Szegö kernels for certain unbounded domains in ℂ^{2}.

Friedrich Haslinger

Abstract. We consider the connection between the Szegö kernel of certain unbounded domains of ℂ^{2} and the Bergman kernels of weighted spaces of entire functions of one complex variable. We apply the results to get explicit expressions for Szegö kernels and Bergman kernels in some special cases and discuss the asymptotic behavior of the kernel functions. Finally , for a special case, we define a pseudometric globally suited to the geometry on the boundary.

1. Introduction

In this paper we consider the connection between the Szegö kernel of certain unbounded domains of ℂ^{2} and the Bergman kernels of weighted spaces of entire functions of one complex variable.

Let p : ℂ ⟶ ℝ_{+} denote a C^{1}–function and define Ω_{p} ⊆ ℂ^{2} by

Ω_{p} = {(z_{1}, z_{2}) ∈ ℂ^{2} : ℑ(z_{2}) > p(z_{1})}.

Weakly pseudoconvex domains of this kind were investigated by Nagel, Rosay, Stein and Wainger [NRSW1],[NRSW2] , where estimates for the Szegö and the Bergman kernel of the domain were made in terms of a nonisotropic pseudometric. For the case where p(z) = |z|^{k} , k ∈ ℕ, Greiner and Stein [GS] found an explicit expression for the Szegö kernel of Ω_{p}, in which one can recognize the form of the pseudometric used for the nonisotropic estimates (see [D],[K]). If p is a subharmonic function, which depends only on the real or only on the imaginary part of z, then one can find analogous expressions and estimates in [N].

Let H^{2}(∂Ω_{p}) denote the space of all functions f ∈ L^{2}(∂Ω_{p}), which are holomorphic in Ω_{p} and such that

sup_{y>0} ∫_{ℂ} ∫_{ℝ} |f(z, t + ip(z) + iy)|^{2} dλ(z)dt < ∞,

1980 Mathematics Subject Classification (1991 Revision). Primary 32A35; Secondary 30D20, 32F40.

Key words and phrases. Szegö kernel, Hilbert spaces of entire functions, Bergman kernel.

where dλ denotes the Lebesgue measure on ℂ. We identify ∂Ω_{p} with ℂ × ℝ, and denote by S((z, t), (w, s)), z, w ∈ ℂ , s, t ∈ ℝ, the Szegö kernel of H^{2}(∂Ω_{p}).

Now let τ > 0. The function e^{2τp} is locally integrable. Hence the space H_{τ} of all entire functions f such that

∫_{ℂ} |f(z)|^{2} exp(−2τp(z)) dλ(z) < ∞ ,

is a Hilbertspace with inner product

(f, g) = ∫_{ℂ} f(z)\overline{g(z)} exp(−2τp(z))dλ(z),

the topology of which is stronger than the compact open topology. Therefore H_{τ} has a Bergman kernel, which we denote by K_{τ} (z, w).

We use the tangential Cauchy–Riemann operator on ∂Ω_{p} to get an expression for the Szegö kernel S((z, t), (w, s)) of H^{2}(∂Ω_{p}) in terms of the Bergman kernels K_{τ} (z, w) of H_{τ} , τ > 0. In this way a problem for two complex variables is reduced to a problem for one complex variable. We suppose that the weight function p behaves in such a way that the corresponding spaces of entire functions are nontrivial, for example if p(z) is a polynomial in ℜz and ℑz.

We apply the main result for special functions p to get generalizations of results in [GS],[K],[N]. In [Han] one can find another approach to get explicit expressions for the Szegö kernel.

Further on the Bergman kernels for the spaces H_{τ}, where p is a function of ℜz, are investigated, especially their asymptotic behavior. The integral expressions which appear in this connection play also a crucial role in [C] and [CG].

Finally we point out that in the case p(z) = x^{2}/2 everything can be computed explictly and we define a pseudometric on ∂Ω_{p} globally suited to the geometry of ∂Ω_{p} (see also [D],[K],[N]).

