ESI
The Erwin Schr¨ odinger International Institute for Mathematical Physics
Boltzmanngasse 9 A-1090 Wien, Austria
Weighted Bergman Kernels and Quantization
Miroslav Engliˇ s
Vienna, Preprint ESI 932 (2000)
Supported by Federal Ministry of Science and Transport, Austria Available via anonymous ftp from FTP.ESI.AC.AT or via WWW, URL: http://www.esi.ac.at
September 5, 2000
WEIGHTED BERGMAN KERNELS AND QUANTIZATION
Miroslav Engliˇ s Abstract. Let Ω be a bounded pseudoconvex domain in C N , φ, ψ two positive functions on Ω such that − log ψ, − log φ are plurisubharmonic, z ∈ Ω a point at which − log φ is smooth and strictly plurisubharmonic, and M a nonnegative integer. We show that as k → ∞, the Bergman kernels with respect to the weights φk ψ M have an asymptotic expansion Kφk ψM (x, y) =
∞ X kN bj (x, y) k−j , π N φ(x, y)k ψ(x, y)M j=0
h ∂ 2 log φ(x) i , b0 (x, x) = det − ∂zj ∂z k
for x, y near z, where φ(x, y) is an almost-analytic extension of φ(x) = φ(x, x) and similarly for ψ. If in addition Ω is of finite type, φ, ψ behave reasonably at the boundary and − log φ, − log ψ are strictly plurisubharmonic on Ω, we obtain also an analogous asymptotic expansion for the Berezin transform and give applications to the Berezin quantization. Finally, for Ω smoothly bounded and strictly pseudoconvex and φ a smooth strictly plurisubharmonic defining function for Ω, we also obtain results about the Berezin-Toeplitz quantization on Ω.
Let Ω be a domain in CN , ρ a positive continuous function on Ω, and Kρ the reproducing kernel of the weighted Bergman space A2 (Ω, ρ) of all holomorphic functions on Ω squareintegrable with respect to the measure ρ(z) dz, dz being the Euclidean volume element in CN ; we call Kρ the weighted Bergman kernel corresponding to ρ, and for ρ ≡ 1 we will speak simply of the Bergman kernel KΩ of Ω. The Berezin transform Bρ is the integral operator defined by Z |Kρ (x, y)|2 ρ(x) dx (1) f (x) Bρ f (y) = Kρ (y, y) Ω for all y for which Kρ (y, y) 6= 0. In terms of the operator Mf of multiplication by f on the space L2 (Ω, ρ dz) this can be rewritten as Bρ f (y) =
hMf Kρ (·, y), Kρ (·, y)i , kKρ (·, y)k2
from which it is immediate that the integral (1) converges, for instance, for any bounded measurable function f . The Berezin transform was first introduced by F.A. Berezin [Ber] in the context of quantization of K¨ahler manifolds. More specifically, let φ be a positive function on Ω such that − log φ is strictly plurisubharmonic, and set gjk = ∂ 2 (− log φ)/∂zj ∂z k
(2)
2000 Mathematics Subject Classification. Primary 32A25, 53D55; Secondary 32A07, 32W05, 47B35. Key words and phrases. Weighted Bergman kernels, Fefferman asymptotic expansion, Berezin transform, Toeplitz operators, star product, Berezin-Toeplitz quantization. ˇ grant A1019701 and A1019005. The author’s research was supported by GA AV CR
1
Typeset by AMS-TEX
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ˇ MIROSLAV ENGLIS
and χ = det(gjk ) (so that ds2 = gjk dzj dz k is the K¨ahler metric with potential − log φ and χ the corresponding volume density). For Ω a bounded symmetric domain in CN and φ(z) = 1/KΩ (z, z) (so that ds2 is the Bergman metric), Berezin showed that for all m ≥ 1 it holds that Kφm χ (x, y) = p(m)φ(x, y)−m (3) where φ(x, y) = 1/KΩ (x, y) is a function on Ω × Ω holomorphic in x, y such that φ(x, x) = φ(x), and p is a polynomial of degree N which depends only on Ω; and that Bφm χ f (y) = f (y) +
³ 1 ´ 1 ˜ ∆f (y) + O m m2
(4)
˜ is the Laplace-Beltrami operator of the metric ds2 on Ω. Using (4), as m → ∞, where ∆ he was then able to construct a nice quantization procedure for mechanical systems whose phase-space is Ω with the Bergman metric. Later the present author showed that to get (4) it suffices that (3) hold only asymptotically as m → ∞ in a certain sense and used this to extend the range of applicability of Berezin’s original procedure to all plane domains with the Poincar´e metric [E1], to some complete Reinhardt domains in C2 with natural rotation-invariant K¨ahler metrics [E2], and finally to any strictly pseudoconvex domain Ω with real-analytic boundary and φ a real-analytic defining function for Ω such that − log φ is strictly plurisubharmonic [E6]. In fact, [E6] even dealt with the more general setting of weights of the form ρ = φm ψ M with −φ, −ψ two C ω defining functions of a strictly pseudoconvex domain Ω such that − log φ, − log ψ are plurisubharmonic, M fixed and m → ∞. Then (3), with φm ψ M in place of φm χ, holds asymptotically for (x, y) near the diagonal, and (4) holds for any f ∈ L∞ (Ω) which is smooth in a neighbourhood of y. The aim of the present paper is to improve these results by relaxing the hypotheses of real-analyticity and of φ, ψ being defining functions. For a function f on a domain in Cn , we say that f is almost analytic at x = a if ∂f /∂xj , j = 1, . . . , n vanish at a together with their partial derivatives of all orders. It is known that any C ∞ function φ(x) possesses a (non-unique) almost analytic extension φ(x, y) such that φ(x, y) is almost-analytic in x and y at all points of the diagonal x = y, and φ(x, x) = φ(x). Further, if φ(x) is real-valued, then the extension may be chosen so that φ(y, x) = φ(x, y) (just replace φ(x, y) by 21 (φ(x, y) + φ(y, x))); in the sequel, we will always assume that an extension with this additional property has been chosen for a real-valued φ(x). We now have the following results. Theorem 1. Let Ω be a bounded pseudoconvex domain in CN , φ, ψ two bounded positive continuous functions on Ω such that − log φ, − log ψ are plurisubharmonic, and let x0 ∈ Ω be a point in a neighbourhood of which φ and ψ are C ∞ and − log φ is strictly plurisubharmonic. Fix an integer M ≥ 0. Then there is a smaller neighbourhood U of x0 such that the asymptotic expansion ∞ X kN bj (x, y)k −j Kφk ψM (x, y) = N · π φ(x, y)k ψ(x, y)M j=0
(5)
holds uniformly for all x, y ∈ U as k → ∞, in the sense that for each m > 0, ¯ ¯ k N φ(x)k/2 φ(y)k/2 sup ¯¯φ(x)k/2 φ(y)k/2 Kφk ψM (x, y) − N π φ(x, y)k ψ(x, y)M x,y∈U
N +m−1 X j=0
bj (x, y)k
¯ ¯ −m ¯ = O(k )
