Abstract. The aim of this paper is to give a sufficient condition for existence and compactness of the ∂Neumann operator Nq on L 2 (0,q) (Ω) in the case Ω is an arbitrary qconvex domain in C nAbstract. The aim of this paper is to give a sufficient condition for existence and compactness of the ∂Neumann operator Nq on L 2 (0,q) (Ω) in the case Ω is an arbitrary qconvex domain in C n
EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN OPERATOR ON q-CONVEX DOMAINS MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN Abstract. The aim of this paper is to give a sufficient condition for existence and compactness of the ∂-Neumann operator Nq on L2(0,q) (Ω) in the case Ω is an arbitrary q-convex domain in Cn . 1. Introduction Let Ω be a domain in Cn . According to the fundamental work of Kohn and H¨ormander in the sixties, if Ω is bounded and pseudoconvex then for every 1 ≤ q ≤ n, the complex Laplacian ✷q on square integrable (0, q) forms on Ω has a bounded inverse, denoted by Nq . This is the ∂-Neumann operator Nq . The most basic property of Nq is that if v is a ∂ closed (0, q + 1)-form, then ∗ u := ∂ Nq+1 v provides the canonical solution to ∂u = v, namely the one orthogonal to the kernel of ∂ and so the one with minimal norm (see Corollary 2.10 in [11]). In this paper, we are interested in the existence and compactness of Nq on (possibly unbounded or non-smooth) q-convex domains, a generalization of pseudoconvex domains in which the existence of plurisubharmonic exhaustion function is replaced by existence of q-subharmonic ones. Our research is motivated from the fact that compactness of Nq implies global regularity in the sense of preservation of Sobolev spaces. Up to now, there is no complete characterization for compactness of Nq even in the case q = 1 and Ω is a smooth bounded domain in C. However, important progresses in this direction of research have been made following the ground breaking paper [2] in which Catlin introduce the notion of domains having the property (Pq ). Recently, K.Gansberger and F. Haslinger studied compactness estimates for the ∂-Neumann operator in weighted L2 -spaces and the weighted ∂-Neumann problem on unbounded domains in Cn (see [5] and [6]). We would like to remark that in [5], instead of using Rellich’s lemma, the author obtained compactness of the weighted Neumann operator Nq,φ under a strong assumption on rapidly increasing of the gradient ∇φ and the Laplacian △φ at the infinite point and at boundary points of a domain 2010 Mathematics Subject Classification. Primary 32W05. Key words and phrases. q-subharmonic functions, q-convex domain, ∂-Neumann operator, compactness of ∂-Neumann operator. 1 2 MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN Ω (Proposition 4.5 in [5]). The main step in Gansberger’s proof is to show that under the above assumption the embedding H01 (Ω, φ, ∇φ) → L2 (Ω, φ) is compact. The paper is organized as follows. In Section 2 we recall basic facts about q-subharmonic functions and q-convex domains. In particular, we note that Corollary 2.13 in [11] is still valid for bounded q-convex domains in Cn . ′ Section 3 is devoted to present the property (Pq ), a slight modification of the property (Pq ) introduced earlier in the case of bounded domains by D. Catlin in [2] and E. J. Straube in [11]. Roughly speaking, we say that a (possibly unbounded) domain Ω has the property (Pq′ ) if Ω admits a real valued, bounded, smooth function φ having the property that the sum of the q smallest eigenvalues of the complex Hessian of φ goes to infinity