2. Szegö and Bergman kernels.

We call the C^{1}–function p : ℂ ⟶ ℝ_{+} \underline{admissible}, if the function

(τ, z, w) ↦ K_{τ} (z, w)

is measureable as function from ℝ_{+} × ℂ^{2} to ℂ and if the corresponding Hilbertspaces H_{τ} , τ > 0 are nontrivial.

Proposition 1. Suppose that the function p is admissible. Then

(1)  S((z, t), (w, s)) = \frac{1}{2π} ∫_{0}^{∞} K_{τ} (z, w)e^{−τ(p(z)+p(w))}e^{−iτ(s−t)} dτ.

Note that the integration in (1) is taken with respect to the index τ of the Bergman kernel K_{τ} (z, w), which makes sense since p is admissible.

Proof. We consider the tangential Cauchy–Riemann operator

L = \frac{∂}{∂\overline{z_{1}}} − 2i\frac{∂p}{∂\overline{z_{1}}} (z_{1})\frac{∂}{∂\overline{z_{2}}}

on ∂Ω_{p}. Then (see [N]) L is a global tangential antiholomorphic vector field, and

H^{2}(∂Ω_{p}) = {f ∈ L^{2}(∂Ω_{p}) : L(f) = 0 as distribution}.

After the usual identification of ∂Ω_{p} with ℂ × ℝ the tangential Cauchy–Riemann operator has the form

L = \frac{∂}{∂\overline{z}} − i\frac{∂p}{∂\overline{z}} \frac{∂}{∂t} .

For a function f ∈ L^{2}(dλ(z)dt) let F denote the Fourier transform with respect to the variable t ∈ ℝ :

(Ff)(z, τ) = ∫_{ℝ} f(z, t)e^{−itτ} dt .

Then

(2)  FLF^{−1} = \frac{∂}{∂\overline{z}} + τ \frac{∂p}{∂\overline{z}} .

F and F^{−1} are to be taken in the sense of the Plancherel theorem.

Now let M denote the multiplication operator

M : L^{2}(dλ(z)dt) ⟶ L^{2}(e^{−2tp(z)}dλ(z)dt)

defined by

(Mf)(z, τ) = e^{τp(z)}f(z, τ) ,

for f ∈ L^{2}(dλ(z)dt) . Then from (2) we get

(3)  FLF^{−1} = M^{−1} \frac{∂}{∂\overline{z}} M .

Let P denote the orthogonal projection

P : L^{2}(dλ(z)dt) ⟶ KerL,

and let P be the orthogonal projection

P : L^{2}(e^{−2tp(z)}dλ(z)dt) ⟶ Ker\frac{∂}{∂\overline{z}} .

The next step is to show that

(4)  P = F^{−1}M^{−1}P MF,

Denote the right side of (4) by Q. We have to show that Q^{2} = Q and Q^{∗} = Q and that

KerL ⊆ L^{2}(dλ(z)dt)

coincides with the image of Q. The first assertion follows directly from the definition of Q, the second assertion from the fact that F^{∗} = F^{−1} and that M^{∗} = M^{−1}. In order to prove the third assertion take a function f ∈ L^{2}(dλ(z)dt) and use (3) to prove that

LQf = F^{−1}M^{−1} \frac{∂}{∂\overline{z}} P MFf,

the last expression is zero, since P MFf ∈ Ker\frac{∂}{∂\overline{z}} , which implies that the image of Q is contained in KerL. To prove the opposite inclusion set g = Qf for f ∈ KerL. We are finish, if we can show that Qg = f. From (3) we get now

MFLf = \frac{∂}{∂\overline{z}} MFf,

which is zero by the assumption on f, hence MFf ∈ Ker\frac{∂}{∂\overline{z}} and therefore

P MFf = MFf.

