On canonical homotopy operators for [delta] in Fock type spaces in Cn

Publ. Mat. 45 (2001), 223–233 ON CANONICAL HOMOTOPY OPERATORS FOR ¯ ∂IN FOCK TYPE SPACES IN Cn J¨ orgen Boo Abstract We show that a certain solution operator for ¯ ∂in a space of forms square integrable against e−|z|2is canonical, i.e., that it gives the minimal solution when applied to a ¯ ∂-closed form, and gives zero when applied to a form orthogonal to Ker ¯ ∂. As an application, we construct a canonical homotopy operator for i∂ ¯ ∂. 0. Introduction One way to solve ¯ ∂equations is to use integral formulas, and wellknown methods have been developed to construct explicit solution operators. Properties of the solutions can be deduced, studying the integral kernels of these operators. Even though the well-known methods in a way seem natural, in some cases the operators obtained are in a certain sense incompatible with the geometry. To make this statement precise, first recall that a solution operator Kis called canonical if u=Kf is the minimal solution to ¯ ∂u =fwhen f∈Ker ¯ ∂and Kf = 0 when fis orthogonal to Ker ¯ ∂. It turns out, that certain solution operators are not canonical with respect to the Euclidean metric; the solution operator in strictly pseudoconvex domains, obtained by Henkin, Skoda and others, see [H], [S], is not canonical with respect to the Euclidean metric. This statement has to be interpreted with some care. First we note, that the operator canonical with respect to the Euclidean metric, the Kohn operator KKis known (see, e.g., [Ha-P]). The Henkin-Skoda operator yields boundary values of solutions, and that it not is canonical means that the boundary values that it produces do not coincide with the boundary values of the solutions produced by KK. 2000 Mathematics Subject Classification. 32A25, 32F20. Key words. ¯ ∂equation, integral formula, canonical homotopy operator, Fock space. 224 J. Boo In [ABO], we studied canonical solution operators in strictly pseudoconvex domains. One major result was that, in the special case of the ball, the Henkin-Skoda operator is canonical with respect to the metric Ω = i(−ρ)∂¯ ∂log (1/−ρ), where −ρis the distance to the boundary. This means that the values given by the Henkin-Skoda operator coincide with the boundary values given by the canonical operator. There were additional advantages of using the non-K¨ahler metric Ω instead of the Euclidean metric, for example the domain of the formal adjoint operator ¯ ∂∗contains all forms that are smooth up to the boundary, contrary to the Euclidean case. This suggests that the Ω metric in some sense is more natural than the Euclidean metric. (The metric Ω is also related to the natural metric on the boundary.) In a general strictly pseudoconvex domain, the Henkin-Skoda operator is only approximately canonical with respect to Ω in a certain sense, see [A-Boo]. In this paper, we