Hilbert-valued forms and barriers on weakly pseudoconvex domains

Publicacions Matem`atiques, Vol 42 (1998), 423–433. HILBERT-VALUED FORMS AND BARRIERS ON WEAKLY PSEUDOCONVEX DOMAINS Vincent Thilliez Abstract We introduce an alternative proof of the existence of certain Ck barrier maps, with polynomial explosion of the derivatives, on weakly pseudoconvex domains in Cn. Barriers of this sort have been constructed very recently by J. Michel and M.-C. Shaw, and have various applications. In our paper, the adaptation of H¨ormander’s L2techniques to suitable vector-valued functions allows us to give a very simple approach of the problem and to improve some aspects of the result of Michel and Shaw, regarding the explosion of the barrier and the regularity assumption on the domain. Introduction and statement of results Let Ω be a bounded weakly pseudoconvex domain in Cnwith Lip1 boundary ∂Ω and let Ω∗be a bounded domain with Lip1boundary such that ¯ Ω⊂Ω∗. Put U=Ω ∗ \ ¯ Ω. The aim of the present paper is to give a simple proof of the following theorem. Main Theorem. Let kbe a positive integer. There exist functions w1,... ,w nbelonging to Ck(Ω ׯ U)and satisfying the following properties: (1) For any ζin ¯ Uand any j=1,... ,n the function wj(·,ζ)is holomorphic in Ω. (2) For any (z,ζ)in Ωׯ U, one has n X j=1 wj(z,ζ)(zj−ζj)=1. Keywords. Barrier maps, weakly pseudoconvex domains, Ckestimates. 1991 Mathematics subject classifications: 32A25, 35N15, 46E99. 424 V. Thilliez (3) There exists a constant Ck, depending only on k,Ωand Ω∗, such that the estimate |D` ζwj(z,ζ)|≤C kdist(z,∂Ω)−((n+1)k+n2+5n+2) holds for any j=1,... ,n, any derivation D` ζof order `≤kwith respect to ζ, and any (z,ζ)in Ωׯ U. It is essential to remark that a quite similar result has been obtained previously by Joachim Michel and Mei-Chi Shaw [8]. However, the proof in [8], using the powerful machinery from [7], seems much more technical than ours. It also requires the boundary of Ω to be C2-smooth, while we require only Lip1regularity. Moreover, the article [8] yelds Ckfunctions wj(z,ζ) having their growth controlled by dist(z,∂Ω)−(ank2+bn) (for suitable an,bn) instead of dist(z,∂Ω)−(ank+bn)as in statement (3). So our version of the theorem improves the main result in [8] with respect to both aspects. It should be also pointed out that a so-called “barrier map” such as (w1,... ,w n) has many applications, for which we refer the reader to [8], [9], [10], [11]. Of course, our improved barrier allows one to obtain, more or less mechanically, some improvements of these applications too (see e.g. the estimates for the solution of ¯ ∂on annuli given by Theorem 2 in [8]). The essential trick of our proof of the main theorem consists in “hiding” the parameter ζ∈Uby means of H¨ormander-type L2techniques applied to square-integrable functions taking their values in a Sobolev space Hm(U). We organize this as follows. Section 1 gathers some properties of Hilbert-valued differential forms needed in the rest of the paper. In section 2, a classical