HILBERT-VALUED FORMS AND BARRIERS ON WEAKLY PSEUDOCONVEX DOMAINS

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 Hörmander’s L2 techniques 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 C n with Lip 1 boundary ∂Ω and let Ω * be a bounded domain with Lip 1 boundary 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 k be a positive integer.There exist functions w 1 , . . ., w n belonging to C k (Ω × Ū ) and satisfying the following properties: (1) For any ζ in Ū and any j = 1, . . ., n the function w j (•, ζ) is holomorphic in Ω. (2) For any (z, ζ) in Ω × Ū , one has 1.The ∂-operator for Hilbert-valued forms 1.1.Notations.Throughout this section, H will denote a complex separable Hilbert space, • | • H its inner product (with the convention that h | h H is linear with respect to h ) and • H the corresponding norm.It is assumed that there exists a conjugation, in other words an antilinear isometric involution h ∈ H → h ∈ H (the notation h should cause no confusion).For (h, h ) ∈ H × H, put Then B is clearly a continuous bilinear symmetric form on H. Now let q be an integer, q ≥ 0. Denote by Λ (0,q) the space of (0, q)-forms on C n .Any element η of Λ (0,q) ⊗ H can be written where the summation is performed for all Q = (j 1 , . . ., j q ), 1 ≤ j 1 < • • • < j q ≤ n, with the standard meaning dz j1 ∧ • • • ∧ dz jq for dz Q , and with each η Q belonging to H. Thus the inner product • | • H , as well as the Hilbert norm • H , extends naturally to Λ (0,q) ⊗ H by means of the formula Let Ω be as in the introduction.We denote by L 2 (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 L 2 -convergence on compact subsets of Ω.The corresponding space of H-valued (0, q)differential forms L 2 (Ω; Λ (0,q) ⊗ H, loc) will be written L 2 (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 Hörmander's L 2 theory for ∂.At last, for any map f : Ω −→ H, any h in H and z in Ω, we put Obviously, if f belongs to L 2 (Ω; H, loc), then f h belongs to L 2 (Ω; C, loc).

A ∂-operator.
The process to define ∂ on H-valued L 2 spaces is the same as for usual L 2 spaces.Let D(Ω; H) be the space of C ∞ , compactly supported, H-valued functions in Ω, endowed with its standard (LF) topology.For any f belonging to L 2 (Ω; H, loc) and any j = 1, . . ., n we define ∂f /∂ zj as the continuous linear form on D(Ω; H) such that Here dV denotes the standard Lebesgue measure and •, • is the duality bracket.If there exists a function g j in L 2 (Ω; H, loc) such that the equality holds for any ϕ in D(Ω; H) (for instance, this is straightforward when f is C 1 ), then ∂f /∂ zj identifies with g j .Hence it is possible to define as an element of L 2 (0,1) (Ω; H, loc).This definition extends in a purely algebraic way to higher degrees, giving rise to a ∂-complex on L 2 (0,q) (Ω; H, loc) spaces.We describe now some basic rules about this complex.
1.3.Lemma.Let f be a function belonging to L 2 (Ω; H, loc) and such that ∂f belongs to L 2 (0,1) (Ω; H, loc).Then for any h in H and any j = 1, . . ., n, one has, in the sense of notation (1.1.1), Thus if ∂f equals zero, the function f h is holomorphic (in the usual sense) in Ω.
Proof: For any ψ in D(Ω; C) and any z in Ω, one has This proves the first assertion of the lemma.In the particular case ∂f = 0, one gets immediately ∂f h = 0 in the usual distribution sense.The second assertion follows.
Now let (e ν ) ν∈Z be a Hilbert basis for H.For any map f : In particular, for q = 0, one has f ν (z) = e ν | f(z) H = f eν (z) in the sense of (1.1.1).The following properties are then easy to check: (i) For any z in Ω, the equality f (z) = ν∈Z f ν (z) ⊗ e ν holds in Λ (0,q) ⊗ H, and consequently (ii) If f belongs to L 2 (0,q) (Ω; H, loc), then each f ν belongs to L 2 (0,q) (Ω; C, loc) and the equality f Although simple, these properties are the key to the following Hörmander-type result.