−j ¯
(6) as k → ∞. Here φ(x, y), ψ(x, y) are fixed almost-analytic extensions of φ(x) and ψ(x) to U × U, respectively. The coefficients bj (x, y) ∈ C ∞ (U × U) are almost-analytic at x = y,
BERGMAN KERNELS
3
and their jets at a point (x, x) on the diagonal depend only on the jets of φ and ψ at x. In particular, h 1 i b0 (x, x) = det ∂∂ log . (7) φ(x) In the situation of the last theorem, consider the domain 2 2 e = {(z1 , z2 , z3 ) ∈ Ω × CM × C : |z3 | + |z2 | < 1}. Ω φ(z1 ) ψ(z1 )
(8)
Recall that a for domain D in Cn , a boundary point z ∈ ∂D is called smooth if in some neighbourhood of z, ∂D is a C ∞ -submanifold of Cn ; the domain D is called smoothly bounded if it is bounded and all its boundary points are smooth. A smooth boundary point z is said to be of finite type ≤ m if there is no complex analytic variety passing through z which has order of contact with ∂D at z bigger than m. (Thus, for instance, a strictly pseudoconvex boundary point is of type 2.) Finally, a smoothly bounded domain is said to be of finite type if all its boundary points are of finite type. Theorem 2. Assume that the hypotheses of Theorem 1 are fulfilled, and that in addition e is smoothly bounded and of finite type. (This implies, in particular, that φ, ψ ∈ C ∞ (Ω).) Ω Then for any f ∈ L∞ (Ω) which is C ∞ in a neighborhood of x0 , there is an asymptotic expansion ∞ X Qj f (y) · k −j , (9) Bφk ψM f (y) = j=0
uniformly for all y in a neighbourhood of x0 , where Qj are linear differential operators whose coefficients involve only the derivatives of φ, ψ at y and Q0 is the identity operator. We remark that in the applications to the Berezin quantization, − log φ is the potential of the K¨ahler metric, and thus is automatically strictly plurisubharmonic on all of Ω. The whole approach can also be adapted to arbitrary K¨ahler manifolds Ω in place of domains in CN [Pe], and sections of line bundles in place of functions. The function φ then defines the metric structure of the line bundle, and ∂∂ log φ is the corresponding curvature form. For compact K¨ahler manifolds and − log φ strictly plurisubharmonic on all of Ω (i.e. the line bundle of strictly negative curvature) the analogue of Theorem 1 has been obtained independently by Zelditch [Ze] for x = y and by Catlin [Ca] for general x, y; and the analogue of Theorem 2 in this setting was established by Karabegov and Schlichenmaier [KS]. In [BMS] and [Sch] the authors also obtain (still in the context of compact manifolds) somewhat stronger results concerning the Berezin-Toeplitz quantization, and we finish by observing that the same results can also be obtained in our noncompact situation. (ρ) Recall that, quite generally, the Toeplitz operator TF with symbol F ∈ L∞ (Ω) is the 2 operator on A (Ω, ρ) given by the recipe (ρ) TF f (y)
=
Z
f (x)F (x)Kρ (y, x)ρ(x) dx, Ω
or, equivalently, (ρ)
TF f = Pρ (F f ) where Pρ is the orthogonal projection of L2 (Ω, ρ) onto A2 (Ω, ρ). For simplicity, we state our result on the Berezin-Toeplitz quantization only for the weights which are of most interest to us, viz. ρ = φm χ with χ = det[−∂∂ log φ].
ˇ MIROSLAV ENGLIS
4
Theorem 3. Let Ω be a smoothly bounded strictly pseudoconvex domain in CN and −φ a smooth defining function for Ω = {φ > 0} such that − log φ is strictly plurisubharmonic. Then: (φm χ)
(i) for any f ∈ C ∞ (Ω), kTf k → kf k∞ as m → ∞; (ii) there exist bilinear differential operators Cj (j = 0, 1, 2, . . . ) such that for any f, g ∈ C ∞ (Ω) and an integer M , M ° ° ° (φm χ) (φm χ) X −j (φm χ) ° m TCj (f,g) ° = O(m−M −1 ) − T T ° f g
(10)
j=0
as m → ∞. Further, C0 (f, g) = f g and C1 (f, g) − C1 (g, f ) = i{f, g}, the Poisson bracket of f and g with respect to the metric (2). The result of the kind appearing in Theorem 3 was first obtained for Ω a domain in C with the Poincar´e metric by Klimek and Lesniewski [KL], using uniformization techniques, and for Ω a bounded symmetric domain with the invariant (Bergman) metric and Ω = Cn with the Euclidean metric by Borthwick, Lesniewski and Upmeier [BLU] and Coburn [Co], respectively, in both cases with the aid of the computational machinery available thanks to the specific nature of the domain and metric. In our case (i) is a fairly straightforward consequence of Theorem 2, while (ii) follows, as in [BMS] and [Sch], from the Boutet de Monvel-Guillemin calculus of generalized Toeplitz operators [BG]; see also [Gu]. As in [E6], our method of proof of Theorem 1 is based on the analysis of the Bergman e of the Forelli-Rudin domain (8) over Ω; (3) is then obtained from Fefferman’s kernel K e near the boundary. This is done in Section 2, after establishing asymptotic expansion of K some localization theorems for the Bergman kernel in Section 1. Theorem 2 is proved in Section 3, and its applications to quantization are described in Section 4. The BerezinToeplitz quantization is discussed in Section 5. At the end of each section we provide various remarks, comments on related developments, open problems, etc. Throughout the paper, “psh” is an abbreviation for “plurisubharmonic”. Acknowledgement. Part of this work was done while the author was visiting Erwin Schr¨odinger Institute for Mathematical Physics in Vienna in November 1999. It is a pleasure for the author to acknowledge its support, and to thank E. Straube for helpful discussions. 1. Preliminaries Our starting point is the following proposition, reproduced here from [E6] (see also [BFS]), which relates the weighted Bergman kernels Kφk ψM on Ω to the unweighted Bergman kernel e of the domain Ω.
Proposition 4. Let Ω be an arbitrary domain in CN (it need not be bounded ), φ, ψ two e the domain defined positive continuous functions on Ω, M a nonnegative integer, and Ω e e by (8). Then the Bergman kernel K := KΩ e of Ω is given by e t) = K(z;
∞ X (k + l + M + 1)! Kψl+M φk+1 (z1 , t1 ) hz2 , t2 il (z3 t3 )k . k! l! π M +1
(11)
k,l=0
e The series converges uniformly on compact subsets of Ω.
e is pseudoconvex if and only Note that by the familiar criterion for Hartogs domains, Ω if Ω is pseudoconvex and − log φ, − log ψ are psh.
