THE ∂-CAUCHY PROBLEM ON WEAKLY q-CONVEX DOMAINS IN C

Let D be a weakly q-convex domain in the complex projective space CP . In this paper, the (weighted) ∂-Cauchy problem with support conditions in D is studied. Specifically, the modified weight function method is used to study the L2 existence theorem for the ∂-Neumann problem on D. The solutions are used to study function theory on weakly q-convex domains via the ∂-Cauchy problem.


Introduction and Main Results
The ∂-problem is one of the important central problems of complex variables. A classical result due to Hörmander tells us that the ∂-problem is solvable in pseudoconvex domains, and hence, pseudoconvex domains has been widely accepted as the standard domain which we can solve the ∂-problem. In [16], Ho extend this problem to weakly q-convex domains. In fact, Ho is the first person to study the ∂-problem in q-convex domains in C n . This paper is devoted to studying the L 2 ∂ Cauchy problem and the ∂-closed extension problem for forms on a weakly q-convex domain D in the complex projective space CP n . These problems were first studied by Kohn and Rossi [20] (see also [12]). They proved the holomorphic extension of smooth CR functions and the ∂-closed extension of smooth forms from the boundary bD of a strongly pseudoconvex domain to the whole domain D. The L 2 theory of these problems has been obtained for pseudoconvex domains in C n or, more generally, for domains in complex manifolds with strongly plurisubharmonic weight functions (see Chapter 9 in [6] and the references therein). The L 2 ∂ Cauchy problem was considered by Derridj [8,9]. In [30,31] Shaw has obtained a solution to this problem on a pseudoconvex domain with C 1 boundary in C n . Also, in the setting of strictly q-convex (or q-concave) domains, this problem has been studied by Sambou in his thesis (see [29]). In [1], Abdelkader-Saber studied this problem on pseudoconvex manifolds satisfing property B. In [26,27], Saber studied this problem on a weakly q-convex domain with C 1 -smooth boundary and on a q-pseudoconvex domain D in C n , 1 q n, with Lipschitz boundary. Recently, Saber [28] studied this result to a q-pseudoconvex domain D in a Stein manifold. On a pseudoconvex domain in CP n , Cao-Shaw-Wang [4] (cf. also [5]) obtained the L 2 existence theorem for the ∂-Neumann operator N and obtained the (weighted) L 2 ∂ Cauchy-problem on such domains. The aim of this paper is to extend this result to the situation in which the boundaries are assumed weakly q-convex domain D in CP n . Moreover, the solutions are used to study function theory on such domains via the ∂-Cauchy problem.