study a space of forms in all of Cnwith growth of infinite order. Using a technique described in [A-Be], a solution operator is obtained. The main result of this paper is that the solution operator is canonical with respect to the Euclidean metric. As an application, following the lines in [ABO], we construct a canonical homotopy operator for i∂ ¯ ∂. The paper is organized like this: In Section 1 we construct the solution operator, and in Section 2 the operator is expressed in terms of the metric, and we can see that the operator is canonical. Finally, in Section 3, we obtain some simple regularity results and construct a homotopy operator for i∂ ¯ ∂. 1. Construction of the operator In this section, we construct a homotopy operator Kfor ¯ ∂. The operator is essentially well known even in a much more general setting, see for instance [A-Be], but nevertheless we sketch the construction in our case. We start with a general process of constructing homotopy operators. Let η=ζ−z. Let Qand Sbe mappings from Cn×Cnto Cn. Define forms qand sby q=Qjdηjand s=Sjdηj.Fort≥0 we let Pt(ζ,z)=Cne(Q+tS)·η(d(q+ts))n, Homotopy Operators for ¯ ∂in Fock Space 225 where C−1 n=(−1)nn!(2πi)n, and S·ηis defined by S·η(ζ,z)= Sj(ζ,z)ηj(ζ,z), and so on. Define the kernel Kby K(ζ,z)=∞ t=0 Pt(ζ,z). Note that d(q+ts)=dq+tds−s∧dt,so(d(q+ts))n=A−n(dq +tds)n−1 ∧ s∧dt, where Acontains no differentials with respect to t. Hence K(ζ,z)=−Cnn∞ 0 e(Q+tS)·ηs∧(dq +tds)n−1dt. Put Ik(ζ,z) =Cn∞ 0 (−1)k+1 n! (n−k−1)!e(Q+tS)·ηs∧(dq +tds)n−k−1∧(ds)k (S·η)kdt and Tk(ζ,z)=Cn(−1)k+1 n! (n−k)!eQ·ηs∧(dq)n−k∧(ds)k−1 (S·η)k. By formally integrating by parts, we see that if 1 ≤k≤n−1, then K(ζ,z)=T1(ζ,z)+···+Tk(ζ,z)+Ik(ζ,z). If we note that In−1=Tn, we get the formula K(ζ,z)= n  k=1 Tk(ζ,z). Change the summation variable and, to let the operator fit into our situation, choose Q(ζ,z)=−¯ ζand S(ζ,z)=¯η. Then: (1.1) K(ζ,z)=Cn n−1  k=0 n! k!ez·¯ ζ−|ζ|2 ×¯ ζ−¯z·(dζ −dz)∧(dζ −dz)·d¯ ζk∧(dζ −dz)·d¯ ζ−d¯zn−k−1 |ζ−z|2n−2k. The kernel Kis of total bidegree (n, n −1). Denote by Kqthe component of Kwhich is of bidegree (0,q)inz, and hence (n, n −q−1) in ζ.We find Kqby expanding dζ ·d¯ ζ−d¯zn−k−1 = n−k−1  q=0 n−k−1 qdζ ·d¯ ζn−k−q−1∧(−dζ ·d¯z)q. 226 J. Boo This gives the formula Kq(ζ,z)=Cn n−q−1  k=0 n! k!n−k−1 q(−1)qez·¯ ζ−|ζ|2 ×¯ ζ−¯z·dζ ∧dζ ·d¯ ζn−q−1∧(dζ ·d¯z)q |ζ−z|2n−2k. (In this formula, konly occurs in the constant and in the exponent of the denominator.) Change the definition of Kby letting K(ζ,z)= n−1 q=0 Kq(ζ,z). This is motivated by the fact that we will integrate K against (0,q)-forms in ζ; we simply ignore the irrelevant parts of K. The leading term in K(ζ,z), corresponding to k= 0, equals φ(ζ,z)= ez·¯ ζ−|ζ|2times the Bochner-Martinelli kernel B(ζ,z), and K(ζ,z)= φ(ζ,z)B(ζ,z)+K(ζ,z), where the kernel Kas well as ¯ ∂Kare integrable. It is well known that ¯ ∂B (ζ,z) = [∆], where ∆ = {ζ=z}is the diagonal and [∆] denotes the current of integration over ∆. Thus (since φ(z,z) = 1 and ¯ ∂zφ(ζ,z)=0) ¯ ∂K (ζ,z)=¯ ∂ζφ(ζ,z)∧B(ζ,z)+¯ ∂K(ζ,z) + [∆] . Let A=n(dq +tds)n−1∧s, so that Pt=CnφA−CnφA∧dt =a−a∧dt (with A=(dq +tds)nas before). Then, since Ptis a closed form, 0=dPt=da −dζ,za∧dt and hence the formula ¯ ∂K =dK =∞ t=0 dζ,zadt =∞ 0 da =−a|t=0 =−Cnφ(ζ,z)(dq)n=−P0(ζ,z) is valid off the diagonal. (Also note that P0(ζ,z)=Cnφ(ζ,z)¯ ∂qn.) By this we will have that ¯ ∂K = [∆] −P0. Let Kand Palso denote the operators associated to the kernels K(ζ,z) and P0(ζ,z); Kf (z)=K(ζ,z)∧f(ζ) and similarily for P. Since the kernel Kis a form of total degree 2n−1, we will have that ¯ ∂Kf =¯ ∂K(ζ,z)∧f(ζ)=¯ ∂K (ζ,z)∧f(ζ)−K(ζ,z)∧¯ ∂f (ζ) =¯ ∂K.f −K¯ ∂f = [∆] .f −Pf −K¯ ∂f =f−Pf −K¯ ∂f. Thus we have obtained the homotopy formula ¯ ∂K +K¯ ∂=I−P,(1.2) Homotopy Operators for ¯ ∂in Fock Space 227 that a priori is valid only for, say, C1-forms with compact support, but as we will see in Section 2, by completeness (1.2) stays valid for all forms square integrable against e−|z|2. 2. Expressing the operator in the metric Let β=i∂ ¯ ∂|z|2/2. Denote by ·,· the pointwise Euclidean metric (for forms) generated by β, and let βk=βk/k!. The Lebesgue volume form equals the form dV =βn.Iffand gare (0,q)-forms, then f,gdV =cqf∧¯g∧βn−q,(2.1) where the constant cqequals 1 if qis even and −iif qis odd. Further, we have that dζ ·d¯ ζ=−2iβ. Let L2 qbe the set of all (0,q)-forms with finite norm with respect to the metric (f,g)=ce−|z|2f,gdV, where c=π−n, which gives the constant function 1 norm 1. Let Kq= L2 q∩Ker ¯ ∂; in particular K0=F2is the Fock space of entire functions, square integrable against e|z|2. The operator Kcan be expressed as inner multiplication by the kernel k(ζ,z)=qkq(ζ,z), where kq(ζ,z)= n−q−1  k=0 Cn,q,kez·¯ ζ¯ ζ−¯z·dζ ∧(dζ ·d¯z)q |ζ−z|2n−2k and Cn,q,k =Cn cq+1 n! k!n−k−1 q(−1)n−12n−q−1in−q−1(n−q−1)! =(−1)n−1 2q+1πniq+1cq+1 (n−k−1)! (n−q−1)! k!(n−k−q−1)!q!, i.e. Kf (z)=f,k(·,z). (Note, that the constant Cn,q,k is real.) Proposition 2.1. The operator Kis L2-bounded. Proof: We have the estimate |k(ζ,z)|eRe z·¯ ζ |ζ−z|2n−1 228 J. Boo for the kernel. Use the Cauchy-Schwarz inequality to obtain Kf2=e−|z|2f,¯ k 2dV (z)2 =e−|ζ−z|2 2e−Re z·¯ ζe−|ζ|2 2f(ζ),k(ζ,z)dV (ζ) 2 dV (z)2 ≤I1(z)I2(z)dV (z), where I1(z)=e−|ζ−z|2 2 |ζ−z|2n−1dV (ζ)=C<∞ and I2(z)=e−|ζ−z|2 2|ζ−z|2n−1e−2Rez·¯ ζe−|ζ|2f(ζ),k(ζ,z) 2dV (ζ) ≤e−|ζ−z|2 2|ζ−z|2n−1e−2Rez·¯ ζe−|ζ|2|f(ζ)|2|k(ζ,z)|2dV (ζ) e−|ζ−z|2 2e−|ζ|2|f(ζ)|21 |ζ−z|2n−1dV (ζ). Hence Kf2e−|ζ|2|f(ζ)|2e−|ζ−z|2 2 |ζ−z|2n−1dV (z)dV (ζ)f2. Remark 1.In [A-Boo], we make extensive use of the fact that a certain homotopy operator is compact. In this situation, however, the operator Kis not compact. This can be seen as follows. Consider one complex variable. The set of all fk=zk √k!d¯z is an orthonormal set in K1. Let uk=Kfk. Since the functions uk=|z|2−k √k!zk−1 also constitute an