at the infinite point and at ∂Ω. We should say that this notion is motivated from Theorem 1.3 in [5] in which the case where q = 1 and ∂Ω is smooth is studied. The main result of the section is Theorem 3.5 which gives sufficient conditions for domain Ω having the property (Pq′ ). The existence and compactness of the ∂-Neumann operator Nq on q-convex domains are discussed in Section 4. We start the section with a geometric necessary condition for compactness of Nq . The main result of the paper is Theorem 4.3. Here we prove that if Ω ⊂ Cn is a q-convex domain having property (Pq′ ) then there exists a bounded ∂-Neumann operator Nq on L2(0,q) (Ω) and Nq is compact. Our proof exploits the property (Pq′ ) of Ω and some techniques from the book [11]. Acknowledgements. This work was partly written during visits of the second named author first at the Max-Planck Institute in the winter of 2011 and later at the Vietnam Institute for Advanced Mathematics in the winter of 2012. He wishes to thank these institutions for financial support and hospitality. We are also grateful to Professor Emile Straube for giving a key idea for the proof of Proposition 4.2. Finally, we are indebted to the referee for his(her) insightful comments, especially for showing us the proof of Theorem 4.3 which is simpler than our original one. Notation. λ2n denotes the Lebesgue measure on Cn and B(a, r) is the ball with center a ∈ Cn and radius r > 0. For a real valued function u ∈ C 2 (Ω) we define Hq (u)(z) to be the sum of the q smallest eigenvalues of the complex Hessian of u acting at the point z. EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN . . . 3 2. Preliminaries A complex-valued differential form u of type (0, q) on an open subset ∑ ′ Ω ⊂ Cn can be expressed as u = uJ dz J , where J are strictly increas|J|=q ing multi-indices with lengths q and {uJ } are defined functions on Ω. Let ∞ C(0,q) (Ω) be the space of complex-valued differential forms of class C ∞ and of type (0, q) on Ω. By C0∞ (Ω) we denote the space of C ∞ functions with compact support in Ω. We use L2(0,q) (Ω) to denote the space of (0, q)-forms on Ω with square-integrable coefficients. If u, v ∈ L2(0,q) (Ω), the weighted L2 -inner product and norms are defined by ∫ ∑ ′ uJ v J dλ2n and ∥u∥2Ω = (u, u)Ω . (u, v)Ω = Ω |J|=q The ∂-operator on (0, q)-forms is given by n (∑ ′ ) ∑′∑ ∂uJ uJ dz J = ∂ dz j ∧ dz J , ∂z j |J|=q |J|=q j=1 ∑′ means that the sum is only taken over strictly increasing multiwhere indices J. The derivatives are taken in the sense of distributions, and the domain of ∂ consists of those (0, q)-forms for which the right hand side belongs to L2(0,q+1) (Ω). So ∂ is a densely defined closed operator, and therefore ∗ has an adjoint operator from L2(0,q+1) (Ω) into L2(0,q) (Ω) denoted by ∂ . For ∑ ′ ∗ uJ dz J ∈ dom(∂ ) one has u= |J|=q+1 n ∑ ′∑ ∂ujK ∂ u=− dz K . ∂z j j=1 ∗ |K|=q The complex Laplacian on (0, q)-forms is defined as ∗ ∗ ✷q := ∂∂ + ∂ ∂, where the symbol ✷q is understood as the maximal closure of the operator initially defined on (0, q)-forms with coefficients in C0∞ (Ω). ✷q is a self adjoint, positive operator. The associated Dirichlet form is denoted by ∗ ∗ Q(f, g) = (∂f, ∂g) + (∂ f, ∂ g), ∗ for f, g ∈ dom(∂) ∩ dom(∂ ). The weighted ∂-Neumann operator Nq is -if it exists-the bounded inverse of ✷q . We refer the reader to the monographs [11] for a complete survey on ∂-Neumann operators and their applications to other problems in several complex variables. Next, we recall the definition of q-subharmonic functions which is an extension of plurisubharmonic functions (see [1], [7], [8]). 