BERGMAN KERNELS
5
Proof. Arguing as in [Lig], Proposition 0 shows that e t) = K(z;
X
β
α
Kgαβ (z1 , t1 ) z2β t2 z3α t3
α,β
where the summation is over all multiindices α ∈ N, β ∈ NM , N = {0, 1, 2, . . . }, and gαβ (z1 ) =
Z
{(z2 , z3 ) :
|z2 |2 ψ(z1 )
+
|z3 |2 φ(z1 )
< 1}
|z2β |2 |z3α |2 dz2 dz3
M +1
= Since
P
|α|=k
π α! β! φ(z1 )|α|+1 ψ(z1 )|β|+M . (|α| + |β| + M + 1)!
xα y α /α! = hx, yik /k!, the required assertion follows.
¤
The construction similar to (8) was first used by Forelli and Rudin [For], [FR], [Rud]; for other applications, see [Lig], [KLR] and the references therein. Let us recall the asymptotic formula for the boundary behaviour of the Bergman kernel due to Fefferman [Fef] and Boutet de Monvel–Sj¨ostrand [BS]. Let Ω be a strictly pseudoconvex domain in Cn with smooth boundary and −φ a C ∞ defining function for Ω, i.e. Ω = {z : φ(z) > 0}, φ is C ∞ in a neighborhood of Ω, ∇φ 6= 0 on ∂Ω, and the Levi matrix (−∂ 2 φ/∂zj ∂z k ) is positive definite on the complex tangent space (the last condition is equivalent to the Monge-Amp´ere matrix in (13) below having n positive and 1 negative eigenvalue, for any z ∈ ∂Ω). Then there exist functions a(x, y), b(x, y), φ(x, y) ∈ C ∞ (Cn × Cn ) such that (a) a(x, y), b(x, y), φ(x, y) are almost-analytic in x, y in the sense that ∂φ(x, y)/∂x and ∂φ(x, y)/∂y have a zero of infinite order at x = y, and similarly for a(x, y) and b(x, y); (b) φ(x, x) = φ(x); (c) for x ∈ ∂Ω, n! (12) a(x, x) = n J[φ](x) > 0 π where J[φ] is the Monge-Amp´ere determinant ·
−φ J[φ] = − det −∂φ/∂zj
−∂φ/∂z k −∂ 2 φ/∂zj ∂z k
¸
(13)
whose positivity follows from the strong pseudoconvexity of ∂Ω; (d) the Bergman kernel of Ω is given by the formula K(x, y) =
a(x, y) + b(x, y) log φ(x, y) φ(x, y)n+1
(14)
for (x, y) ∈ Ω² = {|x − y| < ², dist(x, ∂Ω) < ²}, where ² > 0 is sufficiently small; (e) outside any Ω² the Bergman kernel is C ∞ up to the boundary of Ω × Ω; (f) if the boundary ∂Ω is even real-analytic, then the functions a(x, y), b(x, y) and φ(x, y) can in fact be chosen to be holomorphic in x, y in a neighborhood of the boundary diagonal {(x, x); x ∈ ∂Ω} in Cn , and outside any Ω² the Bergman kernel is holomorphic in x, y in a neighborhood of Ω × Ω.
The original proofs in [Fef] and [BS] deal only with (a)–(e); part (f) is due to Kashiwara [Kas] and Bell [Be1].
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ˇ MIROSLAV ENGLIS
Observe that if φ0 (x, y) is another function satisfying (a) and (b), then h = (φ0 /φ) − 1 vanishes at x = y to an infinite order; thus (14) remains in force with φ0 and a0 = (1 + h)n+1 a + φ0n+1 b log(1 + h) in the place of φ and a. It follows that even for any function φ(x, y) satisfying (a) and (b) there exist a(x, y), b(x, y) such that the conclusions (a)–(d) hold. This allows us to work with a convenient φ(x, y) in concrete situations later on: for instance, if φ(x) is of the form |x1 |2 + (a function of x2 , . . . , xn ), we can take φ(x, y) = x1 y 1 + (a function of x2 , . . . , xn , y2 , . . . , yn ). We will find convenient the following two (probably well-known) “localization lemmas”, which can be used to obtain a local variant of Fefferman’s theorem (see [E6]). Lemma 5. Let Ω1 ⊂ Ω be two bounded pseudoconvex domains and U a neighborhood of a point x0 ∈ ∂Ω such that U ∩ ∂Ω1 = U ∩ ∂Ω and the piece of common boundary U ∩ ∂Ω is smooth and strictly pseudoconvex. Then the difference KΩ1 (x, y) − KΩ (x, y) is C ∞ on (U ∩ Ω1 ) × (U ∩ Ω1 ). Lemma 6. Let Ω be a pseudoconvex domain (possibly unbounded ) and x0 ∈ ∂Ω a strictly pseudoconvex point of its boundary. Then there exists a bounded strictly pseudoconvex domain Ω1 ⊂ Ω such that ∂Ω and ∂Ω1 coincide in a neighborhood of x0 . Further, if x0 is a smooth boundary point, then Ω1 can be chosen to be smoothly bounded. Proof of Lemma 5. For Ω, Ω1 smoothly bounded and strictly pseudoconvex, this is the content of Lemma 1 on p. 6 in [Fef]. The local version given here follows in the same way by J.J. Kohn’s local regularity theorems for the ∂-operator and subelliptic estimates at x 0 [Ko, Theorems 1.13 and 1.16] by the argument as on p. 469 in [Be2], cf. in particular the formula (2.1) there. ¤ Proof of Lemma 6. Let u be a defining function for Ω = {u < 0} strictly-psh in a neighborhood B(x0 , δ) of x0 (see e.g. [Krn], Proposition 3.2.1). Choose a C ∞ function θ : [0, 1) → R+ such that θ ≡ 0 on [0, 1/2], θ00 ≥ 0 on [1/2, 1) and θ(1−) = +∞. Set Ω1 = {x : u(x) + θ(|x − x0 |2 /δ 2 ) < 0}. Then Ω1 ⊂ Ω ∩ B(x0 , δ), ∂Ω1 coincides with ∂Ω in B(x0 , δ/2), and as θ 00 ≥ 0, θ(|x − x0 |2 /δ 2 ) is psh, so Ω1 is strictly pseudoconvex. Finally, if u is C ∞ in B(x0 , δ), then Ω1 is smoothly bounded. ¤ It turns out that the boundedness