Estimates for the Bergman and Szegö projections for pseudoconvex domains of finite type with locally diagonalizable Levi form

Publ. Mat. 50 (2006), 413–446 ESTIMATES FOR THE BERGMAN AND SZEG¨ O PROJECTIONS FOR PSEUDOCONVEX DOMAINS OF FINITE TYPE WITH LOCALLY DIAGONALIZABLE LEVI FORM Philippe Charpentier and Yves Dupain Abstract In this paper, we give precise isotropic and non-isotropic estimates for the Bergman and Szeg¨o projections of a bounded pseudoconvex domain whose boundary points are all of finite type and with locally diagonalizable Levi form. Additional local results on estimates of invariant metrics are also given. 1. Introduction This paper deals with precise mapping properties of the Bergman and Szeg¨o projections of pseudo-convex domains of finite type in Cnwhose Levi form are locally diagonalizable at every point of the boundary (see Section 2 for a precise definition). We obtain sharp estimates for these operators for usual Lp kSobolev spaces, classical Lipschitz spaces Λαand nonisotropic Lipschitz spaces Γαrelated to the geometry of the domain. Our results, in the present paper, are analog to those obtained for convex domains of finite type in [MS94] and [MS97] and extend previously known results for the strictly pseudoconvex case ([AS79] and [PS77]), for the finite type domains of C2([NRSW89], see also [Chr88] and [FK88]) and in the case of pseudoconvex domains of finite type of Cn having a Levi form of rank n−1 ([AC99], see also [Mac88]). Similar results where obtained for pseudoconvex domains in Cnwhose Levi form have comparable eigenvalues (see [Koe02], [Cho02b] and [Cho03]). 2000 Mathematics Subject Classification. Primary: 32H15. Key words. Finite type, Bergman projection, invariant metrics. 414 Ph. Charpentier, Y. Dupain In 1990, C. L. Fefferman, J. J. Kohn and M. Machedon studied the class of domains we consider here and proved local Lipschitz estimate with arbitrary small loss for the ∂Neumann problem and the Szeg¨o projection [FKM90] (see also [Der99]). The present article is the first of two papers. Here, we prove the following results: Theorem. Let Ωbe a bounded pseudo-convex domain of finite type in Cn with locally diagonalizable Levi form. (1) For all p,1< p < +∞, and all s≥0, the Bergman projection of Ωmaps continuously Lp s(Ω) into itself. (2) For 1< p < +∞and s∈N, the Szeg¨o projection maps continuously Lp s(∂Ω) into itself. Theorem. Let Ωbe a bounded pseudo-convex domain of finite type in Cn with locally diagonalizable Levi form. (1) For 0< α < +∞, the Bergman projection maps continuously Λα(Ω) into itself. (2) The Szeg¨o projection maps continuously Λα(∂Ω) into itself, for all α∈]0,+∞[. A consequence of the proof is: Corollary. Under the same conditions, the Bergman projector maps continuously L∞(Ω) into BMO(Ω). Defining non-isotropic Lipschitz spaces Γα, for α < 1/M where Mis the type of Ω, with the pseudo-distance associated to the geometry, we also obtain: Theorem. Let Ωbe a bounded pseudo-convex domain of finite type in Cn with locally diagonalizable Levi form. (1) For α < 1/M, the Bergman projection maps continuously Λαinto Γα. (2) The Szeg¨o projection maps continuously Λα(Ω) into Γα(∂Ω) for 0< α < 1/M. Extending the definition of Γαfor all α > 0, we also give similar results for α≥1/M. In the last section, we indicate some supplementary local results on invariant metrics similar to those obtained by D. W. Catlin in dimension 2 [Cat89] and by J. D. McNeal for convex domains [MN01]. Estimates for the Bergman and Szeg¨ o Projections 415 In a forthcoming paper, we will improve the result of [FKM90] getting local (isotropic and non-isotropic) Lipschitz estimates without loss for the Szeg¨o projection. We essentially use the techniques developed in [CNS92], [NRSW89], [MS94] and [MS97] associated to the geometric properties of the domain and the estimates for the Bergman kernel function proved in [CD]. In Section 2 we quickly describe this geometry and summarize the results necessary to our purpose. Our geometry posses some good properties similar to those of convex domains described by McNeal in [McN94] but they are not completely equivalent and, if we follow the ideas of [MS94] and [MS97], the proof must be modified. For example, to evaluate |f(p)−f(q)|,p, q ∈Ω, we cannot use an integration along the segment [p, q] and we must use a convenient curve essentially