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SZEG¨O KERNELS FOR CERTAIN

arXiv:math/9202201v1 [math.CV] 7 Feb 1992SZEG¨O KERNELS FOR CERTAINUNBOUNDED DOMAINS IN C2.Friedrich Haslinger1. IntroductionIn this paper we consider the connection between the Szeg¨o kernel of certainunbounded domains of C2 and the Bergman kernels of weighted spaces of entirefunctions of one complex variable.Let p : C −→R+ denote a C1–function and define Ωp ⊆C2 byΩp = {(z1, z2) ∈C2 : ℑ(z2) > p(z1)}.Weakly pseudoconvex domains of this kind were investigated by Nagel, Rosay, Steinand Wainger [10,11] , where estimates for the Szeg¨o and the Bergman kernel of thedomain were made in terms of the nonisotropic pseudometric defined in [12,13].For the case where p(z) = |z|k , k ∈N, Greiner and Stein [5] found an explicitexpression for the Szeg¨o kernel of Ωp, in which one can recognize the form of thepseudometric used for the nonisotropic estimates (see [2,8]).

If p is a subharmonicfunction, which depends only on the real or only on the imaginary part of z, thenone can find analogous expressions and estimates in [9].Let H2(∂Ωp) denote the space of all functions f ∈L2(∂Ωp), which are holomor-phic in Ωp and such thatsupy>0ZCZR|f(z, t + ip(z) + iy)|2 dλ(z)dt < ∞,where dλ denotes the Lebesgue measure on C. We identify ∂Ωp with C × R, anddenote by S((z, t), (w, s)), z, w ∈C , s, t ∈R, the Szeg¨o kernel of H2(∂Ωp).We use the tangential Cauchy–Riemann operator on ∂Ωp to get an expressionfor the Bergman kernel Kτ(z, w) in the space Hτ of all entire functions f such thatZ|f(z)|2 exp(−2τp(z)) dλ(z) < ∞,

2FRIEDRICH HASLINGERwhere τ > 0 ; in this connection we suppose that the weight functions p have areasonable growth behavior so that the corresponding spaces of entire functions arenontrivial, for example if p(z) is a polynomial in ℜz and ℑz.On the other hand, if one integrates the Bergman kernels with respect to theparameter τ, one obtains a formula for the Szeg¨o kernel of H2(∂Ωp).We apply the main result for special functions p to get generalizations of resultsin [5,8,9].In [7] one can find another approach to get explicit expressions forthe Szeg¨o kernel.Finally the Bergman kernels for the spaces Hτ, where p is afunction of ℜz, are investigated, especially their asymptotic behavior, which leadsto sharp estimates and applications to problems considered in [7] concerning aduality problem in functional analysis.Proposition 1. Let τ > 0.

Then(1)Kτ(z, w) = eτ(p(z)+p(w))ZRZRS((z, t), (w, s)) eiτ(s−t)p(w) −is dsdt,where the integrals are to be understood in the sense of the Plancherel theorem, i.e.in general one has only L2–convergence of the integrals.The fact that the above formula (1) is not symmetric in z and w is due to theL2– convergence of the integrals.Proposition 2. (2)S((z, t), (w, s)) =Z ∞0Kτ(z, w)e−τ(p(z)+p(w))e−iτ(s−t) dτ.2.

Proofs of Proposition 1. and 2.For the proof we consider the tangential Cauchy–Riemann operatorL =∂∂z1−2i ∂p∂z1(z1) ∂∂z2on ∂Ωp. Then (see [8]) L is a global tangential antiholomorphic vector field, andH2(∂Ωp) = {f ∈L2(∂Ωp) : L(f) = 0 as distribution}.After the usual identification of ∂Ωp with C×R the tangential Cauchy–Riemannoperator has the formL = ∂∂z −i∂p∂z∂∂t .For a function f ∈L2(dλ(z)dt) let F denote the Fourier transform with respect tothe variable t ∈R :(Ff)(z, τ) =Zf(z, t)e−itτ dt .