hypothesis on Ω in Lemma 5 is, in fact, unnecessary: see [E7], Section 4, where also the full details of the proof can be found. The conclusion of the lemma fails, however, if U ∩ ∂Ω = U ∩ ∂Ω1 is only assumed to be weakly pseudoconvex: for instance, take Ω = {max(|z1 |, |z2 |) < 1} ⊂ C2 , Ω1 = {max(|z1 |, 2|z2 |) < 1}, and x0 = (1, 0). Similarly, the hypothesis that Ω be pseudoconvex cannot be dispensed with: an example is Ω1 = {z ∈ C2 : |z| < 2}, Ω = Ω1 ∪ {|z1 | < 3, 1 < |z2 | < 3}, x0 = (2, 0). On the other hand, the hypothesis that Ω1 be pseudoconvex is not needed in the proof and can be omitted (but we won’t have any use for this refinement in the sequel). A similar construction as in Lemma 6 was used by Bell [Be2] (cf. also the references therein). Remark. There is also a “local version” of part (f) of Fefferman’s theorem: namely, if Ω is bounded pseudoconvex and z ∈ ∂Ω is a strictly pseudoconvex and real-analytic boundary point (i.e. ∂Ω is a C ω -submanifold of Cn in some neighbourhood of z), then there exists a neighbourhood U of z and functions a(x, y), b(x, y) and φ(x, y) on U × U , holomorphic in x, y, such that −φ(x, x) is a local defining function for Ω on U and (12) and (14) hold. See [Kan], §9, in particular the Theorem on p. 94. (The author is obliged to M. Kashiwara and Gen Komatsu for this information.) 2. Weighted Bergman kernels We now use Fefferman’s asymptotic expansion together with Proposition 4 to determine
BERGMAN KERNELS
7
the asymptotics of Kφm ψM (z, z) as z and M are fixed and m → ∞. Let us start with two technical lemmas. Let φ(x, y) be a function in C ∞ (Ω × Ω) almost-analytic in x, y on the diagonal and such that φ(x, y) = φ(y, x), φ(x, x) =: φ(x) > 0, and − log φ(x) is strictly psh at some point x 0 . The last condition implies that there exists c > 0 and a small ball U centered at x0 such that φ(x)φ(y) ≤ 1 − c|x − y|2 ∀x, y ∈ U. (15) |φ(x, y)|2 Denote
D = {(x, y, τ ) ∈ U × U × C : |τ |2 < Then |x − y|2
0 such that all points of ∂ Ω 2 2 with |z1 − x0 | + |z2 | < δ are strictly pseudoconvex. We may assume that δ < dist(x0 , ∂Ω). Choose a C ∞ function θ on [0, 1) such that θ 00 ≥ 0, θ ≡ 0 on [0, 2/3] and θ(t) = − log(1 − t) on (3/4, 1), and define u0 (z) = |z3 |2 + g(z1 )|z2 |2 − exp[−θ((|z1 − x0 |2 + |z2 |2 )/δ)]φ(z1 ) e 0 = {u0 < 0}. A similar argument as in the proof of Lemma 6 shows that Ω e 0 is and Ω e and that Ω e and Ω e 0 coincide in a neighborhood of a strictly pseudoconvex domain ⊂ Ω 2 2 Π = {z : |z1 − x0 | + |z2 | ≤ δ/2}. Thus the conclusions (a)–(e) of Fefferman’s theorem are e × Π ∩ Ω. e It follows that e 0 , and also by Lemma 5 K e 0 − K e is C ∞ on Π ∩ Ω applicable to Ω Ω
e is C ∞ on Π ∩ Ω e ×Π∩Ω e minus the boundary diagonal S = {(z, z) : z ∈ Π ∩ ∂ Ω}, e while K near S it is of the form e t) = K(z,
a(z, t) + b(z, t) log[−u(z, t)], [−u(z, t)]N +M +2
(20)
with some almost-analytic C ∞ functions a, b and u satisfying u(z, z) = u(z) and a(z, z) =
(N + M + 1)! J[−u](z), π N +M +1
e z ∈ Π ∩ ∂ Ω.
e Let us now specialize (Here and in (20) we have used the fact that u0 = u on Π ∩ Ω.) e to z, t ∈ Π ∩ Ω with z2 = t2 = 0, i.e. to points of the form (x, 0, z3 ) with (x, z3 ) ∈ D := {|x − x0 |2 < δ/2, |z3 |2 < φ(x)} ⊂ CN +1 . It follows that the function e g(x, z3 ; y, t3 ) = K(x, 0, z3 ; y, 0, t3 )
ˇ MIROSLAV ENGLIS
10
∞ is p C on D × D minus the boundary diagonal ∆ = {(x, z3 ; y, t3 ) : x = y, z3 = t3 , |z3 | = φ(x)}, while near ∆ it is of the form
g(x, z3 ; y, t3 ) =
a0 (x, z3 ; y, t3 ) + b0 (x, z3 ; y, t3 ) log[φ(x, y) − z3 t3 ] [φ(x, y) − z3 t3 ]N +M +2
where a0 , b0 — the restrictions of a, b to z2 = t2 = 0 — are holomorphic in x, y, z3 , t3 on ∆. Switching to the variable z3 t 3 τ= φ(x, y) we thus see that g(x, z3 ; y, t3 ) = F (x, y, τ ) for a function F on the domain D1 = {(x, y, τ ) : x, y ∈ U, |τ |2
0 and an integer m0 such that sup y∈Ω\U
·
|Kφm ψM (x, y)|2 Kφm ψM (x, x)Kφm ψM (y, y)
¸1/m
≤1−δ
∀m ≥ m0 .
(In view of (5), one may replace 1/Kφm ψM (x, x) by φm (x), and similarly for y.) Obviously, this problem is related to the problem of understanding the asymptotic behaviour of |Kφm ψM (x, y)|1/m as m → ∞ away from the diagonal x = y, mentioned in Remark (6.) in Section 2 above. 4. The Berezin star-product As this seems not to be done in full anywhere in the literature, we now describe briefly how to construct a star product from Theorem 2. For the sake of brevity, denote h = 1/m and Ah = A2 (Ω, φm ψ M ), Kh = Kφm ψM , Bh = Bφm ψM (m = 1, 2, . . . ). For T ∈ Bh , the
BERGMAN KERNELS
17
Banach algebra of all bounded linear operators on Ah , define the function T (x, y) — the covariant symbol of T — on Ω × Ω by T (x, y) =
hT Kh (·, y), Kh (·, x)iAh , Kh (x, y)
x, y ∈ Ω.