The last equality yields

Qg = F^{−1}M^{−1}P MFf = F^{−1}MM^{−1}Ff = f,

which proves formula (4).

For fixed τ > 0, let P_{τ} be the orthogonal projection

P_{τ} : L^{2}(e^{−2τp(z)}dλ(z)) ⟶ Ker\frac{∂}{∂\overline{z}} .

Now we claim that

(P f)(z, τ) = \{ \begin{matrix}(P_{τ} f_{τ})(z) & , τ > 0 \\ 0 & , τ ≤ 0 \end{matrix}\} ,

where f_{τ} (z) = f(z, τ), for f ∈ L^{2}(e^{−2tp(z)}dλ(z)dt). In order to see this, first observe that for a function f ∈ L^{2}(e^{−2tp(z)}dλ(z)dt) the orthogonal projection P_{τ} f_{τ} can be written in the form

(P_{τ} f_{τ})(z) = ∫_{ℂ} K_{τ} (z, w)f(w, τ)e^{−2τp(w)} dλ(w),

now it follows that (τ, z) ↦ (P_{τ} f_{τ})(z) is a measureable function, because p is admissible.

Further note that a function f ∈ L^{2}(e^{−2tp(z)}dλ(z)dt) holomorphic with respect to the variable z has the property f(z, t) = 0 , for almost all t ≤ 0, which is a consequence of our assumption on the weight function p.

Since

∫_{ℂ} |(P_{τ} f_{τ})(z)|^{2}e^{−2τp(z)} dλ(z) ≤ ∫_{ℂ} |f(z, τ)|^{2}e^{−2τp(z)} dλ(z),

we obtain

∫_{ℂ} ∫_{ℝ} |(P f)(z, τ)|^{2} dλ(z) dτ ≤ ∫_{ℂ} ∫_{ℝ} |f(z, τ)|^{2} dλ(z) dτ.

All the other properties of an orthogonal projection are easily checked.

Now (1) follows directly from (4). □

Formula (1) can be viewed as a generalization of Theorem 2.8. in [K], as the following arguments show :

Let α ∈ ℝ, α ≥ 2. We consider the function p(z) = |z|^{α} and get from [Has] the following expression for the Bergman kernel K_{τ} (z, w) in the space H_{τ} :

K_{τ} (z, w) = \frac{α}{2π} ∑_{k=0}^{∞}(2τ)^{2(k+1)/α} ( Γ(2(k + 1)/α))^{−1} z^{k}\overline{w}^{k}.

From this it is clear, that p is admissible. Now we apply formula (1) to this sum and get

\begin{matrix}S((z, t), (w, s)) = \\ \frac{α}{4π^{2}} ∑_{k=0}^{∞} (Γ(2(k + 1)/α))^{−1} z^{k}\overline{w}^{k} 2^{2(k+1)/α} ∫_{0}^{∞} τ^{2(k+1)/α}e^{−τ(|z|^{α}+|w|^{α})}e^{−iτ(s−t)} dτ \end{matrix}

evaluation of the last integral gives

Γ (\frac{2(k + 1)}{α} + 1) [|z|^{α} + |w|^{α} + i(s − t)]^{−(2(k+1)/α)−1} ,

by the functional equation of the Γ–function we have

Γ (\frac{2(k + 1)}{α} + 1) = \frac{2(k + 1)}{α} Γ(2(k + 1)/α),

hence

\begin{matrix}S((z, t), (w, s)) \\ = \frac{α}{4π^{2}} ∑_{k=0}^{∞} \frac{2(k + 1)}{α} 2^{2(k+1)/α}z^{k}\overline{w}^{k} [|z|^{α} + |w|^{α} + i(s − t)]^{−(2(k+1)/α)−1} . \end{matrix}

Now we carry out the summation over k and get the following result

Proposition 2. Let p(z) = |z|^{α} , α ∈ ℝ , α ≥ 2. Then the Szegö kernel of H^{2}(∂Ω_{p}) has the form

(5)  S((z, t), (w, s)) = \frac{1}{4π^{2}} A^{−1−2/α} (1 − \frac{z\overline{w}}{A^{2/α}} )^{−2} ,

where

A = \frac{1}{2}(|z|^{α} + |w|^{α} + i(s − t)).