4 MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN Definition 2.1. Let Ω be a domain in Cn . An upper semicontinuous function u : Ω −→ [−∞, ∞), u ̸≡ −∞ is called q-subharmonic if for every q-dimensional complex plane L in Cn , u|L is a subharmonic function on L ∩ Ω. The set of all q-subharmonic functions on Ω is denoted by SH q (Ω). The function u is called to be strictly q-subharmonic if for every U Ω there exists constant CU > 0 such that u(z) − CU |z|2 ∈ SHq (U ). Remark 2.2. (a) q-subharmonicity and strict q-subharmonicity are local properties. (b) 1-subharmonicity (resp. n−subharmonicity) coincides with plurisubharmonicity (resp. subharmonicity). We list below basic properties of q-subharmonic functions that will be used later on (see [7]). Proposition 2.3. Let Ω be an open set in Cn and let q is an integer with 1 q n. Then we have. (a) If u ∈ SH q (Ω) then u ∈ SH r (Ω), for every q r n. (b) If u, v ∈ SH q (Ω) and α, β > 0 then αu + βv ∈ SH q (Ω). (c) If {uj }∞ j=1 is a family of q-subharmonic functions, u = supj uj < +∞ and u is upper semicontinuous then u is a q-subharmonic function. (d) If {uj }∞ j=1 is a family of nonnegative q-subharmonic functions such ∞ ∑ that u = uj < +∞ and u is upper semicontinuous then u is q-subharmonic. j=1 (e) If {uj }∞ j=1 is a decreasing sequence of q-subharmonic functions then so is u = lim uj . j→+∞ (f ) Let ρ ≥ 0 be a smooth radial function in Cn vanishing outside the ∫ unit ball and satisfying Cn ρdV = 1. For u ∈ SH q (Ω) we define ∫ uε (z) := (u ∗ ρε )(z) = u(z − ξ)ρε (ξ)dλ2n (ξ), ∀z ∈ Ωε , B(0,ε) where ρε (z) := and Ωε = {z ∈ Ω : d(z, ∂Ω) > ε}. Then uε ∈ ∞ SHq (Ωε ) ∩ C (Ωε ) and uε ↓ u as ε ↓ 0. (g) Let u1 , . . . , up ∈ SHq (Ω) and χ : Rp → R be a convex function which is non decreasing in each variable. If χ is extended by continuity to a function [−∞, +∞)p → [−∞, ∞), then χ(u1 , . . . , up ) ∈ SHq (Ω). 1 ρ(z/ε) ε2 n The property (g) in the cases q = 1 and q = n are given in Theorem 5.6 and Theorem 4.16 in [4]. These proofs can be easily extended to the general case. We will use (f) and (g) in the proof of Theorem 3.5 to produce a version of Richberg’s regularization lemma for continuous strictly q-subharmonic EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN . . . 5 functions. We should remark that for 2 ≤ q ≤ n, the class of q-subharmonic functions is not invariant under biholomorphic changes of coordinates. We give some equivalent conditions for q-subharmonicity which is similar to plurisubharmonicity (see [1], [8]). Proposition 2.4. Let Ω be a domain in Cn and let q be an integer with 1 q n. Let u be a real valued C 2 -function defined on Ω. Then the following are equivalent: (a) u is a q-subharmonic function Ω. (b) i∂∂u ∧ (i∂∂|z|2 )q−1 0 i.e., Hq (u)(z) ≥ 0 for every z ∈ Ω. ∑ ′ (c) For every (0, q)-form f = fJ dz J we have |J|=q ∑ |K|=q−1 ′ n ∑ ∂ 2u fjK f kK ∂z j ∂z k j,k=1 0. We also have the following simple result about smoothing q-subharmonic functions. Proposition 2.5. Let Ω be an open set in Cn and let u ∈ SHq (Ω) such that u − δ|idCn |2 ∈ SHq (Ω) for some δ > 0. Then for every ε > 0 we have uε − δ|idCn |2 ∈ SHq (Ωε ), where Ωε := {z ∈ Ω : d(z, ∂Ω) > ε}. Proof. By Proposition 2.3 (f) we have (u − δ|idCn |2 )ε ∈ SHq (Ωε ). Since ∫ 2 (u − δ|idCn | )ε (z) = uε (z) − δ |z − w|2 ρε (w)dV (w) B(0,ε) ∫ = uε (z) − δ|z| − δ 2 (2ℜ(z, −w) + |w|2 )ρε (w)dV (w) B(0,ε) = uε (z) − δ|z| − v(ε) (z), ∫ where v(ε) (z) := δ (2ℜ(z, −w) + |w|2 )ρε (w)dV (w) is a pluriharmonic 2 B(0,ε) function in Cn . Hence, uε − δ|idCn |2 = (u − δ|idCn |2 )ε + v(ε) ∈ SHq (Ωε ). This completes the proof. The following definition is an extension of pseudoconvexity. Definition 2.6. A domain Ω ⊂ Cn is said to be q-convex if there exists a q-subharmonic exhaustion function on Ω. Moreover, a C 2 smooth bounded domain Ω is called strictly q-convex if it admits a C 2 smooth defining function which is strictly q-subharmonic on a neighbourhood of Ω. 6 MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN It is not clear if we can find a smooth strictly q-subharmonic exhaustion function on a q-convex domain. However by Proposition 2.7 in [7], every qconvex domain Ω can be written as an increasing union of bounded q-convex domains Ωj such that each Ωj has a smooth strictly q-convex exhaustion function. Using Sard theorem this result can be refined as follows. Proposition 2.7. Let Ω be a q-convex domain in Cn . Then Ω can be written ∞ ∪ as Ω = Ωj such that Ωj Ωj+1 and each Ωj is a strictly q-convex domain. j=1 According to a classical result of Green-Wu, every domain in Cn is n-convex (see Theorem 9.3.5 in [4] for an elegant proof). In the case where 1 < q < n, there exists no geometric characterization for q-convexity. However, in analogy with the classical Kontinuitassatz principle for pseudoconvexity we have the following partial result. Proposition 2.8. Let Ω be a domain in Cn and p ∈ ∂Ω. Assume that there exist r > 0, a sequence {pj } ⊂ Ω, pj → p, and a sequence of q-dimensional complex subspaces Lj satisfying the following conditions: (a) pj ∈ Lj for every j ≥ 1, (b) ∂B(p, r) ∩ Lj is contained in a fixed compact subset K of Ω. Then Ω is not q-convex. Proof. Suppose that there exists a q-subharmonic exhaustion function u on Ω. Since u|Lj is subharmonic, by the maximum principle and the assupmtions (a) and (b) we get for every j ≥ 1 the inequality u(pj ) ≤ sup u. By K letting j → ∞ we arrive at a contradiction. Remark 2.9. (i) Let Ω := {z ∈ C3 : 1 < |z| < 3} and p = (1, 0, 0) ∈ ∂Ω. Denote by L the hyperplane tangent to ∂Ω at the point p. Consider the sequences of points pj := (1 + 1/j, 0, 0), j ≥ 1 tending to p and hyperplanes Lj := L + (1/j, 0, 0) passing through the points pj . Using the above result, we can see that Ω is not 2-convex. (ii) In [8], the following generalization of Levi convexity was introduced. We say that a bounded domain Ω in Cn with a C 2 smooth defining function ρ ∑ ′ fJ dz J is weakly q-convex if for every p ∈ ∂Ω, for every (0, q)-form f = |J|=q satisfying we have n ∑ ∂ρ (p)fiK = 0 ∀|K| = q − 1, ∂z i i=1 ∑ |K|=q−1 ′ n ∑ ∂ 2ρ fjK f kK ∂z j ∂z k j,k=1 0. 7 EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN . . . It follows from Theorem 2.4 in [8] that every C ∞ smooth bounded weakly q-convex domain is q-convex in our sense. Unfortunately, we do not know if the reverse implication is true. The following proposition similar as Corollary 2.13 in [11] is still valid for bounded q-convex domains in Cn . ∑ ′ Proposition 2.10. Let Ω be a bounded q-convex domain in Cn , u = uJ d¯ zJ ∈ |J|=q ¯ ∩ dom(∂¯∗ ) ⊂ L2 (Ω). Then for all b ∈ C 2 (Ω), b 0 the following dom(∂) (0,q) holds n ∫ ∑ ∑ ∂ 2b ′ ujK ukK dλ2n ∥∂u∥2 + ∥∂¯∗ u∥2 . eb ∂zj ∂ z¯k j,k=1 |K|=q−1 Ω 3. The property (Pq′ ) First we recall an important concept introduced and investigated by D. Catlin in [2] and E. J. Straube in [11] (see also [9]). We say that a compact set K in Cn has the property (Pq ) if for every M > 0, there exist a neighborhood UM of K, a C 2 smooth function λM on UM such that 0 λM (z) 1, z ∈ UM and for any z ∈ UM , the sum of the smallest q