(32)
Then T is representable as the integral operator on Ω with kernel T (x, y)Kh (x, y) with respect to the measure φ(x)m ψ(x)M dx, and for T1 , T2 ∈ Bh , (T1 T2 )(x, y) =
Z
T1 (x, z)T2 (z, y) Ω
Kh (x, z)Kh (z, y) φ(z)m ψ(z)M dz. Kh (x, y)
(33)
Set T˜(x) := T (x, x) and let Ah be the vector space of all functions of the form T˜(x) with T ∈ Bh . Since the correspondence T 7→ T˜ is one-to-one, one can transfer the algebraic operations and operator involution from Bh into Ah , which endows Ah with the structure of an involutive complex algebra, with complex conjugation as the involution, and with an (associative, but not commutative) product which we denote by ∗h . See [Ber] or [E2, p. 415] for details. Let A be the algebraic direct sum of all Ah , h = 1, 1/2, . . . , and let A˜ ⊂ A be the subset of all elements of the form f (h; x) = T˜h (x), where Th ∈ Bh for each h, for which there exist functions fj (x, y) on Ω × Ω (j = 1, 2, . . . ), holomorphic in x and y, such that for each N ∈N Th (x, y) = f0 (x, y) + hf1 (x, y) + · · · + hN fN (x, y) + hN +1 FN (h; x, y) where
sup |fj (x, y)| < ∞, x
sup |fj (y, x)| < ∞, x
sup |FN (h; x, y)| < ∞, x,h
∀y ∈ Ω, j = 0, 1, . . . , N, h = 1, 1/2, 1/3, . . . .
(34)
(35)
Clearly A˜ is closed under addition, scalar multiplication and complex conjugation. We denote the product in A by ∗. ˜ Then for each z ∈ Ω, we have an asymptotic expansion as Theorem 12. Let f, g ∈ A. h→0 X Ck (Rfi , Rgj ) hi+j+k (36) (f ∗ g)(h; z) = i,j,k≥0
where R : C ∞ (Ω × Ω) → C ∞ (Ω) is the operator of restriction to the diagonal x = y, and Ck are bilinear differential operators of the form Ck (u, v) =
X
X
αβ T(i,j,k) u/α v/β ,
0≤i,j≤k |α|=i |β|=j
u, v ∈ C ∞ (Ω),
(37)
where α, β are multiindices, the slash stands for covariant differentiation with respect to αβ are, for each fixed i, j, k ≥ 0, i + j ≤ k, tensor fields on Ω, the metric (2), and T(i,j,k) symmetric in the entries of α and of β, of the same form as in Theorem 11. In particular, C0 (u, v) = uv,
C1 (u, v) =
X j,k
g jk u/j v/k ,
C2 (u, v) =
1 X jk lm g g u/jl v/km , 2 j,k,l,m
(38)
ˇ MIROSLAV ENGLIS
18
where as before g jk is the inverse matrix to gjk = −∂∂ log φ. Proof. Denote u(h; x) = f (h; z, x)g(h; x, z), where f (h; z, x) = Th (z, x) and similarly for g. Owing to (33) we then have (f ∗ g)(h; z) =
Z
u(h; x) Ω
|Kh (x, z)|2 φ(x)1/h ψ(x)M dx = (Bh u(h; ·))(z). Kh (z, z)
In view of (34), for each N ∈ N, u(h; x) = u0 (x) + hu1 (x) + · · · + hN uN (x) + hN +1 UN (h; x), where uk (x) =
X
fi (z, x)gj (x, z),
k = 0, 1, . . . , N,
i+j=k
UN (h; x) =
X
hi+j−N −1 fi (z, x)gj (x, z) + f0 (z, x)GN (h; x, z) + FN (h; z, x)g0 (x, z).
i+j>N i,j≤N
The hypotheses on A˜ ensure that u1 , . . . , uN and UN (h; ·) are bounded on Ω, the last uniformly in h. By Theorem 2, we therefore get (Bh u(h; ·))(z) = =
N X
k=0 N X
hk Bh uk (z) + hN +1 O(kUN (h; ·)k∞ ) hk
k=0
=
X
∞ X l=0
hl+k Ql uk (z) + O(hN +1 )
l,k≥0 l+k≤N
=
X
hl Ql uk (z) + hN +1 O(1)
l,i,j≥0 l+i+j≤N
¯ hl+i+j Ql [fi (z, x)gj (x, z)]¯x=z + O(hN +1 ),
where in the last expression the operators Ql act on the x variable. As N is arbitrary, we thus obtain an asymptotic expansion X (f ∗ g)(h; z) = hi+j+l Γl (fi , gj )(z), i,j,l≥0
with
¯ Γk (fi , gj )(z) := Qk [fi (z, ·)gj (·, z)]¯z .
We have to show that this coincides, in fact, with Ck (Rfi , Rgj )(z) for certain differential operators Ck of the form (37). Consider thus, quite generally, two functions f, g on Ω × Ω holomorphic in the first variable and anti-holomorphic in the second, fix a point z ∈ Ω, and let us show that Qk [f (z, ·)g(·, z)], evaluated at z, comes as a sum k X X
i,j=0 |α|=i |β|=j
αβ S(i,j) (Rf )/α (Rg)/β
evaluated at z,
BERGMAN KERNELS
19
αβ where S(i,j) , i, j = 0, . . . , k, are tensor fields on Ω of the form described in the theorem. For brevity, denote F = f (z, ·) and G = g(·, z); thus G is holomorphic and F is antiholomorphic on Ω. In view of Theorem 11, it is sufficient to show that each covariant derivative of the product F G, evaluated at z, is a sum of expressions of the form
¯ S αβ F/α G/β ¯z ,
(39)
with S αβ being components of tensor fields involving only the metric tensor, the curvature tensor, and the latter’s covariant derivatives; the assertion will then follow since, as G is holomorphic, G/β (z) = (Rg)/β (z), and similarly for F/α . Now by the product rule, (F G)/... comes as a sum of products F/... G/... of covariant derivatives of F and G. Using the Ricci formula, which exhibits the ... ... ... )/pq − (T... )/qp of two successive covariant differentiations of a tensor T... commutator (T... ... as a sum of contractions of T... against the curvature tensor (see e.g. [E5], formula (2.17)), we further reduce to showing that F/γα G/δβ |z is of the form (39), for any multiindices α, β, γ and δ. However, in view of the holomorphy of F and G, we have F/l = G/l = 0 for any l, hence the last expression vanishes unless |γ| = |δ| = 0. Thus the required assertion follows. ˜ we have Γ0 (f, g)(z) = F (z)G(z) and Finally, recalling that Q0 = I and Q1 = ∆, P jk Γ1 (f, g)(z) = j,k g F/j G/k |z , hence in terms of u := Rf and v := Rg we get C0 (u, v) = P uv and C1 (u, v) = j,k g jk u/j v/k . The formula for C2 follows similarly from the formula for Q2 in Theorem 11. This completes the proof. ¤ Consider now the ring (over the field of complex numbers) C ∞ (Ω)[[h]] of all formal power series ∞ X f (h; x) = fj (x)hj , fj ∈ C ∞ (Ω) ∀j, j=0
with the usual algebraic operations, and endow it with the C[[h]]-linear product ∗ given by ¡X i≥0
X ¢ ¡X ¢ f i hi ∗ gj hj := Ck (fi , gj ) hi+j+k j≥0
(40)
i,j,k≥0
with the operators Ck from (36). Theorem 13. The product ∗ is a star-product, i.e. it is associative and satisfies (f ∗ g)0 = f0 g0 (the pointwise product in C ∞ (Ω)), (41) ³f ∗ g − g ∗ f ´ = i{f, g} (the Poisson bracket with respect to the metric (2)). h 0 ˜ this is immediate from the preceding theorem; as the term at hm Proof. For f (h; x) ∈ A, P α in (40) — namely, i+j+k=m Ck (fi , gj ) — involves only ∂ fi and ∂ β gj with i, j, |α|, |β| ≤ m, it therefore suffices to check that for each f ∈ C ∞ (Ω)[[h]], z ∈ Ω and N ∈ N, there exists f 0 ∈ A˜ such that α
α
∂ fj0 (z) = ∂ fj (z)
∀ j, |α| ≤ N.