Proposition.
Let ϕ be a continuous plurisubharmonic function in Ω.Let q be an integer, q ≥ 0. Then for any form f in L 2 (0,q+1) (Ω; H, loc) there exists a form g in L 2 (0,q) (Ω; H, loc) such that ∂g = f and where C Ω is a positive constant depending only on the diameter of Ω.
1.5.Remark.When the Hilbert space H is 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 L 2 spaces, with a weight which does not depend on the parameter.Different results for parameter-depending weights can be found in [4].

Construction of a barrier map
2.1.Setting of the construction.Let U and k be as in the introduction.From now on, the Hilbert space H will be the Sobolev space H m (U ) with m = n + k + 1.By Sobolev's lemma, this choice of m ensures the inclusion H m (U ) ⊂ C k ( Ū ) with continuous injection (this is the only reason why some regularity for ∂Ω -namely Lip 1 , see e.g.[1]is required in this work).There exists a positive constant A k depending only on k, Ω and Ω * , such that for any integer with 0 ≤ ≤ k and any derivation D ζ of order , the estimate holds for any ζ ∈ Ū and any F ∈ H m (U ).In particular, F → F (ζ) is a continuous linear form on the Hilbert space H m (U ), so there ex- This, together with Lemma 1.3, implies clearly the following fact: if ∂f (in the sense of section 1) belongs to L 2 (0,1) (Ω; H m (U ), loc), then for any (z, ζ) ∈ Ω × Ū , one can view ( ∂f )(z, ζ) as ∂z (f(z, ζ)) computed in the usual space L 2 (0,1) (Ω; C, loc), with ζ fixed.This remark extends obviously to forms of higher degrees.Now we follow the pattern of the classical work of L. Hörmander [6].For any integer s and any real γ with s ≥ 0, γ ≥ 0, let L s,q γ be the space of families w = (w I ; I ∈ {1, . . ., n} s ) of elements of L 2 (0,q) (Ω; H m (U ), loc) satisfying, for each multi-index I ∈ {1, . . ., n} s , both following properties: (2.1.3) The ∂-operator defined in section 1 extends to L s,q γ by putting ∂w = ( ∂w I ; I ∈ {1, . . ., n} s ).
For j = 1, . . ., n and for any f in L 2 (Ω; H m (U ), loc), let σ j f be the map which, to each point z ∈ Ω, associates the function It is not difficult to see that σ j f itself belongs to L 2 (Ω; H m (U ), loc) and that for any z ∈ Ω, one has with a suitable positive constant C, depending only on m and on the size of Ω, Ω * .In particular, σ j acts componentwise as a continuous linear operator on all the spaces L 2 (0,q) (Ω; H m (U ), loc).Now let w be an element of L s+1,q γ .For each I = (i 1 , . . ., i s ) in {1, . . ., n} s , we put with (I, j) = (i 1 , . . ., i s , j).Note that for (z, ζ) ∈ Ω × U , one gets explicitely Also, by the properties of σ j described above (especially (2.1.5)),this defines an element P w of L s,q γ .It is straightforward to check that the operator P commutes with ∂ and that it satisfies P 2 = 0, giving a double complex just as in [6].

Lemma. Let w be an element of L s,q+1
γ such that ∂w = 0. Then there exists v in L s,q γ such that ∂v = w.