given by the exponential map. Similarly, when we want to apply Cauchy formula, we don’t have directly the existence of polydisc of good weighted size included in the domain. Furthermore, we often need to work in some more local context, which implies that we have to modify the definitions of the concepts we use. In Section 3, devoted to the Bergman projection, we show how to adapt the methods of [MS94] to the geometry of the domain we consider, in a relatively detailed way, to get the wright estimates. Section 4 is devoted to the Szeg¨o projection. Because, on one hand the N.I.S. operators theory is quite well known, and on the other hand, we have detailed the use of the geometry in Section 3, we will only give the main articulations of the proofs. 2. Geometry and Bergman kernel estimates Let us first precise the class of domains we consider in this paper and introduce the basic notations. Definition 2.1. Let Ω be a bounded pseudo-convex domain in Cnwith smooth boundary. Let pbe a point on the boundary of Ω. We say that the Levi form is locally diagonalizable at pif there exist a neighborhood Vof pand a smooth basis Bof sections of the complex tangent bundle T0,1in V∩∂Ω which diagonalizes the Levi form. When this property holds at every point of the boundary, we say that Ω has a locally diagonalizable Levi form. Let ρbe a smooth defining function of Ω (i.e. Ω = {ρ < 0},∇ρ6= 0 on ∂Ω). Let p0be a point of ∂Ω. Let (Li)1≤i≤n−1a family of smooth 416 Ph. Charpentier, Y. Dupain vector fields in a neighborhood Vof p0which is a basis of the complex tangent space at ∂Ω in V∩∂Ω diagonalizing the Levi form, and let Ln=N=4 |∇ρ|2X i ∂ρ ∂zi ∂ ∂zi the complex normal vector field. If L= (L1,...,Lk) is a list of vector fields of B∪B∪ {Ln, Ln}, we denote by |L|the length kof L. For each index i, let cii =∂ρ; [Li,Li]. Let L(i) the set of all lists of vector fields L= (L1,...,Lk) such that Lj=Lior Lj=Lifor all j. If L∈L(i), we denote L(cii) = L1. . . Lk(cii). Recall that if Ω is of finite type there exists M > 0 such that, for all i, there exists a list L∈L(i), |L| ≤ M, such that L(cii)6= 0. Then we define, for p∈V,δ > 0 and i≤n−1, Fi(p, δ) = M−2 X |L|=0 L∈L(i)  L(cii)(p) δ 2/|L|+2 , and put Fn(p, δ) = δ−2. Associated to these functions, we use the following notations: for s= (s1,...,sn), si>0, Fs(p, δ) = Y i Fsi i(p, δ), and, when s= (1,...,1), we simply write Finstead of Fs. Moreover, if L=(L1,...,Lk) is a list if vector fields Lj∈ {L1,L1,..., Ln,Ln}, and if l= (l1,...,ln), libeing the number of indices jsuch that Lj∈ {Li,Li}, we will also write FL=Fl. As usual, the notations .,&,≃mean that the inequality or equivalence holds up to a multiplicative constant depending only on Ω and the choice of the vector fields Li, and .∗,&∗,≃∗mean that the constant depends also on ∗. Shrinking Vif necessary, the following relations hold in V×]0,+∞[: (2.1) δ−2/M .Fi(p, δ).δ−2, and, for α > 1, (2.2) α−2Fi(p, δ)≤Fi(p, αδ)≤α−2/M Fi(p, δ). Estimates for the Bergman and Szeg¨ o Projections 417 2.1. Special coordinate systems. Let (Z) be the canonical coordinate system of Cn. With the previous notations, to each point pof Vand all δ > 0 one can define a change of coordinate Φδ p(sometimes denoted simply Φp), Z= Φδ p(z), such that Φ−1 p(0) = pand Li(p) = ∂ ∂zi, satisfying the following properties. Φpand Φ−1 pare polynomial coordinate systems defined on Cnof degree less than (2M)n−1and: Proposition 2.1. (1) The coefficients of the polynomials defining Φp and Φ−1 pare uniformly bounded for p∈Vand δ > 0. (2) The Jacobians of Φpand Φ−1 pare uniformly bounded from below. Proposition 2.2. If a= (a1,...,an),Dadenotes any derivative of the form ∂|a| ∂zb1 1...∂zbn n∂zc1 1...∂zcn n with bi+ci=ai. Then (1) For all derivative Da,Da(ρ◦(Φδ p)−1)(0).|a|1, for all δ. (2) For all derivative Da,Da(ρ◦(Φδ p)−1(0).|a||ρ(p)|Fa/2(p, |ρ(p)|), if δ=|ρ(p)|. Proposition 2.3. For z∈Φp(V), let ˜zt= (z1,...,zn−t), for t≥0. If ˜zt∈Φp(V), then ρ◦Φ−1 p(z)−ρ◦Φ−1 p(˜zt)≃t. 2.2. Polydisks and pseudo-balls. For 1 ≤i≤n, let Ri(p, δ) = Fi(p, δ)−1/2. Let ε > 0 fixed sufficiently small. For p∈Vand δ > 0, we define the “polydisk” centered at pof radius δby Pp(δ) = (Φδ p)−1{|zi| ≤ εRi(p, δ)}. The volume of Pp(δ) is estimated in terms of the functions Fi: Proposition 2.4. For all α > 0, if p∈V, Vol(Pp(δ)∩ {|ρ|< αδ}).