SZEG¨O KERNELS FOR CERTAIN UNBOUNDED DOMAINS IN C2.3ThenFLF−1 = ∂∂z + τ ∂p∂z .F and F−1 are to be taken in the sense of the Plancherel theorem.Now let M denote the multiplication operatorM : L2(dλ(z)dt) −→L2(e−2tp(z)dλ(z)dt)defined by(Mf)(z, τ) = eτp(z)f(z, τ) ,for f ∈L2(dλ(z)dt) . Then one has(3)FLF−1 = M −1 ∂∂z M .Let P denote the orthogonal projectionP : L2(dλ(z)dt) −→KerL,and let P be the orthogonal projectionP : L2(e−2tp(z)dλ(z)dt) −→Ker ∂∂z .For fixed τ > 0, let Pτ be the orthogonal projectionPτ : L2(e−2τp(z)dλ(z)) −→Ker ∂∂z .Now we claim that(Pf)(z, τ) = (Pτfτ)(z) , τ > 00 , τ ≤0,where fτ(z) = f(z, τ), for f ∈L2(e−2tp(z)dλ(z)dt).

In order to see this it is enoughto observe that a function f ∈L2(e−2tp(z)dλ(z)dt) holomorphic with respect to thevariable z has the property f(z, t) = 0 , for almost all t ≤0, which is a consequenceof our assumption on the weight function p.The next step is to show that(4)P = MFPF−1M −1.Denote the right side of (4) by Q. We have to show that Q2 = Q and thatKer ∂∂z ⊆L2(e−2tp(z)dλ(z)dt)coincides with the image of Q.

The first assertion follows directly from the definitionof Q. For the second assertion take a function f ∈L2(e−2tp(z)dλ(z)dt) and use (3)to prove that∂Qf = MFLPF−1M −1f,

4FRIEDRICH HASLINGERthe last expression is zero, since PF−1M −1f ∈KerL, which implies that the imageof Q is contained in Ker ∂∂z. To prove the opposite inclusion set g = Qf for f ∈Ker ∂∂z.

We are finish, if we can show that Qg = f. From (3) we get nowLF−1M −1f = F−1M −1 ∂∂z f,which is zero by the assumption on f, hence F−1M −1f ∈KerL and thereforePF−1M −1f = F−1M −1f.The last equality yieldsQg = MFPF−1M −1f = MFF−1M −1f = f,which proves formula (4).For a fixed τ > 0 take a function F ∈L2(e−2τp(z)dλ(z)) and definef(z, t) = χ(z)F(z) , t ≥τ0 , t < τ,where χ is a nonnegative, smooth function with the properties (χ(z))2 = p(z), for|z| ≤1 and χ(z) = 1, for |z| ≥2.SinceZCZR|f(z, t)|2e−2tp(z) dtdλ(z) =ZCZ ∞τ|χ(z)F(z)|2e−2tp(z) dtdλ(z)=ZC12p(z)|χ(z)F(z)|2e−2τp(z) dλ(z) ≤Const.ZC|F(z)|2e−2τp(z) dλ(z),it follows thatf ∈L2(e−2tp(z)dλ(z)dt).Now we use formula (4) to obtain (1): application of the operators M −1 andF−1 to the function f from above yieldsF−1M −1f(w, t) =Z ∞τχ(w)F(w)et(iσ−p(w)) dt= χ(w)F(w)e−τ(p(w)−iσ)p(w) −iσ,which is a function in L2(dλ(w)dσ), by the properties of the function χ.The next operator in (4) is now P , which is the Szeg¨o projection, hence anapplication of this operator can be expressed by integration over the Szeg¨o kernelS((z, t), (w, σ)). Finally we carry out the action of the operators F and M andrecall the properties of the operator P on the left side of (4), which imply thatthis operator is for a fixed τ the Bergman projection in a weighted space of entirefunctions in one variable.

The function χ appears on both sides and hence cancelsout. In this way we get formula (1).

In order to prove (2) one writes (4) in theform(5)P = F−1M −1PMF,and applies an analogous procedure as above.

SZEG¨O KERNELS FOR CERTAIN UNBOUNDED DOMAINS IN C2.53. Examples(a) Let α ∈R, α > 0.