Lemma.
Let v be an element of L s,q γ such that P v = 0. Then there exists w in L s+1,q γ+m+1 such that P w = v.If one has additionally ∂v = 0, then ∂w belongs to L s+1,q+1 γ+m+2 .
Proof: Following [6, Lemma 6], we define explicitely with the index i k deleted".Just as in [6], one gets P w = v and part (2.1.4) of the definition of L s+1,q γ is satisfied.The only things to check are the growth conditions.But for any (z, ζ) ∈ Ω × U and any derivation D ζ of order at most m, it is clear that for suitable positive constants C and C , depending only on m and on the size of Ω, Ω * .From these estimates, it is not difficult to prove that w belongs to L s+1,q γ+m+1 .The claim that ∂w belongs to L s+1,q+1 γ+m+2 when ∂v vanishes comes similarly, since one gets then by elementary computations using the remarks in 2.1.
We are now ready to state the key result of this paper.

Theorem.
For every element v of L s,q γ satisfying ∂v = 0 and P v = 0, there exists an element w of L s+1,q γ+m+1+( m Proof: We essentially mimic the proof of Theorem 7 in [6], by decreasing induction on q and 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 > n or s > n.Now assume that it holds for data in L s+1,q+1 γ where γ is an arbitrary positive real number, and s ≤ n, q ≤ n.Let v be as in the statement of the theorem.By 2.3, there exists w in L s+1,q γ+m+1 such that P w = v and ∂w ∈ L s+1,q+1 γ+m+2 .Since ∂( ∂w ) = 0 and P ( ∂w ) = ∂(P w ) = ∂v = 0, the induction assumption applies to the data ∂w with γ = γ + m + 2. One gets w in L s+2,q+1 γ with γ = γ +m+1+( m 2 +1)(2n−(q +1)−(s+1)) = γ + m + 1 + ( m 2 + 1)(2n − q − s), such that P w = ∂w and ∂w = 0.By 2.2, we can find w in L s+2,q γ such that ∂w = w .Put w = w − P w .Since one has w ∈ L s+1,q γ+m+1 ⊂ L s+1,q γ , we see that w belongs to L s+1,q γ .Finally one checks that ∂w = ∂w − ∂(P w ) = P w − P ( ∂w ) = 0 and P w = P w = v.

Proof of the main theorem of the introduction:
The application of 2.4 with q = s = γ = 0 and v ≡ 1 yelds functions w 1 , . . ., w n belonging to L 2 (Ω; H m (U ), loc) and satisfying the following properties: ∂w j = 0 (in the sense of section 1) for j = 1, . . ., n, (2.5.1) Observe that property (2.5.2) is true as well for ζ ∈ Ū , since the functions w j (z, •) extend continuously to Ū for 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 z → w j (z, ζ) is holomorphic in Ω for each ζ ∈ Ū .By the same ideas, this can be shown as well for the functions D ζ w j (•, 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 B z be the euclidean ball with center z and radius 1  2 dist(z, ∂Ω).The plurisubharmonicity of for some constant C. Besides, one has 1 ≤ 2 dist(z, ∂Ω) −1 dist(t, ∂Ω) for all t ∈ B z .From this, and from the estimate (2.1.1)applied to F = w j (t, •), we deduce But in virtue of the holomorphy of w j (t, ζ) with respect to t, of the Cauchy formula and of the estimate (3) of the theorem, we see that C(z 0 , ζ) is bounded uniformly with respect to ζ ∈ Ū .Thus E 1 tends to 0 uniformly with respect to ζ when z tends to z 0 .On the other hand, since w j (z 0 , •) is continuous, it is clear that E 2 tends to 0 when ζ tends to ζ 0 .As announced, one gets finally that w j (z, ζ) tends to w j (z 0 , ζ 0 ) as (z, ζ) tends to (z 0 , ζ 0 ).
2.6.Concluding remark.The results described above suggest a question: is it possible to obtain valuable information about parameter dependence in Skoda's theorem ([13, Théorème 1]) by the kind of argument used in the present paper ?It is known (see e.g.[3], [7], [12]) that such a result would have significant consequences in the construction of integral operators solving the ∂-equation.