αVol(Pp(δ)) and (2.3) Vol(Pp(δ)) ≃F(p, δ). Note that the first inequality follows Propositions 2.3 and 2.1. The change of coordinates Φδ pis “close” to the vector fields in the following sense. Write Li=Pbj i∂ ∂zjand ∂ ∂zj=Pai jLi(in the coordinate system Φδ p) then 418 Ph. Charpentier, Y. Dupain Proposition 2.5. For all A > 0, all p∈V, all q∈Pp(Aδ),Aδ ≤δ0, and all α,β, Dαβaj i(q)≤KαβF(α+β)/2(p, δ)R−1 i(p, δ)Rj(p, δ), and Dαβbj i(q)≤KαβF(α+β)/2(p, δ)R−1 i(p, δ)Rj(p, δ), where Kαβ depends on A,αand β. As a consequence, if L={L1,...,Lk}is a list of vector fields of length k, the vector field L1. . . Lkcan be expressed with derivatives in the coordinate system: if we write L1...Lk(q) = X |s|≤k cs(q)Ds, then Proposition 2.6. For all A > 0, with the above notation, if p∈V, |ρ(p)| ≤ δ≤δ0, for all q∈Pp(Aδ),Aδ ≤δ0,|cs(q)| ≤ KlFL−s 2(p, δ). We now briefly recall the notations about the exponential map associated to the real and imaginary parts of the vector fields Li: we denote by exppthe exponential map centered on p∈Vassociated to the 2nreal fields Yjdefined by Y2k=ℜe(Lk), Y2k−1=ℑm(Lk), for k≤n. Thus, Vbeing sufficiently small, exppis a diffeomorphism of a neighborhood of the origin in R2nonto a neighborhood of pin Vcontaining a fixed ball, and, for all points p1and p2in V, there exists u= (u1,...,u2n) such that p2= expp1(u). Moreover Lemma 2.1. For all points p1and p2in V, if p2= expp1(u)then |ui|.|p1−p2|. Now we recall how the “pseudo-distance” of two points of Vis defined using expp. With the previous notations, we write, for 1 ≤k≤n, R′ 2k(p, δ) = R′ 2k+1(p, δ) = Rk(p, δ), and γ(p1, p2) = inf{t≥0 such that p2= expp1(u1,...,u2n),with |ui| ≤ R′ i(p1, t)}. Estimates for the Bergman and Szeg¨ o Projections 419 Remark 2.1.This function γis independent of the choice of the basis of vector fields diagonalizing the Levi form in V∩∂Ω. If γ′is associated to an other basis diagonalizing the Levi form in V∩∂Ω, there exist two constants K1and K2(depending on the basis) such that K1γ≤γ′≤K2γ. Let the “pseudo-ball” associated to the exponential map is defined by Bexp(p, δ) = {q= expp(u1,...,u2n),|ui| ≤ R′ i(p, δ)}, and is related, for not too small δ, to the polydisks as follows: Proposition 2.7. There exist two constants aand Asuch that, for p∈V,|ρ(p)|< δ ≤δ0, Bexp(p, δ)⊂Pp(aδ)⊂Bexp(p, Aδ). Corollary. If γ(z, w)≤δand |ρ(z)|< δ ≤δ0, then w∈Pz(aδ). We will also use the notion of balls defined by curves. Let p∈V and δ > 0. We denote by BC(p, δ) the set of points q∈Vsuch that there exists a curve ϕ: [0,1] →V, piecewise C1such that ϕ(0) = p,ϕ(1) = q and ϕ′(t) = Piai(t)Yi(ϕ(t)), almost everywhere, with |ai(t)| ≤ R′ i(p, δ). The relation with the “balls” defined with the exponential map is given by the following proposition: Proposition 2.8. There exists K0such that, if p∈V,|ρ(p)|< δ ≤δ0, Bexp(p, δ)⊂BC(p, δ)⊂Bexp(p, K0δ). Corollary. For all B≥1, if w∈BC(z, Bδ),|ρ(z)| ≤ δ≤δ0/B, we have γ(z, w).Bδ. Remark 2.2.γdefines a pseudo-distance on V∩∂Ω but not on Vbecause of the restriction on δin Proposition 2.7 and Proposition 2.8. But these properties will be sufficient for our purpose. If πdenotes the natural projection on ∂Ω, then ∆(p1, p2) = γ(π(p1), π(p2)) + |ρ(p1)−ρ(p2)| is a pseudo-distance on V. The next proposition controls the variations of the weights Fiin the polydisks and balls previously defined: Proposition 2.9. For all B > 0, if p∈V,|ρ(p)| ≤ δ≤δ0/B, and if qbelongs either to Pp(Bδ)or to Bexp(p, Bδ)or to BC(p, Bδ), then, for all i,Fi(p, δ)≃BFi(q, δ). 420 Ph. Charpentier, Y. Dupain 2.3. Point-wise estimates of the Bergman kernel. Proposition 2.10. Let KB(z, w)be the Bergman kernel of Ω. For all points p,p1, and p2in V∩Ω: (1) KB(p, p)≃F(p, |ρ(p)|). (2) For all lists L(resp. L), of length less than N, composed with holomorphic (resp. anti-holomorphic) vector fields Li(resp. Li), li(resp. li) denoting the number of times Li(resp. Li) appears in L(resp. L), LzLwKB(p1, p2).NYFi(p1, δ)1+ li+li 2, where δ=γ(p1, p2) + |ρ(p1)|+|ρ(p2)|. (3) In particular, for all integer m≥0, |∇mKB(p1, p2)|.mF(p1, δ)δ−m. In the calculus in the next sections, we will use the following equivalences: γ(p1, p2) + |ρ(p1)|+|ρ(p2)| ≃ γ(p2, p1) + |ρ(p1)|+|ρ(p2)| ≃γ(p1, p2) + |ρ(p1)|. (2.4) Remark 2.3.(1) The previous estimate does not seem to be symmetric, but, in fact, it is. (2) In [CD] the non-isotropic estimates (2) of the previous proposition was used to prove the continuity of the Bergman projection from L1 to L1∞. In the present paper, the isotropic estimate (3) will be sufficient to estimate the Bergman projection, but we will need the non-isotropic one for the Szeg¨o projection. 