(42)
ˇ MIROSLAV ENGLIS
20
Indeed, one then has Ck (f, g)(z) = Ck (f 0 , g 0 )(z) ∀k = 0, 1, . . . , N , so (f ∗ g)(h; z) = (f 0 ∗ g 0 )(h; z) mod hN +1 , and the validity of (41) for f, g follows from its validity for f 0 , g 0 ; similarly for the associativity. We will search for such an f 0 ∈ A˜ in the form f 0 (h; z) =
N X
pj (x) hj
j=0
with pj bounded holomorphic functions on Ω. Then fj0 (x, y) = pj (y), so (34) and (35) α hold, and (42) is equivalent to ∂ α pj (z) = ∂ fj (z) for all |α|, j ≤ N . Clearly, such functions P 1 α ∂ f j (z)xα ). It remains to pj exist (for instance, take the polynomials pj (x) = |α|≤N α! 0 show that f (h; x) = T˜h (x) for some family of operators Th ∈ Bh (h = 1, 1/2, . . . ). However, observe that for T the operator on Ah of multiplication by a bounded analytic function p, one has from (38) hpKh (·, x), Kh (·, x)i T˜(x) = Kh (x, x) p(x)Kh (x, x) = p(x), = Kh (x, x) by the reproducing property of Kh . It follows that Tf∗ = p. Further, T is bounded if p is. Thus the operators N X ¢∗ ¡ hj pj on Ah Th = multiplication by j=0
do the job we need.
¤
Remark. It follows from (37) that the Berezin star product is an example of a “deformation quantization with separation of variables” in the sense of Karabegov [Kar], i.e. f ∗ g = f g (pointwise product) if f or g is analytic. It is also easy to show that f ∗ g = g ∗ f , i.e. ∗ preserves complex conjugation. 5. Berezin-Toeplitz quantization In this section we restrict our attention to the case of Ω smoothly bounded and strictly pseudoconvex, −φ a smooth defining function for Ω (i.e. φ is C ∞ in a neighbourhood of Ω, φ > 0 on Ω, φ < 0 on the complement of Ω, and φ = 0, k∇φk 6= 0 on ∂Ω) such that − log φ is strictly psh, and to the weights φk χ, χ = det[−∂∂ log φ]. For the proof of part (ii) of Theorem 3, we need to use the theory of Boutet de Monvel-Guillemin Toeplitz operators [BG], in the same way as in [BMS] and [Sch] for compact manifolds; we include a sketch of the proof below for convenience. The point we wish to make here is that the part (i) of Theorem 3 is an easy consequence (not using the Boutet de Monvel-Guillemin theory in any way) of Theorem 2 and the following elementary observation. Lemma 14. Let Ω be a strictly pseudoconvex domain in CN and −φ a smooth defining function for Ω such that − log φ is strictly psh. Then for any positive continuous function g on Ω there exists an integer d > 0 such that − log φ − d1 log g is strictly psh on Ω. Proof. Observe that · ¸ · −φ −φk 1 =φ −φj −φjk − (φ/d)(log g)jk φj /φ
0 1
¸·
−1 0
0 (− log φ − (1/d) log g)jk
¸·
1 0
φk /φ 1
¸
BERGMAN KERNELS
21
where for brevity we have used the subscripts j, k to denote differentiations by z j , z k . Thus − log φ − (1/d) log g is strictly psh at z ∈ Ω if and only if the square matrix ¸ · · ¸ 1 1 0 0 −φ −φk ≡ A(z) − B(z) − −φj −φjk d 0 (log g)jk φ d has 1 negative and N positive eigenvalues. However, in view of the hypotheses on φ and g, both matrix-valued functions A(z) and B(z) are continuous on Ω and A(z) has the required signature (N, 0, 1) for each z ∈ Ω; thus supz∈Ω kB(z)k = b < ∞ and K = {A(z), z ∈ Ω} is a compact subset of the open set N of all Hermitian matrices of signature (N, 0, 1). Taking d so large that b/d < dist(K, ∂N ), the assertion follows. ¤ Proof of part (i) of Theorem 3. Let J[φ] the Monge-Amp´ere determinant (13). As χ = J[φ]/φN +1 , the strict plurisubharmonicity of − log φ implies that J[φ] > 0 on Ω; on the other hand, the fact that φ is a smooth defining function of the strictly pseudoconvex domain Ω implies that J[φ] ∈ C ∞ (Ω) and J[φ] > 0 on ∂Ω. Thus the preceding lemma applies to g = J[φ]. Let d be the constant provided by the lemma, and apply Theorem 2 to Ω, φ, M = d, and ψ = J[φ]1/d φ (thus φm ψ M = φm+d+N +1 χ). This gives Bφm χ f (x) = f (x) + O(m−1 )
as m → ∞,
(43)
for any f ∈ L∞ (Ω) ∩ C ∞ (Ω), uniformly for x in compact subsets. On the other hand, let kx(m) =
Kφm χ (·, x) Kφm χ (x, x)1/2
be the unit vector in the direction of Kφm χ (·, x) (the coherent state). Then by the definition of the Toeplitz operator and of the Berezin transform, (φm χ) (m) (m) kx , kx iL2 (Ω,φm χ) .
Bφm χ f (x) = hTf It follows that
(φm χ)
|Bφm χ f (x)| ≤ kTf
(φm χ)
k kkx(m) k2 = kTf m
(44)
k.
Combining this with (43), we see that lim inf m→∞ kT (φ χ) k ≥ kf k∞ . On the other hand, for any weight ρ, (ρ) kTf hk = kPρ (f h)k ≤ kf hk ≤ kf k∞ khk, (ρ)
whence kTf k ≤ kf k∞ . Thus the desired assertion follows.