orthonormal set, we have an example of a bounded sequence (fk) such that the image sequence (Kfk) has no convergent subsequence. Hence Kis not compact on L2. Homotopy Operators for ¯ ∂in Fock Space 229 By considering the kernel of the operator Pdefined in Section 1, Pis easily seen to be the orthogonal projection from L2 0onto F2. Since Cnequipped with the β-metric is a complete manifold, the smooth, compactly supported forms are dense in the graph norms (see e.g. [B]). We already know that the homotopy formula (1.2) is valid for smooth, compactly supported forms, hence the formula is valid for all forms in Dom ¯ ∂. Remark 2.The L2-boundedness of Khelps to explain why (1.2) is valid for all f∈Dom ¯ ∂, in the following way: If fis in the domain of ¯ ∂, and in particular fitself is in L2, then the terms f,Pf and K¯ ∂f all are in L2. By approximation with smooth, compactly supported forms, we see that ¯ ∂Kf is in L2as well and that (1.2) stays valid in the limit. In particular, we conclude, that Kf ∈Dom ¯ ∂for all f∈Dom ¯ ∂, and that ¯ ∂K: Dom ¯ ∂→Kis a projection. It is easily checked that k(ζ,z)=∂ζh(ζ,z), where h(ζ,z)=n−1 q=0 hq(ζ,z), hq(ζ,z)= n−q−1  k=0 −Cn,q,k n−k−1ez·¯ ζ(dζ ·d¯z)q |ζ−z|2n−2k−2 for q>0 and h0(ζ,z)= n−2  k=0 −Cn,0,k n−k−1·ez·¯ ζ |ζ−z|2n−2k−2+Cn,0,n−1ez·¯ ζlog |ζ−z|2. Since his hermitean, i.e. hq(z,ζ)=(−1)qhq(ζ,z), the operator Hdefined by Hf (z)=f,h(·,z)is self-adjoint. Thus we have seen that K=H¯ ∂∗, where ¯ ∂∗is the formal adjoint of ¯ ∂with respect to (·,·). As an immediate consequence, we get that ¯ ∂K is selfadjoint, and hence is the orthogonal projection onto K. Thus the homotopy formula (1.2) gives the orthogonal decomposition L2=K⊕K⊥. Remark 3.By the method in the proof of Proposition 2.1, one can see that the operator His bounded on L2. Remark 4.By the K¨ahler identities for vector bundles (see for instance [B]), ¯ ∂∗=i[∂,β¬]+∂|z|2¬=1 2[∂,d¯z·dz¬]+¯z·dz¬, where the brackets denote the commutator and ¬denotes interior multiplication with respect to β. When we, as in this context, only let ¯ ∂∗act on (0,q)-forms, this expression reduces to ¯ ∂∗=¯z·dz¬−1 2d¯z·dz¬∂. We conclude: 230 J. Boo Theorem 2.2. Kis the canonical operator with respect to (·,·). Proof: If fis orthogonal to Ker ¯ ∂, then ¯ ∂∗f=0,soKf = 0. If, on the other hand, ¯ ∂f = 0, then f=¯ ∂Kf,soKf =K¯ ∂Kf =Kf − ¯ ∂KKf, and hence ¯ ∂K (Kf) = 0. Since ¯ ∂K is the orthogonal projection onto the kernel, Kf is orthogonal to the kernel, and hence the minimal solution. 