3. The Bergman projection 3.1. Sobolev estimates. The goal of this section is to prove the following: Theorem 3.1. Let Ωbe a bounded pseudo-convex domain in Cnof finite type with locally diagonalizable Levi form. For all p,1< p < +∞, and all s≥0, the Bergman projection of Ωmaps continuously Lp s(Ω) into itself. By interpolation, it suffices to prove the result for s=k∈N. Estimates for the Bergman and Szeg¨ o Projections 421 To each point pof ∂Ω, we associate a orthogonal coordinate system (by a complex linear change) centered at psuch that ∂ρ ∂xn(p) = 1 (xnbeing the real part of the last complex coordinate) and 1 2≤∂ρ ∂xn≤3 2in a neighborhood W(p) of p. For each integer m∈N∗, we cover ∂Ω by a finite number of neighborhood Vk(pk)⊂W(pk), k= 1, . . . , N,pk∈∂Ω, such that, in the previous coordinate system, Vk=Vk(pk) is a polydisk centered at pkof radius ˜r such that the polydisk Uk=Uk(pk) centered at pkof radius 8m˜ris contained in W(pk) and in the neighborhood Vrelated to the point pk defined in Section 2. Let us denote ∆k= ∆k(pk) the polydisk centered at pkof radius 2m˜r. All properties about the geometry and the Bergman kernel, listed in the previous section are then valid in the Uk. We denote then F(k) iand γ(k)the corresponding functions Fiand γdefined in Uk(pk). In [MS94, p. 181], J. D. McNeal and E. M. Stein introduced a notion of B-type kernels. For our purpose we need a small modification of this definition: Definition 3.1. A function B(z, w)∈C∞(Ω ×Ω) is called a B-type kernel if there exist two constants Cand C′such that, for (z, w)∈ Uk×Uk, |B(z, w)| ≤ C n Y i=1 F(k) i(z, δk), where δk=|ρ(z)|+|ρ(w)|+γk(z, w), and, for (z, w)/∈SN k=1 Uk×Uk, |B(z, w)| ≤ C′. Remark. Note that the notion of B-type kernel does not depend neither on the choice of the basis diagonalizing the Levi form nor on the choice of mand the Uk. Lemma 3.1. Let B(z, w)be a B-type kernel. Let |B|the operator associated to |B(z, w)|. Then |B|maps continuously Lp(Ω) into itself for 1< p < +∞. Remark. The proof of this lemma gives an other proof of the Lppart of Theorem 2.2 of [CD]. Lemma 3.1 is an easy consequence of the H¨older inequality and the following lemma: Lemma 3.2. For 0< ε < 1, there exists a constant C1=C1(ε)such that, |B||ρ|−ε(z)≤C1|ρ|−ε(z). 428 Ph. Charpentier, Y. Dupain First, for 0 < α < 1, a function fbelongs to Λα(Cn) if it is bounded and if |f(x)−f(y)|.|x−y|α,x, y ∈Cn. For α > 1, α /∈N,fbelongs to Λα(Cn) if it is bounded and if, Dβf∈Λα−|β|(Cn) for βof length equal to the integral part of α. A function fbelongs to Λ1(Cn) if it is bounded and, for all xand h,|f(x+h) + f(x−h)−2f(x)|.|h|, and fbelongs to Λk(Cn), k∈N∗, if Dβf∈Λ1(Cn) for all βof length k−1. A function fbelongs then to Λα(Ω) if there exists a function F∈ Λα(Cn) whose restriction to Ω is equal to f. In other words, the Banach space Λα(Ω) is defined as a quotient space. We will prove the following result: Theorem 3.2. Let Ωbe a bounded pseudo-convex domain in Cnof finite type with locally diagonalizable Levi form. For 0< α < +∞, the Bergman projection maps continuously Λα(Ω) into itself. The proof is based on a result essentially due to Hardy and Littlewood: Proposition 3.1. Let fbe a bounded C∞function on Ω. If there exists an integer m > α such that |∇mf(z)|.|ρ(z)|−m+αon Ω, then f∈Λα(Ω). To see that PBfsatisfies this sufficient condition when f∈Λα(Ω), we use the following characterisation: Proposition 3.2. For a function fdefined on Ωthe three following properties are equivalent: (1) f∈Λα(Ω). (2) There are functions fk, such that f=P∞ 0fkand (a) kfkkL∞(Ω) .f2−kα, (b) for all integer m > α,k∇mfkkL∞(Ω) .f,m 2mk2−kα. (3) For all integers kand m,m > α, there exist two functions gk and bksuch that f=gk+bkand (a) kbkkL∞(Ω) .f2−kα, (b) k∇mgkkL∞(Ω) .f,m 2mk2−kα. Choose an integer m > α + 1 and use the decomposition of fgiven by property (3) of Proposition 3.2 with ksuch that |ρ(z)| ≃ 2−k. Then ∇mPBf(z) = ∇mPBbk(z) + ∇mPBgk(z). To estimate these quantities, we first prove the two following lemmas: Lemma 3.5. For all m≥1, there exits a constant Cdepending only on Ωand msuch that ZΩ |∇m zKB(z, w)|dw ≤C|ρ(z)|−m. Estimates for the Bergman and Szeg¨ o Projections 429 Lemma 3.6. Let Dbe a derivation of order m≥2. There exist operators B0,...,Bm−1defined by kernels B0,...,Bm−1and vector fields X1,...,Xm−1, of order 1, such that: (1) for i= 1,...,m−1,RΩ|Bi(z, w)|dw .