This generalizes a result of Greiner and Stein [GS], where the same formula appears for α ∈ ℕ (see also [D],[K]).

Proposition 3. If the weight function p depends only on the real part of z and satisfies

∫_{ℝ} e^{−2p(x)+2yx} dx < ∞,

for each y ∈ ℝ, then the Bergman kernel of H_{τ} is given by

(6)  K_{τ} (z, w) = \frac{1}{2π} ∫_{ℝ} \frac{exp(η(z + \overline{w}))}{∫_{ℝ} exp(2(rη − τp(r))) dr} dη,

or

(6’)  K_{τ} (z, w) = \frac{τ}{2π} ∫_{ℝ} \frac{exp(τη(z + \overline{w}))}{∫_{ℝ} exp(2τ(rη − p(r))) dr} dη.

Formula (6) shows, that also in this case p is admissible.

Proof. Here we use a modification of methods developed in [N].

In sake of simplicity we set τ = 1. Similar to the proof of Proposition 1 we consider the multiplication operator

M_{p} : L^{2}(dλ(z)) ⟶ L^{2}(e^{−2p(x)}dλ(z)),

defined by (M_{p}f)(z) = e^{p(x)}f(z) , f ∈ L^{2}(dλ(z)). Now a computation shows that

\frac{∂}{∂\overline{z}} (e^{p(x)}f(z)) = e^{p(x)} (\frac{1}{2} \frac{∂p}{∂x}f + \frac{∂f}{∂\overline{z}} ) ,

which can be expressed by the operator identity

L(f) := (M_{−p} \frac{∂}{∂\overline{z}} M_{p}) (f) = \frac{1}{2} \frac{∂p}{∂x}f + \frac{∂f}{∂\overline{z}} .

Let F denote the Fouriertransform with respect to y :

Ff(x, η) = ∫_{−∞}^{∞} f(x, y)e^{−iyη} dy.

Then in the sense of distributions we have

FL(f)(x, η) = \frac{1}{2} (e^{−p(x)+ηx} \frac{∂}{∂x} (e^{p(x)−ηx}Ff(x, η))) .

We set ψ(x, η) = e^{p(x)−ηx} and define the multiplication operator

M_{ψ} : L^{2}(dλ(z)) ⟶ L^{2}(e^{−2p(x)+2yx}dλ(z))

by (M_{ψ}g)(x, η) = ψ(x, η)g(x, η), for g ∈ L^{2}(dλ(z)). Combining this with the last results we get

L = \frac{1}{2}F^{−1}M_{−ψ} \frac{∂}{∂x}M_{ψ}F,

and finally

\frac{∂}{∂\overline{z}} = \frac{1}{2}M_{p}F^{−1}M_{−ψ} \frac{∂}{∂x}M_{ψ}FM_{−p}.

In this context we consider differentiation with respect to x as an operator

\frac{∂}{∂x} : L^{2}(e^{−2p(x)+2yx}dλ(z)) ⟶ L^{2}(e^{−2p(x)+2yx}dλ(z)),

in the sense of distributions.

Further we remark that Ker\frac{∂}{∂x} consists of all functions g ∈ L^{2}(e^{−2p(x)+2yx}dλ(z)), which are constant in x.

By our assumption on the weight function p the space L^{2}(e^{−2p(x)+2yx}dx) contains the constants for each y ∈ ℝ.