We consider the function p(z) = |z|α and get from [6] thefollowing expression for the Bergman kernel Kτ(z, w) in the space Hτ :Kτ(z, w) = 2πα∞Xk=0(2τ)2(k+1)/α ( Γ(2(k + 1)/α))−1 zkwk.Now we apply formula (2) to this sum and getS((z, t), (w, s))= 2πα∞Xk=0(Γ(2(k + 1)/α))−1 zkwk 22(k+1)/αZ ∞0τ 2(k+1)/αe−τ(|z|α+|w|α)e−iτ(s−t) dτ,evaluation of the last integral givesΓ2(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 haveΓ2(k + 1)α+ 1= 2(k + 1)αΓ(2(k + 1)/α),henceS((z, t), (w, s)) = 2πα∞Xk=02(k + 1)α22(k+1)/αzkwk [|z|α + |w|α + i(s −t)]−(2(k+1)/α)−1 .Now we setA = 12(|z|α + |w|α + i(s −t))and carry out the summation over k with the resultS((z, t), (w, s)) = 2πα2 A−1−2/α1 −zwA2/α−2.This generalizes a result of Greiner and Stein [5], where the same formula appearsfor α ∈N (see also [2,8]). (b) If the weight function p depends only on the real part of z and satisfiesZRe−2p(x)+2yx dx < ∞,for each y ∈R, then the Bergman kernel of Hτ is given by(6)Kτ(z, w) = 12πZexp(η(z + w))Rexp(2(rητp(r))) dr dη,

6FRIEDRICH HASLINGERor(6’)Kτ(z, w) = τ2πZRexp(τη(z + w))RR exp(2τ(rη −p(r))) dr dη.This follows by a modification of methods developed in [9]. To show (6) we proceedin the following way:In sake of simplicity we set τ = 1.

Similar to the proofs of Proposition 1 and 2we consider the multiplication operatorMp : L2(dλ(z)) −→L2(e−2p(x)dλ(z)),defined by (Mpf)(z) = ep(x)f(z) , f ∈L2(dλ(z)). Now a computation shows that∂∂zep(x)f(z)= ep(x)12∂p∂xf + ∂f∂z,which can be expressed by the operator identityL(f) :=M−p∂∂z Mp(f) = 12∂p∂xf + ∂f∂z .Let F denote the Fouriertransform with respect to y :Ff(x, η) =Z ∞−∞f(x, y)e−iyη dy.Then in the sense of distributions we haveFL(f)(x, η) = 12e−p(x)+ηx ∂∂xep(x)−ηxFf(x, η).We set ψ(x, η) = ep(x)−ηx and define the multiplication operatorMψ : L2(dλ(z)) −→L2(e−2p(x)+2yxdλ(z))by (Mψg)(x, η) = ψ(x, η)g(x, η), for g ∈L2(dλ(z)).

Combining this with the lastresults we getL = 12F−1M−ψ∂∂xMψF,and finally∂∂z = 12MpF−1M−ψ∂∂xMψFM−p.In this context we consider differentiation with respect to x as an operator∂∂x : L2(e−2p(x)+2yxdλ(z)) −→L2(e−2p(x)+2yxdλ(z)),in the sense of distributions.Further we remark that Ker ∂∂x consists of all functions g ∈L2(e−2p(x)+2yxdλ(z)),which are constant in x

SZEG¨O KERNELS FOR CERTAIN UNBOUNDED DOMAINS IN C2.7By our assumption on the weight function p the space L2(e−2p(x)+2yxdx) con-tains the constants for each y ∈R. Let Py denote the orthogonal projection ofL2(e−2p(x)+2yxdx) onto the constants and P the orthogonal projection of L2(e−2p(x)+2yxdλ(z))onto Ker ∂∂x.