|ρ(z)|−1, (2) for all C∞function f, ZΩ DzKB(z, w)f(w)dw =Bm−1f(z) + Bm−2(X1f)(z) + ···+B0(X1. . . Xm−1f)(z). Proof of Lemma 3.5: We use the covering {Vk}of ∂Ω, associated to m, defined in the previous section. If z /∈Sk≥1Vkthen |ρ(z)|is bounded from below and ∇zKB(z, w) is bounded. Suppose now z∈Vk0and write ZΩ |∇m zKB(z, w)|dw=ZΩ\Uk0 |∇m zKB(z, w)|dw + ZUk0∩Ω |∇m zKB(z, w)|dw. In the first integral the distance from zto wis uniformly bounded from below and, thus, ∇m zKB(z, w) is uniformly bounded. It suffices then to estimate the second integral. As in the proof of Lemma 3.2 we use the following partition of Uk0: E0={w∈Uk0∩Ω,such that γ(z, w)≤ |ρ(z)|} and, for k≥1, Ek=w∈Uk0∩Ω,such that 2k−1|ρ(z)|< γ(z, w)≤2k|ρ(z)|. By the estimates on the derivatives of the Bergman kernel (Proposition 2.10) and the relations between Fi(z, αδ) and Fi(z, δ) ((2.2)), we get ZEk |∇m zKB(z, w)|dw .m n Y i=1 Fi(z, 2k|ρ(z)|)1 (2k|ρ(z)|)mVol(Ek), and the proof is finished recalling (as it has been established in the proof of Lemma 3.2) that Vol(Ek).m n Y i=1 Fi(z, 2k|ρ(z)|)−1. 430 Ph. Charpentier, Y. Dupain Proof of Lemma 3.6: We use the partition of unity defined in Lemma 3.4, and, by the corollary of that lemma for m′=m−1 DzPBf(z) = ZΩ DzKB(z, w)f(w)ψ0(w)dw + n X k=1 ZΩ Z5˜r 0 ···Z5˜r 0 DzKB z, w′, wn− m−1 X i=1 ti!m−1 X l=0 × Ψ(k) m−1−l w′, wn− m−1 X i=1 ti !Tlf w′, wn− m−1 X i=1 ti !!dt1...dtm−1 !dw. RΩ|DzKB(z, w)f(w)|ψ0(w)dw is uniformly bounded (see remark after the corollary) and the functions Ψ(k) ibeing uniformly bounded it then suffices to prove that ZΩ∩∆kZ5˜r 0 ···Z5˜r 0 DzKB z, w′, wn− m−1 X i=1 ti ! dt1...dtm−1dw.m|ρ(z)|−1. As in Lemma 3.5, this is trivial if z /∈Uk. Suppose z∈Ukand w∈Ω∩∆k. Then (w′, wn−Pti) belongs also to Ukand we use the estimates of the derivatives of the Bergman kernel (Proposition 2.10) and the estimate of δ(t) obtained in the proof of Lemma 3.3 δ(t) = |ρ(z)|+|ρ( ˜wt)|+γ(z, ˜wt)&|ρ(z)|+|ρ(w)|+γ(z, w) + m−1 X i=1 ti. The properties of functions Fi(2.2) implies  DzKB z, w′, wn− m−1 X i=1 ti! .mQn i=1 Fi(z, |ρ(z)|+|ρ(w)|+γ(z, w)) (|ρ(z)|+|ρ(w)|+γ(z, w) + t1+···+tm−1)m and thus Z5˜r 0 ···Z5˜r 0 DzKB z, w′, wn− m−1 X i=1 ti!dt1. . . dtm−1 .mQn i=1 Fi(z, |ρ(z)|+|ρ(w)|+γ(z, w)) |ρ(z)|+|ρ(w)|+γ(z, w). Estimates for the Bergman and Szeg¨ o Projections 431 Now, to finish, it suffices to consider the partition of Ukdefined in the proof of Lemma 3.5. End of the proof of Theorem 3.2: By Lemma 3.5 |∇mPBbk(z)| ≤ C|ρ(z)|−m2−kα .fC|ρ(z)|−m+α, and by Lemma 3.6 |∇PBgk(z)|.|ρ(z)|−1[kgkk∞+···+kX1...Xm−1gkk∞]. But kgkk∞+···+kX1. . . Xm−1gkk∞.kgkk∞+ ∇m−1gk ∞, and, because m−1> α,  ∇m−1gk ∞.f,m 2(m−1)k2−kα .f,m |ρ(z)|−m+1+α. This completes the proof. As an immediate corollary of Lemma 3.5 and the fact that a C1function satisfying |∇f(z)|.|ρ(z)|is in BMO(Ω) (Lemma 7 of [MS94]), we have: Corollary. Under the conditions stated in Theorem 3.1, the Bergman projector maps continuously L∞(Ω) into BMO(Ω). 3.3. Nonisotropic H¨older estimates. Let V1,...,VNbe a covering of the boundary as defined in Section 3.1 (for m= 1). For each k, if pis a point of Vk, let Φpbe the change of variables associated to pand δ(p); we denote by (zi) the corresponding coordinates. Recall that Vkis chosen so that, if ˜ρ=ρ◦Φ−1 p, and w, z ∈Φp(Vk) such that wj=zjfor j < n and wn=zn+t,t∈R, then ˜ρ(w)−˜ρ(z)≃t (Proposition 2.3). In Section 2.2, we defined the function γk(p, q) for points pand qin the euclidean ball B(pk, rk). Let us define a global version γof these functions putting, for arbitrary points pand qin Ω, γ(p, q) = inf {γk(p, q) such that p, q ∈Vk,1≤k≤N}, if there exists ksuch that p, q ∈Vkand γ(p, q) = 1 if not. Then we denote ρ(p, q) = min {γ(p, q) + |ρ(p)|+|ρ(q)|,|p−q|} . With these notations, we define now the space Γα(Ω): 432 Ph. Charpentier, Y. Dupain Definition 3.2. For α < 1/M, Γα(Ω) is the space of continuous functions fon Ω such that |f(p)−f(q)|.