3. Application to solving i∂ ¯ ∂problems Note, that the kernel K(ζ,z) from (1.1) almost is a convolution kernel; K(ζ,z)=φ(ζ,z −ζ)A(z−ζ), where φ(ζ,z)=ez·¯ ζand A(η)isan (integrable) convolution kernel whose coefficients roughly are ¯ηk/|η|∗, where ∗denotes an exponent not higher than 2n. By performing an appropriate change of variables in the integral defining Kf and differentiating under the integral sign, and then substituting back, we see that Kf has partial derivatives with respect to zkand ¯zkif fhas, and furthermore Kcommutes with the holomorphic derivatives in the sense that ∂ ∂zk Kf =K∂f ∂ζk (3.1) (where we let the derivative act as a Lie derivative on forms, i.e. so that it only affects the coefficient functions). As a consequence, we have that Kpreserves regularity. By similar arguments, Ppreserves regularity and satisfies the same commuting rule (3.1) as K. In particular, the homotopy formula (1.2) gives a C∞-smooth orthogonal decomposition of L2∩C∞. Remark 5.For antiholomorphic derivatives, there is no rule as simple as (3.1). Instead, we have the formula ∂ ∂¯zk Kf =K∂f ∂¯ ζk +zkKf −K(ζkf). However, the main reason for using (3.1) is to prove Proposition 3.1 below. The corresponding commutation rule for ¯ ∂would be hard to prove using the above formula for the antiholomorphic derivatives, and anyway the rule for ¯ ∂is known; it is just the homotopy formula (1.2). Recall that we restricted the operator Kto operate on (0,q)-forms only. Now we extend Kto an operator operating on (p, q)-forms by demanding that the (p, 0) part should be ignored; more precisely we let KaI, ¯ Jd¯ ζJ∧dζI=KaI, ¯ Jd¯ ζJ∧dzI. The homotopy formula (1.2) and the commutation rule (3.1) still hold for this extended K.IfΦ(ζ,z)= Homotopy Operators for ¯ ∂in Fock Space 231 IdzI∧d¯ ζI, then the kernel for this extended Kis k(ζ,z)∧Φ(ζ,z), and thus the kernel for the corresponding operator Hsuch that K=H¯ ∂∗is h(ζ,z)∧Φ(ζ,z). In particular, His self-adjoint, which in turn implies that the operator ¯ ∂K, acting on (p, q)-forms, is the orthogonal projection onto the kernel of ¯ ∂. The observation above concerning holomorphic derivatives yields the following proposition. Proposition 3.1. ∂K =−K∂ and ∂P =P∂. Proof: If fis a q-form, then ∂f =(−1)q∂ ∂zkf∧dzk.Thus ∂K αd¯ ζJ∧dζI=∂K αd¯ ζJ∧dzI =(−1)|J|−1∂ ∂zk Kαd¯ ζJ∧dzk∧dzI =(−1)|J|−1K∂α ∂ζk d¯ ζJ∧dzk∧dzI =(−1)|J|−1K∂α ∂ζk d¯ ζJ∧dζk∧dζI =−K∂αd¯ ζJ∧dζI. That proves the statement for K, and by using (1.2) twice we get that ∂Pf =∂f−K¯ ∂f=∂f +K∂¯ ∂f =∂f −K¯ ∂∂f =P∂f, which proves the statement for P. Remark 6.This proposition and the homotopy formula (1.2) gives the corresponding homotopy formula dK +Kd =I−Pfor d. In addition, we define operators ¯ Kand ¯ Pby ¯ Kf =K¯ fand analogously for ¯ P. The operator ¯ Kobviously takes (p+1,q)-forms into (p, q)-forms. Note that the operator K¯ Ksolves the ∂¯ ∂equation: If fis ad-closed (q,q)-form, then fis both ∂- and ¯ ∂-closed. Hence v=¯ Kf satisfies ∂v =fand (by Proposition 3.1) ¯ ∂v = 0. Let u=Kv. Then ¯ ∂u =v, hence ∂¯ ∂K ¯ Kf =∂¯ ∂u =∂v =f. Thus, we can easily find an operator that solves ∂¯ ∂equations. However, to get a homotopy operator, we need a little extra effort. (Also note, that the solutions were minimal in each step, but that the resulting solution will not necessarily be minimal.)