Then it is easily seen that(Pg)(x, y) = Pygy(x),for g ∈L2(e−2p(x)+2yxdλ(z)), where gy(x) = g(x, y).For a fixed y ∈R and a function h ∈L2(e−2p(x)+2yxdx) one hasPyh = (h, 1)(1, 1)1 =ZRe−2p(x)+2yx dx−1 ZRh(x)e−2p(x)+2yx dx.Finally let P denote the orthogonal projection of L2(e−2p(x)dλ(z)) onto H1 =Ker ∂∂z.With the help of the above operator identities we readily establish nowP = MpF−1M−ψPMψFM−p.This identity, together with the above remarks on the orthogonal projection P,implies formula (6).Using (2) one getsS((z, t), (w, s)) = 12πZ ∞0ZRτ exp(τ(η(z + w) −p(z) −p(w) −i(s −t)))RR exp(2τ(rη −p(r))) drdη dτ,which is similar to an expression in [9].Now we investigate the asymptotic behavior of the integral(7)ZRexp(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 formp(r) = |r|αα , α > 1, r ∈R.Let p∗denote the Young conjugate of p which is given by(8)p∗(η) = supx≥0[x|η| −p(x)] = |η|α′α′ ,where 1α +1α′ = 1. Note that p∗∗= p. Now we can estimate the integral (7) fromabove.Zexp(2τ(rη −p(r))) dr =Z ∞exp(2τ(rη −p(r))) dr +Z 0exp(2τ(rη −p(r))) dr.

8FRIEDRICH HASLINGERLet λ > 1. Then we have for η ≥1Z ∞0exp(2τ(rη −p(r))) dr ≤Z ∞0exp(2τ(rη −ληr + p∗(λη))) dr= exp(2τ(p∗(λη))Z ∞0exp(−2τ(λ −1)rη) dr= exp(2τp∗(λη)2τ(λ −1)η ,and for the second part of the integralZ 0−∞exp(2τ(rη −p(r))) dr =Z ∞0exp(2τ(−rη −p(r))) dr≤Z ∞0exp(−2τrη) dr=12τη .For η ≤−1 we estimate in the analogous way.Finally for |η| < 1 we getZ ∞0exp(2τ(rη −p(r))) dr ≤Z ∞0exp(2τ(r −p(r))) dr,Z 0−∞exp(2τ(rη −p(r))) dr =Z ∞0exp(2τ(−rη −p(r))) dr≤Z ∞0exp(2τ(r −p(r))) dr.Hence for each η ∈R we obtainZRexp(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 functionof 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 thesupremum in formula (8) is attained in the point µ(η), henceZRexp(2τ(rη −p(r))) dr ≥Z ∞0exp(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)

SZEG¨O KERNELS FOR CERTAIN UNBOUNDED DOMAINS IN C2.9for each η ≥0, where D(τ, λ) > 0 is a constant depending on τ and λ.To see this we remark thatη(µ(η) + 1) −p(µ(η) + 1) = ηα/(α−1) + η −1/αη1/(α−1) + 1α,andληµ(λη) −p(µ(λη)) = (1 −1/α)λα/(α−1)ηα/(α−1).It suffices to show that1 −(1 −1/α)λα/(α−1)ηα/(α−1) + η ≥1/αη1/(α−1) + 1α−D˜(λ),for each η ≥0, where D˜(λ) > 0 is a constant depending on λ. But this followseasily from the fact that1 −(1 −1/α)λα/(α−1) > 1/α.For η < 0 we argue in a similar way.On the whole we have now proved that(10) D(τ, λ) exp(2τp∗(η/λ)) ≤ZRexp(2τ(rη −p(r))) dr ≤C(λ, τ) exp(2τp∗(λη)),for each η ∈R and λ > 1.For the conjugate function p∗one obtains by the same methods(11) D1(τ, λ) exp(2τp(r/λ)) ≤ZRexp(2τ(rη −p∗(η))) dη ≤C1(λ, τ) exp(2τp(λr)),for each r ∈R and λ > 1.The asymptotic behavior of (7) as a function of τ, τ →∞, can be derived from[1], pg.

65 :ZRexp(2τ(rη −p(r))) dr ≍τp′′(µ(η))2π1/2exp(2τp∗(η)).Letexp(2τ℘∗(η)) =ZRexp(2τ(rη −p(r))) dr.Then formula (6’) can be written in the form(12)Kτ(z, w) = τ2πZRexp2τ(η(z + w2) −℘∗(η))dη.In view of (10) and (11) this means that the Bergman kernel Kτ(z, w) is in acertain sense an analytical continuation of the original weight exp(2τp(r)), namelyin the formexp2τ℘(z + w2).