ρ(p, q)α. Remark. Note that with this definition, Γα(Ω) is independent of the choice of the covering Vk(Remark 2.1). In the previous section, we proved that the Bergman projection maps continuously Λαinto itself. We will now see, that, the properties of holomorphic functions in Ω related to the geometry give a better: Theorem 3.3. Let Ωbe a bounded pseudo-convex domain of finite type in Cnwith locally diagonalizable Levi form. For α < 1/M, the Bergman projection maps continuously Λαinto Γα. By Theorem 3.2, it suffices to show that a holomorphic function in Λα belongs automatically to Γα. We prove this in two steps. First, we show that the derivatives of a holomorphic function in Λαsatisfy some non-isotropic estimates in terms of the functions Fi. The proof of the theorem is then done in Section 3.3.2. In Section 3.3.3 we indicate briefly how the theorem can be extended for α≥1/M. 3.3.1. Nonisotropic estimates of the derivatives of a holomorphic function in Λα.For a= (a1,...,an)∈Nn,Dadenotes the derivative, in the (zi) coordinate system, Da=∂|a| ∂za1 1...∂zan n. Recall that we denote Fa/2=Qn i=1 Fai/2 i. Proposition 3.3. Let f∈Λα(Ω) ∩H(Ω). Let p∈Vk,Φpthe change of coordinates corresponding to pand δ=|ρ(p)|, and Daa derivative. Then there exits a constant C, depending only on Ω,fand a, such that Da(f◦Φ−1 p)(0)≤ChF(p, |ρ(p)|)a/2|ρ(p)|α+ 1i. The starting point of the proof is the following lemma which is valid for any domain Ω (c.f. [MS94, Lemma 8, p. 197]): Lemma 3.7. Let f∈Λα(Ω)∩H(Ω). Let ∂sbe a derivative of length |s|> α(in the canonical coordinate system). There exists a constant Cdepending only on f,|s|and Ωsuch that, for all q∈Ω, |∂sf(q)| ≤ C|ρ(q)|−|s|+α. Let p∈Vk. This lemma and the regularity properties of the change of variables z= Φp(Z) (Proposition 2.1) imply that, if Dadenotes a derivative with respect to the zvariable of length |a|> α, then there Estimates for the Bergman and Szeg¨ o Projections 433 exists a constant Cdepending only on f,|a|and Ω such that, for all z∈ Φp(Ω), (3.2) Daf◦Φ−1 p(z)≤Cρ(Φ−1 p(z)) −|a|+α. We want to use this inequality through Cauchy’s formula, so, we need to know that certain polydisks are contained in Φp(Ω): Lemma 3.8. There exists two constants c < 1and ν > 0, depending only on Ω, such that, for 0< t < δ0, if zbelongs to the polydisk (|zi| ≤ cF−1/2 i(p, |ρ(p)|)|ρ(p)|+t |ρ(p)|1 2M2 ,for i < n, |zn+t| ≤ c|ρ(p)||ρ(p)|+t |ρ(p)|1 2M2), then ρ◦Φ−1 p(z) = ˜ρ(z)≤1 4[ρ(p)−νt]. Proof: Let zbe a point in the polydisk described in the lemma and z′=z+ (0,...,0, t). Taylor’s formula gives |˜ρ(z′)−˜ρ(0)| ≤ X |s|≤M ∗c|s|Ds˜ρ(0)F−s/2(p, |ρ(p)|)|ρ(p)|+t |ρ(p)||s| 2M2 +A, where ∗are absolute constants and, by Proposition 2.2 (1), and (2.1), Asatisfies A≤K1cM+1 |ρ(p)|M+1 M−M+1 2M2≤K2|ρ(p)|c. Moreover, Proposition 2.2 (2) implies |Ds˜ρ(0)| ≤ K|ρ(p)|Fs/2(p, |ρ(p)|) and |˜ρ(z′)−˜ρ(0)| ≤ cK3(|ρ(p)|+t). On the other hand, by Proposition 2.3, there exists a constant νsuch that ˜ρ(z)−˜ρ(z′)<−νt, which implies the lemma, for csmall enough. Proof of Proposition 3.3: Let l > α be an integer. Using (3.2) and the previous lemma, Cauchy’s formula applied at the point (0,...,0,−t) 434 Ph. Charpentier, Y. Dupain shows that there exists a constant Kdepending only on f,l, Ω and |s| such that (3.3)  Da∂l ∂zl n f◦Φ−1 p(0,...,0,−t) ≤KFa/2(p, |ρ(p)|)|ρ(p)| |ρ(p)|+t1 2M21 |ρ(p)|+t−α+l . We choose l= [α]+1. The function f◦Φ−1 pbeing holomorphic, we have ∂l ∂zl nf◦Φ−1 p=∂ ∂xn ∂l−1 ∂zl−1 nf◦Φ−1 p, and Da∂l−1 ∂zl−1 nf◦Φ−1 p(0,...,0,−t) = Da∂l−1 ∂zl−1 nf◦Φ−1 p(0,...,0,−δ0) −Zδ0 t Da∂l ∂zl nf◦Φ−1 p(0,...,0,−u)du =A+B. Note first that |A|is bounded by a constant depending only on |a|and f, because Φ−1 p(0,...,0,−δ0) belongs to a fixed compact of Ω. If αis not an integer, (3.3) implies |B| ≤ KFa/2(p, ρ(p)) Zδ0 t du (|ρ(p)|+u)l−α ≤K1Fa/2(p, |ρ(p)|)1 (|ρ(p)|+t)l−α−1, and a simple iteration gives the estimate of the proposition. If αis an integer (then l=α+ 1), the same inequality implies |B| ≤ KFa/2(p, ρ(p)) |ρ(p)|1/2M2Zδ0 t du (|ρ(p)|+u)1+ 1 2M2 ≤KFa/2(p, |ρ(p)|)|ρ(p)| (|ρ(p)|+t)1/2M2 , and we may follow the same proof as before. 3.3.2. Proof of Theorem 3.3. The result is a trivial consequence of Theorem 3.2 and the following lemma: Lemma 3.9. For α < 1/M, a holomorphic function in Λα(Ω) verifies |f(p)−f(q)|.