¤
Proof of part (ii) of Theorem 3. Let us now consider the domain (8) with M = 0: e = {(z, w) ∈ Ω × C : |w|2 < φ(z)}. Ω
(45)
e is a smoothly bounded, strictly pseudoconvex The hypotheses then mean precisely that Ω n e domain in C , n = N + 1, and r(z, w) := |w|2 − φ(z) is a smooth defining function for Ω. e is a compact manifold, and we denote by α the restriction to X of Its boundary X = ∂ Ω the 1-form Im ∂r = (∂r − ∂r)/2i. Then α is a contact form, i.e. α ∧ (dα)n−1 determines a nonvanishing volume form on X. Following [BG], let L2 be the Lebesgue space on X with respect to this measure, and H 2 the closure in L2 of the subspace of all functions e For each (z, w) ∈ Ω, e the evaluation in C ∞ (X) which extend holomorphically into Ω. functional f 7→ f (z, w) turns out to be continuous on H 2 , hence is given by the scalar product with a certain element k(z,w) ∈ H 2 . The function e Szeg¨o (z1 , w1 ; z2 , w2 ) := hk(z ,w ) , k(z ,w ) iH 2 K 2 2 1 1
ˇ MIROSLAV ENGLIS
22
e ×Ω e minus the boundary e ×Ω e is called the Szeg¨o kernel; it extends smoothly to all of Ω on Ω e Szeg¨o on X is the Szeg¨o projection π, i.e. diagonal, and the integral operator determined by K the orthogonal projection in L2 onto H 2 . For F ∈ L∞ (X), the (global) Toeplitz operator TF is the operator on H 2 defined by TF f = π(F f ). Let H(m) ⊂ H 2 (m = 0, 1, 2, . . . ) be the subspace of all functions of the form f (z, w) = f (z)w m
((z, w) ∈ X).
A routine computation shows that (see e.g. [Ran], p. 291) α ∧ (dα)n−1 = (n − 1)!
J[r] dS, k∂rk
dS being the surface element (i.e. (2n − 1)-dimensional Hausdorff measure) on X. Introducing the coordinates p (z ∈ Ω, θ ∈ [0, 2π]) (z, w) = (z, eiθ φ(z)) p p on X, we have dS = φ + k∂φk2 dz dθ; and as φ + k∂φk2 = k∂rk and J[r] = J[φ] = φN +1 χ, we thus have Z
X
|f (z)wm |2 α ∧ (dα)n−1 = (n − 1)!
Z Z Ω
= 2π (n − 1)!
Z
2π
|f (z)|2 φ(z)m J[r] dz dθ
0
Ω
|f |2 φm+N +1 χ dz.
It follows that H(m) is isometrically (up to the immaterial factor (n − 1)!2π) isomorphic to the Bergman space A2 (Ω, φm+N +1 χ). As the subspaces H(m) are pairwise orthogonal and span all of H 2 , we therefore arrive at the following analogue of Proposition 4 (cf. [Lig], [BFS], [KLR]): e Szeg¨o (z1 , w1 ; z2 , w2 ) = K
∞ 1 X K m+N +1 χ (z1 , z2 ) · (w1 w2 )m . 2πN ! m=0 φ
(46)
For f ∈ L∞ (Ω), let fˆ ∈ L∞ (X) be defined by fˆ(z, w) = f (z)
(47)
and consider the Toeplitz operator Tfˆ. Then all the subspaces H(m) are invariant under Tfˆ, and the restriction of Tfˆ to H(m) is (up to the unitary equivalence above) precisely the (φm+N +1 χ)
Toeplitz operator Tf
. That is, Tfˆ '
∞ M
T (φ
m
χ)
.
m=N +1
Finally, a generalized Toeplitz operator on H 2 is the operator TP given by TP = πP π
(48)
BERGMAN KERNELS
23
where P is a pseudodifferential operator on the compact manifold X. The order ord(TP ) and the symbol σ(TP ) of TP are defined as the order of P and the restriction of the symbol σ(P ) of P to the submanifold Σ := {(x, ξ) : ξ = tαx , t > 0} of the cotangent bundle of X, respectively. It was then shown in [BG] that these two definitions are unambiguous, and (P1) the generalized Toeplitz operators form an algebra under composition, (P2) ord(T1 T2 ) = ord(T1 ) + ord(T2 ), σ(T1 T2 ) = σ(T1 )σ(T2 ), (P3) σ([T1 , T2 ]) = {σ(T1 ), σ(T2 )} (the Poisson bracket), (P4) if ord(T ) = 0, then T is a bounded operator on H 2 , and (P5) if ord(T1 ) = ord(T2 ) = k and σ(T1 ) = σ(T2 ), then ord(T1 − T2 ) ≤ k − 1. Let T be the subalgebra of all generalized Toeplitz operators which commute with the circle action Uθ : f (z, w) 7→ f (z, eiθ w) ((z, w) ∈ X, θ ∈ R)
on H 2 ; clearly, the operators Tfˆ with fˆ as in (47) belong to T . Let D be the infinitesimal generator of the semigroup Uθ ; one has Dh = imh
∀h ∈ H(m) ,
(49)
and D = T∂/∂θ is a generalized Toeplitz operator of order 1. Using (P1)–(P5) it was then shown in [BMS] (see also [Sch] and [Gu]) that if T ∈ T is of order 0, then T = Tfˆ + D−1 R for some (uniquely determined) f ∈ L∞ (Ω) and R ∈ T of order 0. Repeated application of this formula reveals that, for each k ≥ 0, T =
k X
D−j Tfˆj + D−k−1 Rk ,
j=0
with fj ∈ L∞ (Ω) and Rk ∈ T of order 0. Invoking the boundedness of zeroth order operators, it follows that k ´ ³ X D−j Tfˆj Dk+1 T − j=0
is a bounded operator on H 2 , that is, by (49) and (48), k ° ° X ° (φm χ) ° m−j Tfj °T |H(m) − ° = O(m−k−1 ). j=0
Taking for T the product TfˆTgˆ with f, g ∈ L∞ (Ω) ∩ C ∞ (Ω), we obtain (10). Finally, the assertions concerning C0 and C1 follow from the properties (P2) and (P3) of the symbol; see the references mentioned above for the details. ¤ Remarks. (1.) The same comment as in Remark (2.) at the end of Section 3 applies here as well: namely, the machinery of [BG] makes heavy use of the compactness of the e so the last proof does not easily generalize to unbounded domains Ω. manifold X = ∂ Ω, The problem is that, first, zeroth order pseudodifferential operators no longer need to be
24
ˇ MIROSLAV ENGLIS