homological device leads to a Hm(U)-valued barrier map. Then Ckregularity and estimates for the barrier are obtained by elementary arguments, for a suitable choice of m. 1. The ¯ ∂-operator for Hilbert-valued forms 1.1. Notations. Throughout this section, Hwill denote a complex separable Hilbert space, h·|·i Hits inner product (with the convention that hh|h0iHis linear with respect to h0) and k·k Hthe corresponding norm. It is assumed that there exists a conjugation, in other words an antilinear isometric involution h∈H7→¯ h∈H(the notation ¯ hshould cause no confusion). For (h, h0)∈H×H, put B(h, h0)=h ¯ h|h 0 i H . Then Bis clearly a continuous bilinear symmetric form on H. Now let qbe an integer, q≥0. Denote by Λ(0,q)the space of (0,q)-forms on Cn. Hilbert-valued forms and barriers 425 Any element ηof Λ(0,q)⊗Hcan be written η=X Q d¯zQ⊗ηQ where the summation is performed for all Q=(j 1 ,... ,j q), 1 ≤j1< ··· <j q≤n, with the standard meaning d¯zj1∧···∧d¯z j qfor d¯zQ, and with each ηQbelonging to H. Thus the inner product h·|·i H , as well as the Hilbert norm k·k H, extends naturally to Λ(0,q)⊗H by means of the formula hη|η0iH=X Q hηQ|η0 QiH. Let Ω be as in the introduction. We denote by L2(Ω; H,loc) the space of locally square-integrable H-valued functions in Ω (see [2] for the main facts about such spaces), endowed with the topology of L2-convergence on compact subsets of Ω. The corresponding space of H-valued (0,q)- differential forms L2(Ω; Λ(0,q)⊗H,loc) will be written L2 (0,q)(Ω; H,loc) for short. When H=C, this is the familiar space of (0,q)-forms with locally square-integrable coefficients used in H¨ormander’s L2theory for ¯ ∂. At last, for any map f:Ω−→ H ,anyhin Hand zin Ω, we put (1.1.1) fh(z)=hh|f(z)i H . Obviously, if fbelongs to L2(Ω; H,loc), then fhbelongs to L2(Ω; C,loc). 1.2. A ¯ ∂-operator. The process to define ¯ ∂on H-valued L2spaces is the same as for usual L2spaces. Let D(Ω; H) be the space of C∞, compactly supported, H-valued functions in Ω, endowed with its standard (LF) topology. For any fbelonging to L2(Ω; H,loc) and any j=1,... ,n we define ∂f/∂¯zjas the continuous linear form on D(Ω; H) such that ¿∂f ∂¯zj ,ϕ À=−ZΩ Bµf(z),∂ϕ ∂¯zj (z)¶dV (z) for all ϕ∈D(Ω; H). Here dV denotes the standard Lebesgue measure and h·,·i is the duality bracket. If there exists a function gjin L2(Ω; H,loc) such that the equality ZΩ B(gj(z),ϕ(z)) dV (z)=−ZΩ Bµf(z),∂ϕ ∂¯zj (z)¶dV (z) holds for any ϕin D(Ω; H) (for instance, this is straightforward when f is C1), then ∂f/∂¯zjidentifies with gj. Hence it is possible to define ¯ ∂f = n X j=1 d¯zj⊗∂f ∂¯zj 426 V. Thilliez as an element of L2 (0,1)(Ω; H,loc). This definition extends in a purely algebraic way to higher degrees, giving rise to a ¯ ∂-complex on L2 (0,q)(Ω; H,loc) spaces. We describe now some basic rules about this complex. 