ρ(p, q)α,p, q ∈Ω. Estimates for the Bergman and Szeg¨ o Projections 435 Proof: Let us denote by Z= (Zi) the coordinate system associated to each Vkas in the beginning of Section 3.1. We showed, at the beginning of the proof of Lemma 3.3, that if Z∈Vk0∩Ω, the point Zt= (Z1,...,Zn−1, Zn−t) belongs to Uk0∩Ω for 0 ≤t≤˜r. To prove the lemma, fbeing in Λα, it suffices to establish the estimate when ρ(p, q) = γ(p, q)+ρ(p)+ρ(q), and, fbeing bounded, when pand q are in some Vlfor l≥1 and so that ρ(p, q) is small (i.e. ≪˜r). To simplify the notations, let τ=ρ(p, q), and consider the associated points pτand qτ(in the Zcoordinates). By Proposition 2.8, qτ∈Bexp(pτ, Kτ), and there exist coefficients (ui)1≤i≤2nsuch that qτ= exppτ(u1,...,u2n) and |ui|.R′ i(pτ, τ). Thus, if we denote fτ(p) = f(p)−f(pτ), and the similarly for q, we have f(p)−f(q) = fτ(p)−fτ(q) + f(pτ)−f(qτ) =fτ(p)−fτ(q) + Z1 0XuiYif(exppτ(tu1,...,tu2n)) dt. The function fbeing holomorphic the last integral can be written Z1 0XviLif(exppτ(tu1,...,tu2n)) dt, where |vi|.Ri(pτ, τ). Now, in the coordinate system defined by Φw(t), w(t) = exppτ(tu1,...,tu2n), Li(w(t)) = ∂ ∂zi, and Proposition 3.3 gives (recall α < 1/M) |Lif(w(t))|.F1/2 i(w(t),|ρ(w(t))|)|ρ(w(t))|α. But, ρ(w(t)) belongs to [ρ(pτ), ρ(qτ)], and thus, by the properties of coordinate system (Zi), |ρ(w(t))| ∼ τ. Then, Proposition 2.9 gives Fi(wt,|ρ(w(t))|)∼Fi(pτ, τ) and |f(pτ)−f(qτ)|.τα. fbeing holomorphic, d dt (f(pt)) = ∂f ∂Zn(pt), then, if we write fτ(p) as an integral of d dt (f(pt) between 0 and τ,α < 1 and Lemma 3.7 imply (recall |ρ(pt)|&t) |fτ(p)| ≤ C(f)Zτ 0 tα−1dt =C(f)τα. The similar inequality is clearly true replacing pby q. Then the lemma is proved. 436 Ph. Charpentier, Y. Dupain 3.3.3. The case α≥1/M.The spaces Λαare well adapted to our purpose for all α > 0, but those given by Definition 3.2 cannot be used, for α≥1/M, to obtain results on the Bergman projection (for example, if α > 1/M it forces the function to be constant in certain directions). To get through the general case we have to define the spaces Γαin another way. The functions in Λαcan be characterized by the existence of a decomposition in a sum of functions whose derivatives are well controlled in an isotropic way (Proposition 3.2). Similarly, for α < 1/M, the space Γα can be characterized (see Lemma 3.9 for the fact that if fsatisfy Definition 3.3 below then f∈Γα(Ω), and the methods of [MS94] and Proposition 2.5 for the converse) using such a decomposition but with non-isotropic estimates for the derivatives. It is then natural to define the space Γα, for all α > 0, as follows (recall that we denote by Z= (Zi) the coordinate system associated to each Vlas in the beginning of Section 3.1). Definition 3.3. A function fbelongs to Γα, 0 < α < +∞, if it is in Λα(V0) and, for every l≥1 and every integer kthere exists functions fkand gk, defined in Vl∩Ω, such that f=fk+gkin Vl∩Ω and: (1) kfkkL∞.f2−kα. (2) If Z∈Vl∩Ω and |ρ(Z)| ≥ 2−k, for all integer m≥Mα, |∇mgk(Z)|.f,m 2mk2−kα. (3) If Z∈Vl∩Ω and |ρ(Z)|<2−k,Dabeing a derivative of length |a| ≥ Mα with respect to the coordinate system Φ = ΦZassociated to Z and δ=|ρ(Z)|, Da(gk◦Φ−1)(0).f,a Fa/2(Z, 2−k)2−kα. Theorem 3.4. Let Ωbe a bounded pseudo-convex domain of finite type in Cnwith locally diagonalizable Levi form. For 0< α < +∞, the Bergman projection maps continuously Λαinto Γα. Once again, the result follows the next lemma: Lemma 3.10. For all α > 0, each function in Λα(Ω) belongs to Γα(Ω). Proof: Recall that we showed, at the beginning of the proof of Lemma 3.3, that if Z∈Vl∩Ω, the point Zt= (Z1,...,Zn−1, Zn−t) belongs to Ul∩Ω for 0 ≤t≤˜r. Estimates for the Bergman and Szeg¨ o Projections 437 Let s > α be an integer. By integration by parts, we have, for Z∈ Vl∩Ω and fholomorphic in Ul∩Ω, f(Z) = (−1)s−1 (s−1)! Z˜r 0 ts−1ds dtsf(Zt)dt +E(Z), where Eis C∞on Vl∩Ω. For ksuch that 2−k≤˜r, we define fkand gkby fk(Z) = (−1)s−1 (s−1)! Z2−k 0 ts−1ds dtsf(Zt)dt, gk(Z) = (−1)s−1 (s−1)! Z˜r 2−k ts−1ds dtsf(Zt)dt +E(Z) so that f=fk+gk. Then Lemma. With the previous notations, if f∈Λα(Ω) is holomorphic, we have |fk(Z)| ≤ C(f)2−kα and, if Φpis the change of coordinates associated to p=Z2−kand δ= |ρ(p)|and z= Φp(Z), for all derivative Da, Da(gk◦Φ−1 p)(z)≤C(f, a)hFa/2(Z2−k,|ρ(Z2−k)|)|ρ(Z2−k)|α+ 1i. Proof: By the choice of the coordinate system, |ρ(Zt)|&t, and, fbeing holomorphic, ds dts(f(Zt)) = ∂sf ∂Zs n(Zt), then s > α and Lemma 3.7 imply |fk(Z)| ≤ C(f)Z2−k 0 t−s+α+s−1dt =C(f)2−kα. To estimate the derivatives of gk, let us first integrate by parts: gk(z) = ˜ E(Z) + s−1 X l=0 ∗2−kl dl dtlf(Zt)|t=2−k=˜ E(Z) + s−1 X l=0 ∗2−kl dlf dZl n (Z2−k), where the ∗are absolute constants depending only on sand ˜ Eis C∞ on Vl∩Ω. Then, Proposition 3.3, (2.1) and the fact that |ρ(Z2−k)|&2−kimply