bounded, and second, that the various smoothing operators which arise as error terms need not be bounded either. The former can be coped with by means of the CalderonVaillancourt theorem (by replacing L∞ (Ω) ∩ C ∞ (Ω) by the space of all functions on Ω all of whose derivatives up to a certain order are continuous and uniformly bounded); the latter has so far been dealt with successfully only in the case of Ω = Cn with the Euclidean metric (hence with the potential − log φ(z) = |z|2 , giving rise to the Segal-Bargmann spaces 2 of entire functions square-integrable with respect to the Gaussian measures e−m|z| dz), see [Bw]. e be smooth prevents us from dealing with the case Similarly, the requirement that X = ∂ Ω of φ not being a defining function (or a power of one), thus excluding the most interesting cases such as the Bergman and the Cheng-Yau (K¨ahler-Einstein) metric. Apparently, this difficulty will probably be even harder to overcome than the previous one. We remark that, on the other hand, to some extent the above approach can be generalized to some non-K¨ahler compact manifolds as well; see [BU]. (2.) On a purely formal level, the Berezin-Toeplitz quantization has been carried out on any K¨ahler manifold by Reshetikhin and Takhtajan [RT]; however, it is not clear whether these arguments (involving a formal application of the stationary-phase method) can be made rigorous. Cf. also the similar formal expansion by Cornalba and Taylor [CT]. For bounded symmetric domains in Cn (Hermitian symmetric spaces) with the invariant metric, a similar line of attack — namely, a study of the asymptotics of the weighted Bergman projections Pφm : L2 (Ω, φm ) → A2 (Ω, φm ) as m tends to infinity — has been initiated by Arazy and Ørsted [AO]. (3.) The proofs in Sections 2 and 3 feature an interplay between two subjects: the ∂techniques due to Kohn, Catlin, and others, on the one hand, and the theory of Fourier integral operators on the other hand. The latter is more powerful for the applications we have — in particular, the Boutet de Monvel-Guillemin theory of generalized Toeplitz operators in this section relies on it completely, and it is also hidden in Boutet de Monvel and Sj¨ostrand’s proof of the Fefferman theorem which was our departing point for most of the developments in Section 2. On the other hand, the former lend themselves much more easily to localization — they work in a neighbourhood of a smooth strictly pseudoconvex point even if the boundary is bad (nonsmooth, weakly pseudoconvex, even unbounded) away from it. From this point of view, it would be very desirable to have at least the following “noncompact version” of the Boutet de Monvel-Guillemin theory. Let Ω be a bounded pseudoconvex domain in CN and φ a function on Ω which is positive and C ∞ on Ω, vanishes at ∂Ω (but it is not assumed to be C ∞ up to the boundary), e as in (45), and let and such that − log φ is strictly psh on Ω. Consider the domain Ω 2 e X = {(z, w) ∈ Ω × C : |w| = φ(z)} be the part of ∂ Ω which lies over Ω. Then X is a noncompact strictly pseudoconvex CR-manifold; the form α ∧ (dα)N is, as before, a nonvanishing surface element on X, and we define H 2 as the closure in L2 of the subspace of functions that extend holomorphically to Ω × C. The corresponding Szeg¨o kernel will then still satisfy (46), and one can again introduce the global Toeplitz operators TF , F ∈ L∞ (X), and Tfˆ, f ∈ L∞ (Ω), and the generalized Toeplitz operators TP , with P a pseudodifferential operator on X which we now assume in addition to be bounded on L2 . Let again T be the set of all generalized Toeplitz operators commuting with the circle action Uθ . By the Calderon-Vaillancourt theorem, Tfˆ ∈ T if f ∈ BC ∞ (Ω), the subspace in C ∞ (Ω) of functions all partial derivatives of which belong to L∞ (Ω). Problem. Do the properties (P1)–(P5) prevail in this setting? As has already been mentioned above, this was settled in the affirmative in the case of Ω = CN and φ(z) = |z|2 in [Bw].
BERGMAN KERNELS
25
(4.) We conclude by exhibiting how the Berezin-Toeplitz star product implicit in Theorem 3, namely, ∞ X hj Cj (f, g), f, g ∈ C ∞ (Ω)[[h]], (50) f ∗ g := j=0
with Cj given by (10) and extended C[[h]]-linearly from C ∞ (Ω) to C ∞ (Ω)[[h]], is related to the Berezin star product constructed in Section 4. Let us for the moment write ∗BT for (h) (φm χ) the former and ∗B for the latter, and as in Section 4 set again h = 1/m and Tf = Tf , (h)
(m)
kx = kx , etc. For f, g ∈ C ∞ (Ω), we thus have (Bh f ∗B Bh g)(x) =
∞ X
CkB (Bh f, Bh g)(x) hk ,
k=0
where CkB are the operators Ck from (40), extended to C ∞ (Ω)[[h]] by C[[h]]-linearity, and we regard ∞ X Qj f h j (51) Bh f = j=0
∞
as an element of C (Ω)[[h]]. On the other hand, by the definition of ∗B (cf. (33), (44) and (36)) we have (h) (Bh f ∗B Bh g)(x) = hTf Tg(h) kx(h) , kx(h) iAh . But in view of (44) and (50), the last expression is equal to Bh
∞ X
CkBT (f, g) hk
k=0
where
CkBT
are the operators Ck from (50). Thus the two star products are related by ∞ X
Bh CkBT (f, g) hk =
∞ X
CkB (Bh f, Bh g) hk .
(52)
k=0
k=0
As Q0 = I, it follows from (51) that Bh , regarded as a formal power series in h with differential operators as coefficients, has an inverse Bh−1 ; one can thus rewrite (52) as ∞ X
CkBT (f, g) hk = Bh−1
k=0
∞ X
CkB (Bh f, Bh g) hk ,
k=0 ∞
or, regarding Bh as a linear operator on C (Ω)[[h]] (i.e. as a formal differential operator), Bh (f ∗BT g) = Bh f ∗B Bh g.
(53)
Thus the two star products differ only by a “change of ordering” f ↔ Bh f . In the terminology of [Kar], (53) says that the star products ∗B and ∗BT are duals of one another. (5.) Using the formulas for C1B and C2B in Theorem 12, it is possible to derive from (53) analogous formulas for the operators C1BT and C2BT : X C1BT (u, v) = − g jk u/k v/j , j,k
C2BT (u, v) =
X 1 X jk lm Ricjk u/k v/j . g g u/km v/jl − 2 j,k,l,m
j,k
(6.) Again, the Berezin-Toeplitz star product is an example of a deformation quantization with the separation of variables, but this time with the role of holomorphic and anti-holomorphic variables swapped (i.e. f ∗BT g = f g if g is holomorphic or f is antiholomorphic); see [KS] for the identification of ∗BT within this scheme.
26
ˇ MIROSLAV ENGLIS
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[email protected] WWW: http://www.math.cas.cz/~englis/