10FRIEDRICH HASLINGERFor p(z) = x2/2 everything can be computed explicitly:ZRexp(2τ(rη −r2/2)) dr = (π/τ)1/2 exp(τη2),(13)Kτ(z, w) = τ2π expτ4(z + w)2and(14)S((z, t), (w, s)) = 12π14(z + w)2 −18(z + z)2 −18(w + w)2 −i(s −t)−2Applying formula (1) to the expression for the Szeg¨o kernel in (14), we arrive againat (13), now the integral with respect to s converges only in L2.Results of this type have also been obtained by Gindikin (see [4] or [3] ).Finally we mention an estimate for the Bergman kernel, which plays an importantrole in the duality problem of [7] and which, in itself, seems to be interesting.For the Bergman kernel in formula (13) the following condition is satisfied: foreach τ1 > τ there exists τ0, 0 < τ0 < τ, such thatZCZC|Kτ(z, w)|2 exp(−2τ1p(z) −2τ0p(w)) dλ(z) dλ(w) < ∞.This follows by a direct computation using (13). In the general case the integrationwith respect to the variable z causes no problems, as the function z 7→Kτ(z, w)belongs to the Hilbertspace Hτ1, for each fixed w. But, afterwards, the integrationwith respect to the variable w makes difficulties, because τ0 < τ.

SZEG¨O KERNELS FOR CERTAIN UNBOUNDED DOMAINS IN C2.11Acknowledgment. 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.References1. N.G.

de Bruijn, Asymptotic methods in analysis, North-Holland PublishingCo., Amsterdam, 1958.2. K.P.

Diaz, The Szeg¨o kernel as a singular integral kernel on a family of weaklypseudoconvex domains, Trans. Amer.

Math. Soc.

304 (1987), 147–170.3. B.A.

Fuks, Introduction to the theory of analytic functions in several complexvariables, (in Russian) M.,Fizmatgiz, Moscow, 1962.4. S.G. Gindikin, Analytic functions in tubular regions, Sov.

Math.–Doklady 3(1962), 1178–1182.5. P.C.

Greiner and E.M. Stein, On the solvability of some differential operatorsof type □b, Proc. Internat.

Conf., (Cortona, Italy, 1976–1977), Scuola Norm. Sup.Pisa, Pisa, 1978, pp.

106–165.6. N. Hanges, Explicit formulas for the Szeg¨o kernel for some domains in C2, J.Functional Analysis 88 (1990), 153–165.7.

F. Haslinger, The Bergman kernel and duality in weighted spaces of entirefunctions, preprint PAM–310, Berkeley, 1986.8. H. Kang, ∂b–equations on certain unbounded weakly pseudoconvex domains,Trans.

Amer. Math.

Soc. 315 (1989), 389–413.9.

A. Nagel, Vector fields and nonisotropic metrics,Beijing Lectures in Har-monic Analysis, E.M. Stein, Ed., Princeton Univ. Press, 1986.10.

A. Nagel, J.P. Rosay, E.M. Stein and S. Wainger, Estimates for the Bergmanand Szeg¨o kernels in certain weakly pseudoconvex domains,Bull. Amer.

Math.Soc. 18 (1988), 55–59.11.

A. Nagel, J.P. Rosay, E.M. Stein and S. Wainger, Estimates for the Bergmanand Szeg¨o kernels in C2, Ann. of Math.

129 (1989), 113–149.12. A. Nagel, E.M. Stein and S. Wainger, Boundary behavior of functions holo-morphic in domains of finite type, Proc.Nat.Acad.Sci.U.S.A.

78 (1981),6596–6599.13. A. Nagel, E.M. Stein and S. Wainger, Balls and metrics defined by vectorfields I: basic properties, Acta Math.

155 (1985), 103–147.INSTITUT F¨UR MATHEMATIK, UNIVERSIT¨AT WIEN, STRUDLHOFGASSE4,A–1090 WIEN, AUSTRIA.

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