Da(gk◦Φ−1 p)(z)≤C(f)hFa/2(Z2−k,|ρ(Z2−k)|)|ρ(Z2−k)|α+1i. 444 Ph. Charpentier, Y. Dupain bumping is given by a theorem of S. Cho [Cho92] which is valid in any finite type domain. As shown by J. D. McNeal in [MN01], Theorem 5.1 and the estimates on the Bergman kernel recalled in Section 2.3 give an estimate for the ∂problem for the norms associated to an invariant metric: Theorem 5.3. Let Ωbe a domain in Cnwhich satisfies the hypothesis of Theorem 3.1. Then there exists a constant C > 0such that, for any (n, 1)-form α,∂-closed, there exists a solution uof the equation ∂u =α satisfying kukI≤CkαkI, where k.kIdenotes the norm associated to any of the metrics of Caratheodory, Bergman or Kobayashi. References [AS79] P. Ahern and R. Schneider, Holomorphic Lipschitz functions in pseudoconvex domains, Amer. J. Math. 101(3) (1979), 543–565. [AC99] H. Ahn and S. Cho, On the mapping properties of the Bergman projection on pseudoconvex domains with one degenerate eigenvalue, Complex Variables Theory Appl. 39(4) (1999), 365–379. [Cat89] D. W. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200(3) (1989), 429–466. [CG94] D.-C. Chang and S. Grellier, Estimates for the Szeg¨o kernel on decoupled domains, J. Math. Anal. Appl. 187(2) (1994), 628–649. [CNS92] D. C. Chang, A. Nagel and E. M. Stein, Estimates for the ∂-Neumann problem for pseudoconvex domains in C2 of finite type, Proc. Nat. Acad. Sci. U.S.A. 85(23) (1988), 8771–8774. [CD] Ph. Charpentier and Y. Dupain, Geometry of pseudoconvex domains of finite type with locally diagonalizable Levi form and Bergman kernel, J. Math. Pures Appl. (9) 85(1) (2006), 71–118. [Cho92] S. Cho, Extension of complex structures on weakly pseudoconvex compact complex manifolds with boundary, Math. Z. 211(1) (1992), 105–119. Estimates for the Bergman and Szeg¨ o Projections 445 [Cho02a] S. Cho, Estimates of invariant metrics on pseudoconvex domains with comparable Levi form, J. Math. Kyoto Univ. 42(2) (2002), 337–349. [Cho02b] S. Cho, Estimates of the Bergman kernel function on pseudoconvex domains with comparable Levi form, J. Korean Math. Soc. 39(3) (2002), 425–437. [Cho03] S. Cho, Boundary behavior of the Bergman kernel function on pseudoconvex domains with comparable Levi form, J. Math. Anal. Appl. 283(2) (2003), 386–397. [Chr88] M. Christ, Regularity properties of the ∂bequation on weakly pseudoconvex CR manifolds of dimension 3, J. Amer. Math. Soc. 1(3) (1988), 587–646. [DJS85] G. David, J.-L. Journ´ e and S. Semmes, Op´erateurs de Calder´on-Zygmund, fonctions para-accr´etives et interpolation, Rev. Mat. Iberoamericana 1(4) (1985), 1–56. [Der99] M. Derridj, R´egularit´e h¨old´erienne pour b, sur des hypersurfaces de Cn, `a forme de Levi d´ecomposable en blocs, J. Geom. Anal. 9(4) (1999), 627–652. [FK88] C. L. Fefferman and J. J. Kohn, H¨older estimates on domains of complex dimension two and on three-dimensional CR manifolds, Adv. in Math. 69(2) (1988), 223–303. [FKM90] C. L. Fefferman, J. J. Kohn and M. Machedon, H¨older estimates on CR manifolds with a diagonalizable Levi form, Adv. Math. 84(1) (1990), 1–90. [Koe02] K. Koenig, On maximal Sobolev and H¨older estimates for the tangential Cauchy-Riemann operator and boundary Laplacian, Amer. J. Math. 124(1) (2002), 129–197. [Mac88] M. Machedon, Szeg¨o kernels on pseudoconvex domains with one degenerate eigenvalue, Ann. of Math. (2) 128(3) (1988), 619–640. [McN94] J. D. McNeal, Estimates on the Bergman kernels of convex domains, Adv. Math. 109(1) (1994), 108–139. [MN01] J. D. McNeal, Invariant metric estimates for ∂on some pseudoconvex domains, Ark. Mat. 39(1) (2001), 121–136. [MS94] J. D. McNeal and E. M. Stein, Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J. 73(1) (1994), 177–199. [MS97] J. D. McNeal and E. M. Stein, The Szeg¨o projection on convex domains, Math. Z. 224(4) (1997), 519–553. 446 Ph. Charpentier, Y. Dupain [NRSW89] A. Nagel, J.-P. Rosay, E. M. Stein and S. Wainger, Estimates for the Bergman and Szeg¨o kernels in C2,Ann. of Math. (2) 129(1) (1989), 113–149. [PS77] D. H. Phong and E. M. Stein, Estimates for the Bergman and Szeg¨o projections on strongly pseudo-convex domains, Duke Math. J. 44(3) (1977), 695–704. Institut de Math´ematiques Universit´e de Bordeaux I 33405 Talence France E-mail address:Philippe.Charpentie[email protected] E-mail address:[email protected] Rebut el 7 de setembre de 2005.