1.3. Lemma. Let fbe a function belonging to L2(Ω; H,loc) and such that ¯ ∂f belongs to L2 (0,1)(Ω; H,loc). Then for any hin Hand any j=1,... ,n, one has, in the sense of notation (1.1.1), ∂fh ∂¯zj =µ∂f ∂¯zj¶h in L2(Ω; C,loc). Thus if ¯ ∂f equals zero, the function fhis holomorphic (in the usual sense) in Ω. Proof: For any ψin D(Ω; C) and any zin Ω, one has fh(z)∂ψ ∂¯zj (z)=¿∂ψ ∂¯zj (z)h|f(z)ÀH =¿∂ψ ∂¯zj (z)¯ h|f(z)ÀH =Bµf(z),∂(ψ⊗¯ h) ∂¯zj (z)¶, hence ¿∂fh ∂¯zj ,ψÀequals ¿∂f ∂¯zj ,(ψ⊗¯ h)À. On the other hand, one has similarly µ∂f ∂¯zj¶h (z)ψ(z)=¿ψ(z)h|∂f ∂¯zj (z)ÀH =¿ψ(z)¯ h|∂f ∂¯zj (z)ÀH =Bµ∂f ∂¯zj (z),(ψ⊗¯ h)(z)¶, hence ¿µ∂f ∂¯zj¶h ,ψÀalso equals ¿∂f ∂¯zj ,(ψ⊗¯ h)À. This proves the first assertion of the lemma. In the particular case ¯ ∂f = 0, one gets immediately ¯ ∂fh= 0 in the usual distribution sense. The second assertion follows. Now let (eν)ν∈Zbe a Hilbert basis for H. For any map f:Ω−→ Λ(0,q)⊗H,f=PQd¯z Q⊗f Q , we define the νth component fνof fby fν(z)=X Q he ν|f Q (z)i Hd¯z Q ,z∈Ω. Hilbert-valued forms and barriers 427 In particular, for q=0,one has fν(z)=he ν|f(z)i H=f e ν(z)inthe sense of (1.1.1). The following properties are then easy to check: (i) For any zin Ω, the equality f(z)=P ν∈Z f ν (z)⊗e νholds in Λ(0,q)⊗H, and consequently kf(z)k2 H=Pν∈Z|fν(z)|2. (ii) If fbelongs to L2 (0,q)(Ω; H,loc), then each fνbelongs to L2 (0,q)(Ω; C,loc) and the equality f=Pν∈Zfν⊗eνholds in L2 (0,q)(Ω; H,loc). (iii) If moreover ¯ ∂f belongs to L2 (0,q+1)(Ω; H,loc), then for each νthe form ¯ ∂(fν) belongs to L2 (0,q+1)(Ω; C,loc) and satisfies ¯ ∂(fν)= ( ¯ ∂f)ν. Although simple, these properties are the key to the following H¨ormander-type result. 1.4. Proposition. Let ϕbe a continuous plurisubharmonic function in Ω.Letqbe an integer, q≥0. Then for any form fin L2 (0,q+1)(Ω; H,loc) satisfying ¯ ∂f =0and ZΩ kf(z)k2 He−ϕ(z)dV (z)<∞, there exists a form gin L2 (0,q)(Ω; H,loc) such that ¯ ∂g =fand ZΩ kg(z)k2 He−ϕ(z)dV (z)≤CΩZΩ kf(z)k2 He−ϕ(z)dV (z), where CΩis a positive constant depending only on the diameter of Ω. Proof: By properties (i)-(iii) above, it is easily seen that for each ν, the form fνlies in L2 (0,q+1)(Ω; C,loc), is ¯ ∂-closed, and has finite weighted norm RΩ|fν(z)|2e−ϕ(z)dV (z). Applying a standard result of H¨ormander ([5, 4.4.2]), we solve ¯ ∂componentwise: one gets a form gν in L2 (0,q)(Ω; C,loc) such that ¯ ∂gν=fνand RΩ|gν(z)|2e−ϕ(z)dV (z)≤ CΩRΩ|fν(z)|2e−ϕ(z)dV (z). This, together with the elementary fact kgν(z)⊗eνkH=|gν(z)|, allows easily to check that the series Pν∈Zgν⊗eν converges in L2 (0,q)(Ω; H,loc) to a form gsatisfying all the required properties. 428 V. Thilliez 1.5. Remark. When the Hilbert space His chosen as a Sobolev space (this choice will be adopted in the next section), Proposition 1.4 can be viewed as a result on parameter dependence for ¯ ∂in weighted L2 spaces, with a weight which does not depend on the parameter. Different results for parameter-depending weights can be found in [4]. 