Let P_{y} denote the orthogonal projection of L^{2}(e^{−2p(x)+2yx}dx) onto the constants and P the orthogonal projection of L^{2}(e^{−2p(x)+2yx}dλ(z)) onto Ker\frac{∂}{∂x}. Then it is easily seen that

(P g)(x, y) = P_{y}g_{y}(x),

for g ∈ L^{2}(e^{−2p(x)+2yx}dλ(z)), where g_{y}(x) = g(x, y).

In fact for a fixed y ∈ ℝ and a function h ∈ L^{2}(e^{−2p(x)+2yx}dx) one has

P_{y}h = \frac{(h, 1)}{(1, 1)}1 = (∫_{ℝ} e^{−2p(x)+2yx} dx)^{−1} ∫_{ℝ} h(x)e^{−2p(x)+2yx} dx.

Finally let P denote the orthogonal projection of L^{2}(e^{−2p(x)}dλ(z)) onto H_{1} = Ker\frac{∂}{∂\overline{z}} .

With the help of the above operator identities we readily establish now

P = M_{p}F^{−1}M_{−ψ}P M_{ψ}FM_{−p}.

This identity, together with the above remarks on the orthogonal projection P, implies formula (6). □

If we apply now (1) we get

Proposition 4. Let p be as in the last proposition. Then the Szegö kernel of H^{2}(∂Ω_{p}) has the form

S((z, t), (w, s)) = \frac{1}{4π^{2}} ∫_{0}^{∞} ∫_{ℝ} \frac{τ exp(τ(η(z + \overline{w}) − p(z) − p(w) − i(s − t)))}{∫_{ℝ} exp(2τ(rη − p(r))) dr} dη dτ.

(see also [N])

Now we investigate the asymptotic behavior of the integral

(7)  ∫_{ℝ} exp(2τ(rη − p(r))) dr,

which appears in formula (6), first as a function of η, for |η| → ∞.

We restrict our attention to the case where the weight function p is of the form

p(r) = \frac{|r|^{α}}{α} , α > 1, r ∈ ℝ.

Let p^{∗} denote the Young conjugate of p which is given by

(8)  p^{∗}(η) = sup_{x≥0} [x|η| − p(x)] = \frac{|η|^{α′}}{α′} ,

where \frac{1}{α} + \frac{1}{α′} = 1. Note that p^{∗∗} = p.

Now we can estimate the integral (7) from above. ∫_{ℝ} exp(2τ(rη − p(r))) dr = ∫_{0}^{∞} exp(2τ(rη − p(r))) dr + ∫_{−∞}^{0} exp(2τ(rη − p(r))) dr. Let λ > 1. Then we have for η ≥ 1

∫_{0}^{∞} exp(2τ(rη − p(r))) dr ≤ ∫_{0}^{∞} exp(2τ(rη − ληr + p^{∗}(λη))) dr

= exp(2τ(p^{∗}(λη)) ∫_{0}^{∞} exp(−2τ(λ − 1)rη) dr

= \frac{exp(2τp^{∗}(λη)}{2τ(λ − 1)η},

and for the second part of the integral

∫_{−∞}^{0} exp(2τ(rη − p(r))) dr = ∫_{0}^{∞} exp(2τ(−rη − p(r))) dr

≤ ∫_{0}^{∞} exp(−2τrη) dr

= \frac{1}{2τη} .

For η ≤ −1 we estimate in the analogous way.

Finally for |η| < 1 we get

∫_{0}^{∞} exp(2τ(rη − p(r))) dr ≤ ∫_{0}^{∞} exp(2τ(r − p(r))) dr,

∫_{−∞}^{0} exp(2τ(rη − p(r))) dr = ∫_{0}^{∞} exp(2τ(−rη − p(r))) dr
≤ ∫_{0}^{∞} exp(2τ(r − p(r))) dr.

Hence for each η ∈ ℝ we obtain

∫_{ℝ} exp(2τ(rη − p(r))) dr ≤ C(λ, τ) exp(2τp^{∗}(λη)),

for each λ > 1, where C(λ, τ) > 0 is a constant depending on λ and τ.