Notation and Preliminaries
Let (x 0 , x 1 , . . . , x n ) be a (fixed) homogeneous coordinates of CP n . If U 0 is the open set in CP n defined by x 0 = 0 and if (z 1 , z 2 , . . . , z n ), where z i = x i /x 0 , is the homogeneous coordinates of U 0 , we assume that The Fubini-Study metric of CP n determined by (x 0 , x 1 , . . . , x n ). This is well-known standard Kähler metric of CP n .
Let D be a bounded domain in CP n and let C ∞ p,q (D) be the space of complex-valued differential forms of class C ∞ and of type (p, q) on D. Denote by L 2 (D) the space of square integrable functions on D with respect to the Lebesgue measure in CP n , L 2 p,q (D) the space of (p, q)-forms with coefficients in L 2 (D) and L 2 p,q (D, φ) the space of (p, q)-forms with coefficients in L 2 ( D) with respect to the weighted function e −φ . For u, v ∈ L 2 p,q (D), the inner product u, v and the norm u are denoted by: where is the Hodge star operator. Let dist(z, bD) be the Fubini distance from z ∈ D to the boundary bD and let δ be a C 2 defining function for D normalized by |dδ| = 1 on bD such that Let φ t = −t log |δ|, t 0, for u, v ∈ L 2 p,q (D, φ t ), the inner product u, v φt and the norm u φt are denoted by: where (t) = δ t = δ t . Since φ t is bounded on D, the two norms · and · t are equivalent. Let ∂ : dom ∂ ⊂ L 2 p,q (D, φ t ) → L 2 p,q+1 (D, φ t ) be the maximal closure of the Cauchy-Riemann operator and ∂ * φ be its Hilbert space adjoint.
Denote by ∇ the Levi-Civita connection of CP n with the standard Fubini-Study metric ω. Let {e i } be an orthonormal basis of vector fields. For any two vector fields f, g, the curvature operator of the connection ∇ is denoted by By setting R ijkl = ω(R(e i , e j )e k , e l ), the Ricci tensor R ij is denoted by which turns out to be self-adjoint with respect to ω and the scalar curvature as the trace of the Ricci tensor.
Definition 2.1. Let D be an open set in an n-dimensional complex manifold X, let k be an integer with 1 ≤ k ≤ n − 1 and put E = X\D. The set D is said to be pseudoconvex of order k in X if, for every b ∈ E and for every coordinate neighborhood (U, (z 1 , . . . , z n )) which contains b as the origin, the set contains no points of E for some t > 0, then there exists > 0 such that for each  [16], D is said to be a Moreover, D is weakly q-convex if and only if for any z ∈ bD the sum of any q eigenvalues δ i 1 , . . . , δ iq , with distinct subscripts, of the Levi-form at z satisfies q j=1 δ i j 0 (cf. [15] and Lemma 4.7 in [34]). 16]). Let D be a smooth domain in C n and ρ be its defining function.
The following two conditions are equivalent.
(2) For any z ∈ bD the sum of any q eigenvalues ρ i 1 , . . . , ρ iq , with distinct subscripts, of the Levi-form at z satisfies q j=1 ρ i j 0. It follows from Lemma 2.1 that D is weakly q-concave if and only if for any q eigenvalues ρ i 1 , . . . , ρ iq of the Levi-form at z ∈ bD with distinct subscripts we have q j=1 ρ i j 0. Example 2.1. Let D be an open subset of an n-dimensional complex manifold X and suppose that the boundary bD is a real hypersurface of class C 2 in X, that is, there exist, for each z ∈ bD, a neighborhood U of z and a C 2 function ρ : U → R such that dρ(z) = 0 and D ∩ U = {z ∈ U : ρ(z) < 0}. Then D is pseudoconvex of order n − q in X, if and only if the Levi form ∂∂ρ has at least n − q non-negative eigenvalues on T z (bD) for each defining function ρ of D near z, where T z (bD)(⊂ T z (bD)) is the holomorphic tangent space of the real hypersurface bD at z (cf. [10,35] called such a subset D a (q − 1)-pseudoconvex open subset with C 2 boundary).
If an open set D in an n-dimensional complex manifold X is weakly q-convex, 1 ≤ q ≤ n, then D is pseudoconvex of order n − q in X. However, the converse is not valid even if X = C n (see [10] and [22]). By Fujita [13], an open subset D of C n is pseudoconvex of order n − q in C n , if and only if D has an exhaustion function which is pseudoconvex of order n − q on D. Thus, by the approximation theorem of Bungart [3], an open subset D of X is pseudoconvex of order n − q in X, if and only if D is locally q-complete with corners in X in the sense of Peternell [24].

Proposition 2.1 (Bochner-Hörmander-Kohn-Morrey formula). Let D be a compact domain with C 2 -smooth boundary bD and δ(x) = −d(x, bD). Suppose that Θ is the curvature term defined in (2.1) with respect to the Fubini-Study metric ω. Then, for any
This formula is known (cf. [2,7,15,18,19,32,36]) for some special cases, although it has not been stated in the literature in the form (2.2). If u has compact support in the interior of D, the (2.2) was proved in [2], Chapter 8 of [7] and (2.12) of [36]. The boundary term had been computed in [14], Chapter 3 by combining the Morrey-Kohn technique on the boundary with non-trivial weight function. If one combines the results of [15] and [37] with the interior formulae discussed above, one can prove that (2.2) holds for the general case with a weight function e −φ and the curvature term. Specially, for φ = 0, (2.2) was proved in [32]. The statement for (0, q)-forms and (n, q)-forms was computed in [32] and [36]. Also, following Lemma 3.3 of Henkin-Iordan [14] and its proof showed that the curvature operator Θ acting on L 2 p,q (D) is a non-negative operator.