2. Construction of a barrier map 2.1. Setting of the construction. Let Uand kbe as in the introduction. From now on, the Hilbert space Hwill be the Sobolev space Hm(U) with m=n+k+ 1. By Sobolev’s lemma, this choice of m ensures the inclusion Hm(U)⊂Ck(¯ U) with continuous injection (this is the only reason why some regularity for ∂Ω —namely Lip1, see e.g. [1]— is required in this work). There exists a positive constant Akdepending only on k, Ω and Ω∗, such that for any integer `with 0 ≤`≤kand any derivation D` ζof order `, the estimate (2.1.1) |D` ζF(ζ)|≤A k kFk H m (U) holds for any ζ∈¯ Uand any F∈Hm(U). In particular, F7→ F(ζ) is a continuous linear form on the Hilbert space Hm(U), so there exists h(ζ)inH m (U) such that one has F(ζ)=hh(ζ)|Fi H m (U)for any F∈Hm(U). Consequently, any map fbelonging to L2(Ω; Hm(U),loc) satisfies, with the notation (1.1.1), (2.1.2) f(z,ζ)=f h(ζ) (z) for all (z,ζ)∈Ωׯ U. This, together with Lemma 1.3, implies clearly the following fact: if ¯ ∂f (in the sense of section 1) belongs to L2 (0,1)(Ω; Hm(U),loc), then for any (z,ζ)∈Ωׯ U, one can view (¯ ∂f)(z,ζ)as¯ ∂ z (f(z,ζ)) computed in the usual space L2 (0,1)(Ω; C,loc), with ζfixed. This remark extends obviously to forms of higher degrees. Now we follow the pattern of the classical work of L. H¨ormander [6]. For any integer sand any real γwith s≥0, γ≥0, let Ls,q γbe the space of families w=(w I ;I∈{1,... ,n} s) of elements of L2 (0,q)(Ω; Hm(U),loc) satisfying, for each multi-index I∈{1,... ,n} s, both following properties: (2.1.3) ZΩ kwI(z,·)k2 Hm(U)dist(z,∂Ω)2γdV (z)<∞, (2.1.4) the map (t1,... ,t s)7−→ X I∈{1,... ,n}s t1 i1...t s i sw I is s-linear alternating with respect to (t1,... ,t s)∈(C n) s. Hilbert-valued forms and barriers 429 The ¯ ∂-operator defined in section 1 extends to Ls,q γby putting ¯ ∂w = (¯ ∂wI;I∈{1,... ,n} s). For j=1,... ,n and for any fin L2(Ω; Hm(U),loc), let σjfbe the map which, to each point z∈Ω, associates the function (σjf)(z,·):ζ∈U7−→ (zj−ζj)f(z,ζ). It is not difficult to see that σjfitself belongs to L2(Ω; Hm(U),loc) and that for any z∈Ω, one has (2.1.5) k(σjf)(z,·)kHm(U)≤Ckf(z,·)kHm(U) with a suitable positive constant C, depending only on mand on the size of Ω, Ω∗. In particular, σjacts componentwise as a continuous linear operator on all the spaces L2 (0,q)(Ω; Hm(U),loc). Now let wbe an element of Ls+1,q γ. For each I=(i 1 ,... ,i s)in {1,... ,n} s,weput (Pw)I= n X j=1 σjw(I,j), with (I,j)=(i 1 ,... ,i s,j). Note that for (z,ζ)∈Ω×U, one gets explicitely (Pw)I(z,ζ)= n X j=1 (zj−ζj)w(I,j)(z,ζ). Also, by the properties of σjdescribed above (especially (2.1.5)), this defines an element Pw of Ls,q γ. It is straightforward to check that the operator Pcommutes with ¯ ∂and that it satisfies P2= 0, giving a double complex just as in [6]. 2.2. Lemma. Let wbe an element of Ls,q+1 γsuch that ¯ ∂w =0. Then there exists vin Ls,q γsuch that ¯ ∂v =w. Proof: Obvious by Proposition 1.4 applied with ϕ(z)=−2γLog dist(z,∂Ω). 