To estimate the integral in (7) from below we denote by μ the inverse function of the derivative p′

μ(η) := (p′)^{−1} (η) = |η|^{1/(α−1)}.

First suppose that η ≥ 0 and observe that p′ is strictly increasing and that the supremum in formula (8) is attained in the point μ(η), hence

∫_{ℝ} exp(2τ(rη − p(r))) dr ≥ ∫_{0}^{∞} exp(2τ(rη − p(r))) dr
≥ exp(2τ(η(μ(η) + 1) − p(μ(η) + 1))).

Next we claim that for each λ, 0 < λ < 1, the following inequality holds

(9)  2τ(η(μ(η) + 1) − p(μ(η) + 1)) ≥ 2τ(λημ(λη) − p(μ(λη))) − D(τ, λ),

for each η ≥ 0, where D(τ, λ) > 0 is a constant depending on τ and λ.

To see this we remark that

η(μ(η) + 1) − p(μ(η) + 1) = η^{α/(α−1)} + η − 1/α (η^{1/(α−1)} + 1)^{α} ,

and

λημ(λη) − p(μ(λη)) = (1 − 1/α)λ^{α/(α−1)}η^{α/(α−1)}.

It suffices to show that

(1 − (1 − 1/α)λ^{α/(α−1)}) η^{α/(α−1)} + η ≥ 1/α (η^{1/(α−1)} + 1)^{α} − D˜(λ),

for each η ≥ 0, where D˜(λ) > 0 is a constant depending on λ. But this follows easily from the fact that

1 − (1 − 1/α)λ^{α/(α−1)} > 1/α.

For η < 0 we argue in a similar way.

On the whole we have now proved

Proposition 5. Let p(r) = r^{α}/α , α > 1, r ∈ ℝ. Then

(10) D(τ, λ) exp(2τp^{∗}(η/λ)) ≤ ∫_{ℝ} exp(2τ(rη − p(r))) dr ≤ C(λ, τ) exp(2τp^{∗}(λη)),

for each η ∈ ℝ and λ > 1.

For the conjugate function p^{∗} we get by the same methods

(11) D_{1}(τ, λ) exp(2τp(r/λ)) ≤ ∫_{ℝ} exp(2τ(rη − p^{∗}(η))) dη ≤ C_{1}(λ, τ) exp(2τp(λr)),

for each r ∈ ℝ and λ > 1.

This is a genealization of Lemma 11.3 in [C].

Remark. The asymptotic behavior of (7) as a function of τ, τ → ∞, can be derived from [B], pg. 65 :

∫_{ℝ} exp(2τ(rη − p(r))) dr ≍ (\frac{τp′′(μ(η))}{π})^{−1/2} exp(2τp^{∗}(η)).

Let

exp(2τ℘^{∗}(η)) = ∫_{ℝ} exp(2τ(rη − p(r))) dr.

Then formula (6’) can be written in the form

(12)  K_{τ} (z, w) = \frac{τ}{2π} ∫_{ℝ} exp (2τ(η(\frac{z + \overline{w}}{2} ) − ℘^{∗}(η))) dη.

In view of (10) and (11) this means that the Bergman kernel K_{τ} (z, w) is in a certain sense an analytical continuation of the original weight exp(2τp(r)), namely in the form

exp (2τ℘(\frac{z + \overline{w}}{2} )) .

Example. For p(z) = x^{2}/2 everything can be computed explicitly:

∫_{ℝ} exp(2τ(rη − r^{2}/2)) dr = (π/τ)^{1/2} exp(τη^{2}),

(13)  K_{τ} (z, w) = \frac{τ}{2π} exp (\frac{τ}{4} (z + \overline{w})^{2})

and

(14) S((z, t), (w, s)) = \frac{1}{4π^{2}} (\frac{1}{4}(z + \overline{w})^{2} − \frac{1}{8}(z + \overline{z})^{2} − \frac{1}{8}(w + \overline{w})^{2} − i(s − t))^{−2}

Applying formula (1) to the expression for the Szegö kernel in (14), we arrive again at (13), now the integral with respect to s converges only in L^{2}.