The ∂-Cauchy Problem on Weakly q-Convex Domains
This section is devoted to showing the existence of the ∂-Neumann operator on a weakly q-convex domain D in CP n , 1 q n, and by applying these existence to solve the ∂ problem with support conditions on D. The boundary integral in (2.2) is non-negative for q 1 by the assumption on D. Also, by taking φ ≡ 0 in (2.2) and using Proposition 2.2, we find the fundamental estimate This means that has closed range and ker = {0}. Thus, one can establish the L 2 -existence theorem of the ∂-Neumann operator N .
(v) N , ∂N and ∂ * N are bounded linear operators on L 2 p,q (D).
Using the duality relations pertaining to the ∂-Neumann problem, one solve the L 2 ∂ Cauchy problem on weakly q-convex domains in CP n , 1 q n. This method was first used by Kohn-Rossi [20] for smooth forms on strongly pseudoconvex domains. More precisely, we prove the following L 2 Cauchy problem for ∂ in CP n : Proof. Let f ∈ L 2 p,q (CP n ), supp f ⊂ D, then f ∈ L 2 p,q (D). From Theorem 3.1, N n−p,n−q exists for n − q 1. Since N n−p,n−q = −1 n−p,n−q on Range n−p,n−q and Range N n−p,n−q ⊂ dom n−p,n−q , then N n−p,n−q f ∈ dom n−p,n−q ⊂ L 2 n−p,n−q (D), for q n − 1. Thus, we can define u ∈ L 2 p,q−1 (D) by Thus supp u ⊂ D and u vanishes on bD. Now, we extend u to CP n by defining u = 0 in CP n \ D. It follows from the same arguments of Theorem 9.1.2 in [6] and Theorem 2.2 in [1] that the form u satisfies the equation ∂u = f in the distribution sense in CP n . Thus the proof follows.

The Weighted ∂-Cauchy Problem
In this section, we assume that D is a weakly q-convex domain, 1 q n, with C 2 smooth boundary in CP n . Also, we will choose φ t = −t log |δ|, t > 0 in (2.2), and using Remark 2.3 and by using Proposition 2.2, the inequality (2.2) implies the weighted L 2 -existence for the ∂. Also, for u ∈ Dom( t ) of degree q 1 and for t > 0, t u t t u t . Since t is a linear closed densely defined operator, then, from [15, Theorem 1.1.1], Range( t ) is closed. Thus, from (1.1.1) in [15] and the fact that t is self adjoint, we have the Hodge decomposition . Since t is one to one on dom( t ) from (1.5.3) in [15], then there exists a unique bounded inverse operator . Therefore, we can establish the existence theorem of the inverse of t the so called weighted ∂-Neumann operator N t .
Proof. Following Theorem 4.1, N t exists for forms in L 2 n−p,n−q (D, φ t ). Thus, one can defines u t ∈ L 2 p,q−1 (D, φ t ) by Thus supp u t ⊂ D and u t vanishes on bD. Now, we extend u t to CP n by defining u t = 0 in CP n \ D. We want to prove that the extended form u t satisfies the equation Since ϑ| D = ∂ * | D , when ϑ acts in the distribution sense (see [15]), then we obtain ∂η, (t) Thus, from (4.1) and (4.2), one obtains in the distribution sense in D. Since u = 0 in CP n \D, then for u ∈ L 2 p,q (CP n )∩dom ∂ * , where the third equality holds since u = (−1) q+1 ∂N n−p,n−q f ∈ dom ∂ * . Thus ∂u t = f in the distribution sense in CP n .
Proof. Since D has C 2 smooth boundary, there exists a bounded extension operator from W s p,q (Ω) to W s p,q (CP n ) for all s 0 (cf. e.g. [33]). Letf ∈ W 1+ε p,q (CP n ) be the extension of f so thatf | Ω = f with f W 1+ε (CP n ) C f W 1+ε (Ω) .

Corollary 4.2.
Let D CP n be a weakly q-concave domain, n 2 with C 2 smooth boundary. Then, for any f ∈ W 1+ε p,q (D), where 0 ≤ p ≤ n, 1 q n − 2, p = q, and 0 ε < 1 2 , such that ∂f = 0 in D, there exists u ∈ W 1+ε p,q−1 (D) such that ∂u = f in D. Proof. If p = q, we have that F = ∂u for some U ∈ W 1 p,q−1 (CP n ). Let u = U on D, we have u ∈ W 1 p,q−1 (D) satisfying ∂u = f in D.