430 V. Thilliez 2.3. Lemma. Let vbe an element of Ls,q γsuch that Pv =0. Then there exists win Ls+1,q γ+m+1 such that Pw =v. If one has additionally ¯ ∂v =0, then ¯ ∂w belongs to Ls+1,q+1 γ+m+2 . Proof: Following [6, Lemma 6], we define explicitely wI(z,ζ)= s+1 X k=1 (−1)s+1−k¯zik−¯ ζik |z−ζ|2vI\(ik)(z,ζ) for I=(i 1 ,... ,i s+1)∈{1,... ,n} s+1,(z,ζ)∈Ω×Uand I\(ik)=“I with the index ikdeleted”. Just as in [6], one gets Pw =vand part (2.1.4) of the definition of Ls+1,q γis satisfied. The only things to check are the growth conditions. But for any (z,ζ)∈Ω×Uand any derivation D` ζof order at most m, it is clear that ¯¯¯¯ D` ζµ¯zik−¯ ζik |z−ζ|2¶¯¯¯¯ ≤C0dist(z,∂Ω)−(m+1), hence kwI(z,·)k2 Hm(U)≤C00 dist(z,∂Ω)−2(m+1) s+1 X k=1 kvI\(ik)(z,·)k2 Hm(U), for suitable positive constants C0and C00, depending only on mand on the size of Ω, Ω∗. From these estimates, it is not difficult to prove that wbelongs to Ls+1,q γ+m+1. The claim that ¯ ∂w belongs to Ls+1,q+1 γ+m+2 when ¯ ∂v vanishes comes similarly, since one gets then (¯ ∂wI)(z,ζ)= s+1 X k=1 (−1)s+1−k¯ ∂zµ¯zik−¯ ζik |z−ζ|2¶∧vI\(ik)(z,ζ), by elementary computations using the remarks in 2.1. We are now ready to state the key result of this paper. 2.4. Theorem. For every element vof Ls,q γsatisfying ¯ ∂v =0and Pv =0, there exists an element wof Ls+1,q γ+m+1+( m 2+1)(2n−q−s)such that ¯ ∂w =0and Pw =v. Hilbert-valued forms and barriers 431 Proof: We essentially mimic the proof of Theorem 7 in [6], by decreasing induction on qand s. The only different point is that a special attention has to be paid to the evolution of the index γ(invisible in [6]) during the induction. First, the result is trivial for q>nor s>n. Now assume that it holds for data in Ls+1,q+1 γ0where γ0is an arbitrary positive real number, and s≤n,q≤n. Let vbe as in the statement of the theorem. By 2.3, there exists w0in Ls+1,q γ+m+1 such that Pw0=vand ¯ ∂w0∈Ls+1,q+1 γ+m+2 . Since ¯ ∂(¯ ∂w0)=0andP( ¯ ∂w0)=¯ ∂(Pw0)=¯ ∂v = 0, the induction assumption applies to the data ¯ ∂w0with γ0=γ+m+ 2. One gets w00 in Ls+2,q+1 γ00 with γ00 =γ0+m+1+(m 2+1)(2n−(q+1)−(s+1)) = γ+m+1+(m 2+ 1)(2n−q−s), such that Pw00 =¯ ∂w0and ¯ ∂w00 =0. By 2.2, we can find w000 in Ls+2,q γ00 such that ¯ ∂w000 =w00. Put w=w0−Pw000. Since one has w0∈Ls+1,q γ+m+1 ⊂Ls+1,q γ00 , we see that wbelongs to Ls+1,q γ00 . Finally one checks that ¯ ∂w =¯ ∂w0−¯ ∂(Pw000)=Pw00 −P(¯ ∂w000)=0and Pw =Pw0=v. 2.5. Proof of the main theorem of the introduction: The application of 2.4 with q=s=γ= 0 and v≡1 yelds functions w1,... ,w nbelonging to L2(Ω; Hm(U),loc) and satisfying the following properties: ¯ ∂wj= 0 (in the sense of section 1) for j=1,... ,n,(2.5.1) n X j=1 wj(z,ζ)(zj−ζj) = 1 for any (z,ζ)∈Ω×U,(2.5.2) ZΩ kwj(z,·)k2 Hm(U)dist(z,∂Ω)2(n+1)(m+2)−2dV (z)<∞(2.5.3) for j=1,... ,n. Observe that property (2.5.2) is true as well for ζ∈¯ U, since the functions wj(z,·) extend continuously to ¯ Ufor all fixed z∈Ω (see 2.1). Also, in virtue of Lemma 1.3, of the representation formula (2.1.2) and of (2.5.1), the function z7→ wj(z,ζ) is holomorphic in Ω for each ζ∈¯ U.By the same ideas, this can be shown as well for the functions D` ζwj(·,ζ), `≤k. Thus we have proved part (1) and (2) of the main theorem. We proceed now to prove part (3). For z∈Ω, let Bzbe the euclidean ball with center zand radius 1 2dist(z,∂Ω). The plurisubharmonicity of |D` ζwj(·,ζ)| 2yelds |D` ζwj(z,ζ)|2≤Cdist(z,∂Ω)−2nZBz |D` ζwj(t, ζ)|2dV (t)