Results of this type have also been obtained by Gindikin (see [F] or [G] ).

With expression (14) in mind we now define a pseudometric on ∂Ω_{p} , p(z) = x^{2}/2, globally suited to the complex geometry on ∂Ω_{p} (see [D] for other special cases and [N] for a more general setting).

We show that

d((z, t), (w, s)) = |x^{2} + u^{2} − \frac{1}{2}(z + \overline{w})^{2} − i(s − t)|^{1/2}

defines a pseudometric on ∂Ω_{p}, where (z, t), (w, s) ∈ ∂Ω_{p} and z = x + iy, w = u + iv.

It is easily seen that d((z, t), (w, s)) = 0 if and only if (z, t) = (w, s) and that

d((z, t), (w, s)) = d((w, s), (z, t)).

Let (ζ, r) ∈ ∂Ω_{p} be a further boundary point. We have to show that

(15)  d((z, t), (w, s)) ≤ C[d((z, t), (ζ, r)) + d((ζ, r), (w, s))],

where C > 0 is a constant.

By a straigthforward computation we obtain

d((z, t), (w, s)) = |\frac{1}{2}(|z|^{2} + |w|^{2}) − i(s − t) − z\overline{w} + i(uv − xy)|^{1/2} .

We set

P = \frac{1}{2} (|z|^{2} + |w|^{2}) − i(s − t)

and

Q = z\overline{w} − i(uv − xy).

Let d_{1}((z, t), (w, s)) denote the infimum over all δ > 0 such that

|P − Q| < δ^{2} and |z − w| < δ.

We claim that

(16)  |P − Q| ≈ d_{1}^{2}.

In order to show (16) we only have to consider the case where

|P − Q| ≤ δ^{2} and |z − w| = δ.

We get

|P − Q| ≥ |ℜ(P − Q)| = |\frac{1}{2}(|z|^{2} + |w|^{2}) − xu − yv|

= \frac{1}{2}|x^{2} − 2xu + u^{2} + y^{2} − 2yv + v^{2}| = \frac{1}{2}|z − w|^{2} = \frac{1}{2}δ^{2},

and hence |P − Q| ≳ δ^{2}, which proves (16).

In the proof of (15) we can now use d_{1} instead of d. We set ζ = ξ + iη and

D((z, t), (ζ, r)) = \frac{1}{2}(|z|^{2} + |ζ|^{2}) − i(r − t) − [z\overline{ζ} − i(ξη − xy)].

Now, if |D((z, t), (ζ, r))| < δ^{2} and |z −ζ| < δ, and |D((ζ, r), (w, s))| < δ^{2} and |ζ −w| < δ, then we get |z − w| < 2δ and since

D((z, t), (ζ, r))+D((ζ, r), (w, s)) = \frac{1}{2}(|z|^{2} +|w|^{2})+|ζ|^{2} −i(s−t)−z\overline{ζ} −ζ\overline{w}−i(uv −xy)

= D((z, t), (w, s)) + |ζ|^{2} + z\overline{w} − z\overline{ζ} − ζ\overline{w} = D((z, t), (w, s)) + (z − ζ)(\overline{w} − \overline{ζ})

we obtain

|D((z, t), (w, s))| = |D((z, t), (ζ, r)) + D((ζ, r), (w, s)) − (z − ζ)(\overline{w} − \overline{ζ})|
< 2δ^{2} + δ^{2} = 3δ^{2},

and hence (15) is proved.

Acknowledgment. The author would like to express his sincere thanks to A. Nagel for valuable discussions during a conference at the M.S.R.I. in Berkeley.

References

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Institut für Mathematik , Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria E-mail: HAS@PAP.UNIVIE.AC.AT