}{sinh[(2β − π)(ξ − ik)] sinh(πξ)}e^{−i\overline{z}_{1}ξ}

may be followed to show that the operator from (4.10) with kernel

∫_{D_{β}} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ} \frac{(ζ − ik)ζ}{sinh[(2β − π)(ζ − ik)] sinh(πζ)}\overline{z}_{1}^{−iζ−1}w_{1}^{iζ−1}dζ

maps L^{2}(D_{β}) to L_{−1}^{2}(D_{β}). Similarly, the proof of Proposition 3.1 shows that the operator with kernel

\frac{1}{2π} \frac{1}{\overline{z}_{2}w_{2}} ∫_{ℝ} \frac{ζ}{sinh(πζ)}g(ζ − ik, z_{1})z_{1}^{−iζ/2−1}\overline{z}_{1}^{−iζ/2}w_{1}^{iζ−1}dζ

occurring in (4.11) maps L^{2}(D_{β}) to L_{−1}^{2}(D_{β}). We have estimates for the term in (4.11) when f ∈ W_{𝒟}^{s}(D_{β}):

∑_{α≤k} ∥|z_{1}|^{−k+α}Λ_{t}^{α}f_{−1} ∥_{L^{2}(D_{β})}
≲ ∑_{α≤k} ∥|z_{1}|^{−k+α}Λ_{t}^{α}f_{−1}∥_{L^{2}((𝔻_{1}×ℂ)∩D_{β})} + ∥f_{−1}∥_{W^{k}(D_{β})}
≲ ∑_{α≤k} ∥t_{1}^{−k+α}Λ_{t}^{α}f_{−1}∥_{L^{2}((𝔻_{1}×ℂ)∩D_{β})} + ∥f_{−1}∥_{W^{k}(D_{β})}
≲∥f_{−1}∥_{W^{k}(D_{β})},

where 𝔻_{1} := {|z_{1}| ≤ 1}, the variable t_{1} is as in (4.7), and the last step follows from Theorem 1.4.4.4 in [6] (with a slight variation in the argument we can also apply Theorem 11.8 in [9] which holds for smooth domains). Then, together (4.10) and (4.11) show

∥\frac{∂^{k}}{∂w_{1}^{k}} T_{−1}f∥_{L^{2}(D_{β})} ≲∥f_{−1}∥_{W^{k}(D_{β})} + ∑_{α≤k} ∥|z_{1}|^{−k+α}Λ_{t}^{α}f_{−1}∥_{L^{2}(D_{β})}
≲∥f_{−1}∥_{W^{k}(D_{β})}.

The estimate in (4.2) is verified, completing the proof of the theorem. □

5. The case j ≠ −1

We construct operators

T_{j} : W_{𝒟}^{k}(D_{β}) → W_{j}^{k}(D_{β}) ∀k,

for the cases j ≠ −1 as follows. We let Q_{j} be the projection from L^{2}(D_{β}) to L_{j}^{2}(D_{β}) given by

Q_{j}f(z_{1}, z_{2}) = \frac{1}{2π} ∫_{−π}^{π} f(z_{1}, e^{iθ}z_{2})e^{−ijθ}dθ.

Then we take the operator T_{j} to be given by

T_{j}f = w_{2}^{j+1}T_{−1}(z_{2}^{−j−1}Q_{j}f).

For each T_{j}, due to properties of the operator T_{−1}, we have a theorem similar to Theorem 4.1:

Theorem 5.1. Let β > π/2, and D_{β} be defined as above. For all j ∈ ℤ there exists a bounded linear projection

T_{j} : L^{2}(D_{β}) → B_{j}(D_{β})

which satisfies

T_{j} : W_{𝒟}^{k}(D_{β}) → W_{j}^{k}(D_{β}) ∀k,

and

∥T_{j}f∥_{W_{j}^{k}(D_{β})} ≲ ∥f∥_{W^{k}(D_{β})}.

This proves the Main Theorem.

6. Remarks

We end with a few remarks. We first note that in our proof of Theorem 4.1, we worked with Sobolev spaces, W^{k} for integer k. The general case for all s ≥ 0 follows by interpolation. Secondly, there are infinitely many projection operators which have the same regularity properties as our constructed projection in the Main Theorem. Other projections can be constructed for instance by changing the factor τ_{k}(ξ) in Section 2 with the replacement of the term e^{−ξ^{2}} with another e^{−mξ^{2}} for any positive integer m. Then the rest of the arguments could be followed verbatim. By [2], if the Bergman projection were to map C_{0}^{∞}(\overline{D_{β}}) continuously into C^{∞}(\overline{D_{β}}) (it does not) then we would automatically have continuity from the larger space C^{∞}(\overline{D_{β}}) as well. Thus, it would be of interest to find an improvement to the projection, along the lines presented here, which preserves W_{j}^{s}(D_{β}) for all s ≥ 0. We lastly note that, while it would be ideal to obtain an operator which would map W^{s} to itself, without the restriction to the space W_{j}^{s}, by summing the operators in Main Theorem over j, the dependence of the norms in Theorem 5.1 on j prohibit the convergence of such a summation. Following the calculations of the proof of Proposition 3.1 leads to the estimates for the norms of T_{j}:

∥T_{j}∥ ≲ \frac{sinh[(j + 1)(β − π/2)]}{j + 1}.

This exponential growth of the estimates thus prohibits us from using results such as the Cotlar–Stein almost orthogonality lemma to conclude any convergence of a sum over the operators T_{j}.

References

[1] D. Barrett. Behavior of the Bergmann projection on the Diederich-Fornæss worm. Acta. Math., 168:1–10, 1992.

[2] D. Barrett. Duality between A^{∞} and A^{−∞} on domains with nondegenerate corners. Contemp. Math., 185:77–87, 1995.

[3] S. Bell and Ligocka. A simplification and extension of Fefferman’s theorem on biholomorphic mappings. Invent. Math., 57(3):283–289, 1980.

[4] M. Christ. Global C^{∞} irregularity of the ¯∂-Neumann problem for worm domains. J. Amer. Math. Soc., 9(4):1171–1185, 1996.

[5] K. Diederich and J. E. Fornæss. Pseudoconvex domains: an example with nontrivial Nebenhulle. Math. Ann., 225:275–292, 1977.

[6] P. Grisvard. Elliptic problems in nonsmooth domains. Number 24 in Monographs and studies in mathematics. Pitman Advanced Pub. Program, 1985.

[7] C. Kiselman. A study of the Bergman projection in certain Hartogs domains. Proc. Symposia in Pure Math., 52:219–231, 1991.

[8] S. Krantz and M. Peloso. The Bergman kernel and projection on non-smooth worm domains. Houston J. Math., 34(3):873–950, 2008.

[9] J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. Springer-Verlag, New York, 1972.

David E. Barrett Department of Mathematics University of Michigan - Ann Arbor 2074 East Hall Ann Arbor Michigan 48109 barrett@umich.edu Dariush Ehsani Hochschule Merseburg Eberhard-Leibnitz-Str. 2 D-06217 Merseburg Germany dehsani.math@gmail.com

Marco M. Peloso

Dipartimento di Matematica Università degli Studi di Milano Via C. Saldini 50 I-20133 Milano marco.peloso@unimi.it 1226

Documenta Mathematica 20 (2015)