eigenvalues of the complex Hessian of λ is at least M (or, equivalently, λM − Mq |z|2 ∈ SHq (UM )). Moreover, given a closed set E (not necessarily bounded), we say that E locally has the property (Pq ) if for every z0 ∈ E we can find a compact neighborhood K of z0 in E such that K has the (Pq ) property. Using Kohn-Morrey-H¨ormander formula in [11], it is not hard to prove (see [2] for the case q = 1 and [11] for general q) that if Ω is a smoothly bounded pseudoconvex domain in Cn with the boundary ∂Ω having the property (Pq ) then the ∂-Neumann operator Nq is compact. The following notion is the key to our research on compactness of Nq in the case where Ω is unbounded. Definition 3.1. Let Ω be a domain in Cn . We say that Ω has the property (Pq′ ) if there exists a real valued, bounded C 2 smooth function φ on Ω such that for every positive number M , we have φ(z) − M |z|2 ∈ SHq (Ω \ KM ) for some compact subset KM of Ω. Remark 3.2. (i) The function φ is not assumed to be q-subharmonic on the whole Ω. We will prove, however, that φ can be chosen to have this additional property. (ii) If Ω has the property (Pq′ ) then for every complex space L of dimension q, Ω ∩ L is quasibounded (in L) i.e., Ω ∩ L contains only a finite number of 8 MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN disjoint balls with fixed radii (see Definition 1.4 in [5]). Indeed, it suffices to prove the statement for the case q = n. Assume for the sake of seeking a contradiction that we can find a sequence of disjoint balls B(zj , r) contained in Ω with |zj | → ∞. By passing to a subsequence, we can find a sequence mj of real numbers such that mj → +∞ and φ(z) − mj |z|2 ∈ SHn (B(zj , r)) for every j ≥ 1. Now we let θ ≥ 0 be a smooth function with compact support in B(0, r) such that θ = 1 on B(0, r/2). Set θj (z) := θ(z − zj ). By Stoke’s theorem∫we have ∫ B(zj ,r/2) i∂∂φ ∧ (i∂∂|z|2 )n−1 ≤ ∫ θj i∂∂φ ∧ (i∂∂|z|2 )n−1 B(zj ,r) = B(zj ,r) iφ∂∂θj ∧ (i∂∂|z|2 )n−1 ≤ C∥φ∥Ω λ2n (B(zj , r)). Here C > 0 is a constant depends only on the second derivatives of θ. It follows that there exists C ′ > 0 depends only on n such that mj ≤ C ′ ∥φ∥Ω for every j ≥ 1. By letting j → ∞ we get a contradiction. It is easy to see that finite intersection of domains possessing the (Pq′ ) property still has this property. The main result of the section provides a substantial class of domains satisfying the property (Pq′ ). More precisely, we have Theorem 3.3. Let Ω be an open set in Cn with ∂Ω locally has the property (Pq ). Assume that there exist negative q-subharmonic functions ρ, ρ˜ on Ω satisfying the following conditions. (a) ρ ∈ C 2 (Ω), ρ(z) − |z|2 ∈ SHq (Ω). q (ρ)(z) (b) lim|z|→∞ H1+ρ(z) 2 = ∞. (c) ρ˜ is strictly q-subharmonic on Ω and satisfies lim ρ˜(z) = 0 for every z→ξ ξ ∈ ∂Ω. Then Ω has the property (Pq′ ). We first need the following result which generalizes in part Theorem 2.1 in [9] where the case q = 1 was treated. Lemma 3.4. Let Ω be a bounded domain in Cn with ∂Ω has the property (Pq ). Assume that there exists a negative strictly q-subharmonic exhaustion function φ of Ω. Then for every real valued continuous function f on ∂Ω the function P Bf,Ω (z) := sup{u(z) : u ∈ SHq (Ω), lim sup u(x) ≤ f (ξ) ∀ξ ∈ ∂Ω} x→ξ belongs to SHq (Ω) ∩ C(Ω) and P Bf,Ω |∂Ω = f. EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN . . . 9 Proof. Since ∂Ω has the property (Pq ), by Proposition 4.10 in [11], there exists a sequence fj of continuous q−subharmonic functions on neighborhoods of ∂Ω such that fj converges uniformly to f on ∂Ω. By Proposition 2.3 (f), after taking convolution with a smoothing kernel we may achieve that fj is q−subharmonic and C 2 smooth near ∂Ω for every j. Now we fix j ≥ 1 and choose a real valued C 2 smooth function θj on Cn with compact support such that θj = 1 on a small neighborhood of ∂Ω. Since φ is strictly q-subharmonic on Ω, by taking a constant Mj > 0 large enough the function Fj (z) := Mj φ(z) + θj (z)fj (z) will belong to SHq (Ω) ∩ C(Ω) and satisfies Fj |∂Ω = fj . So the function P Bfj ,Ω (z) := sup{u(z) : u ∈ SHq (Ω), lim sup u(x) ≤ fj (ξ) ∀ξ ∈ ∂Ω} x→ξ satisfies lim inf P Bfj ,Ω (z) ≥ lim Fj (z) = fj (ξ) ∀ξ ∈ ∂Ω. z→ξ z→ξ On the other hand, since φ is a negative subharmonic exhaustion function of Ω, by a well known result in potential theory we know that Ω is regular with respect to the Dirichlet problem for Laplacian. So we can find a real valued continuous function Hj on Ω which is harmonic on Ω and satisfies Hj = fj on ∂Ω. By the maximum principle for subharmonic functions we obtain P Bfj ,Ω ≤ Hj on Ω. Therefore lim supP Bfj ,Ω (z) ≤ lim Hj (z) = fj (ξ) ∀ξ ∈ ∂Ω. z→ξ z→ξ Summing up, we have proved that lim P Bfj ,Ω = fj (ξ) ∀ξ ∈ ∂Ω. z→ξ Now we apply Lemma 1 in [12] to conclude that P Bfj ,Ω is in fact continuous on Ω. Notice that Walsh’s lemma is proved only in the case q = 1, however since q-subharmonicity is invariant both under taking finite maximum and translates of variables we can check that his proof works also for general q. Finally, from the definition of the envelopes we deduce easily that ∥P Bfj ,Ω − P Bfk ,Ω ∥Ω = ∥fj − fk ∥∂Ω . It follows that P Bfj ,Ω converges uniformly to P Bf,Ω on Ω. In particular P Bf,Ω ∈ SHq (Ω) ∩ C(Ω) and satisfies P Bf,Ω |∂Ω = f. We are done. Remark. The above lemma is false if the assumption on the existence of φ is omitted. Indeed, consider the punctured disk Ω := {z ∈ C : 0 < |z| < 1}. Then ∂Ω has the property (P1 ). Now we let f (z) = 0 for |z| = 1 and f (0) = 1. Suppose that there exists u ∈ SH1 (Ω) ∩ C(Ω) such that u = f on 10 MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN ∂Ω. Since u is bounded near 0, it extends to a subharmonic function on the whole disk {z : |z| < 1}. Notice that u(0) = 1 > 0 = sup u(z). |z|=1 This is a violation of the maximum principle for subharmonic function. Proof. We split the proof into three steps. Step 1. We show that there exists a real valued, bounded function ψ ∈ SHq (Ω) ∩ C(Ω) having the following property: For every M > 0 and every point ξ ∈ ∂Ω there exists rξ,M > 0 such that ψ(z) − M |z|2 is q-subharmonic on Ω ∩ B(ξ, rξ,M ). For this, we fix M, ξ as above and define for j ≥ 1 the domain Ωj := Ω∩B(0, j). Since ∂Ω locally has the (Pq ) property and since this property is preserved after taking finite union of compact sets, we infer that the compact set ∂Ωj has the (Pq ) property as well. In order to apply Lemma 3.4 we observe that the function max{˜ ρ(z), |z|2 − j 2 } is negative strictly qsubharmonic exhaustion for Ωj . Now we define the following function on a part of ∂Ωj φj (z) = −|z|2 for z ∈ B(0, j −1)∩∂Ω; φj (z) = ρ(z)−|z|2 for z ∈ ∂B(0, j)∩Ω. Extend φj to a real valued continuous function (still denoted by φj ) on ∂Ωj such that ρ(z) − |z|2 ≤ φj (z) ≤ −|z|2 , z ∈ ∂Ωj . It follows from Lemma 3.4 that P Bφj ,Ωj ∈ SHq (Ωj ) ∩ C(Ωj ) and P Bφj ,Ωj = φj on ∂Ωj . Furthermore, using the assumption (a) we also get P Bφj ,Ωj (z) ≥ ρ(z) − |z|2 on Ωj . We define ρj = P Bφj ,Ωj on Ωj and ρj (z) = ρ(z) − |z|2 on Ω \ Ωj . Thus for j ≥ 2 we have ρj ∈ SHq (Ω) ∩ C(Ω). Moreover ρj (z) + |z|2 < 0 ∀z ∈ Ω, ρj (z) + |z|2 = 0 ∀z ∈ B(0, j − 1) ∩ ∂Ω. Set ρ˜j (z) := ρj (z)+|z|2 on Ω. Following a construction given in the example after Proposition 5.3 in [5] we define ψ(z) := ∑ e2j ρ˜j (z) j≥2 2j . It is easy to see that ψ is real valued, bounded, continuous and strictly qsubharmonic on Ω. For j ≥ 2, we have the following simple estimate on Ωj which is taken in the sense of distribution ( e2j ρ˜j (z) ) 2 q−1 2 q−1 2j ρ˜j (z) 2 q 2j ρ˜j (z) i∂∂ ∧(i∂∂|z| ) ≥ i∂∂ ρ ˜ ∧(i∂∂|z| ) e ≥ (i∂∂|z| )e . j 2j EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN . . . 11 Observe that lim ρ˜j (z) = 0 for every j such that j > |ξ| + 1, so the above z→ξ inequalities imply that there exists rξ,M > 0 such that ψ(z) − M |z|2 is q-subharmonic on Ω ∩ B(ξ, rξ,M ). Step 2. Define 1 θ(z) = + ψ(z), ∀z ∈ Ω. 1 − ρ(z) 1 Since the function x → 1−x , x < 0 is convex, increasing, we see that θ is bounded, continuous and strictly q-subharmonic on Ω. We will prove that θ satisfies the condition given in Definition 3.1. For this, we note that the following inequalities holds ( ) 1 Hq (ρ)(z) Hq (ρ)(z) i∂∂ ∧(i∂∂|z|2 )q−1 ≥ (i∂∂|z|2 )q ≥ (i∂∂|z|2 )q . 2 1 − ρ(z) (1 − ρ2 (z)) 2(1 + ρ(z)2 ) So using assumption (b) we infer that for every M > 0 there exists rM > 0 1 − M |z|2 is q-subharmonic on Ω \ B(0, rM ). Now using the such that 1−ρ(z) result obtained in Step 1 and a standard compactness argument, we can find a compact KM ⊂ Ω such that θ(z) − M |z|2 is q-subharmonic on Ω \ KM . Step 3. In this last step, we will regularize θ to get a smooth function φ satisfying the condition of Definition 3.1. To this end, it suffices to follow closely the proof of Richberg’s regularization theorem given in Theorem I.5.21 in [4]. First we introduce the maximum regularization operator as follows. Fix a function λ ∈ C ∞ (R), λ ≥ 0 having compact support in [−1, 1]. Define for ζ = (ζ1 , . . . , ζn ), ζj > 0 the following function ∫ ∏ Mζ (t1 , . . . , tn ) = max{t1 + h1 , . . . , tn + hn } λ(hj /ζj )dh1 · · · dhn . Rn 1≤j≤n Then Mζ is non decreasing in all variables, smooth and convex on Rn . Moreover, if u1 , . . . , un are functions on an open subset D ⊂ Cn such that uj (z) − M |z|2 ∈ SHq (D), then Mζ (u1 , . . . , un )(z) − M |z|2 ∈ SHq (D). The last property in the case q = 1 is given in Lemma I.5.18 in [4]. For general q, the proof is the same. Now, let Ωα be a locally finite open covering of Ω consisting of relatively compact balls. By Step 2, for each α, there exists mα > 0 such that θ(z) − mα |z|2 ∈ SHq (Ωα ). Moreover limα→∞ mα = +∞. Choose concentric balls Ω′′α ⊂ Ω′α ⊂ Ωα of radii rα′′ < rα′ < rα such that Ω′′α still covers Ω. We set for εα , δα > 0 small enough θα (z) := (θ ∗ ρεα )(z) + δα (rα′ − |z|2 ), z ∈ Ωα . 2 Let ηα := δα min{rα′ − rα′′ , (rα 2 − rα′ )/2}. 2 2 2 12 MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN Choose δα < mα /2 such that ηα < 1, and then εα so small that u ≤ u ∗ ρεα ≤ u + ηα on Ωα . It follows that θα − mα /2|z|2 ∈ SHq (Ωα ) and θα < θ − ηα on ∂Ωα , θ + ηα < θα on Ω′′α . We define φ(z) = M(ηα ) (uα ) on Ω. Then we have θ ≤ φ ≤ θ + 2 on Ω. Moreover, by Corollary I.5.19 in [4], the function φ is smooth, q-subharmonic on Ω and satisfies φ(z) − mα /2|z|2 ∈ SHq (Ωα ) for every α. Thus Ω satisfies property (Pq′ ). The proof is complete. Example 3.5. For n ≥ 2 and 1 ≤ q ≤ n, using Theorem 3.3 we will construct an unbounded q-convex domain Ω in Cn having the property (Pq′ ). Let ψ1 , ψ2 ∈ C ∞ (Cn ) be defined by ψ1 (z) := 2 n ∑ (x2j +1)yj2 + j=1 ∑ |zn |2 |zi zj | , ψ2 (z) = |z1 | +· · ·+|zn−q+1 | − , 2 1≤i n(M − 1). Indeed, for any vector λ = (λ1 , . . . , λn ) we have ∑ ∑ ∂ψ1 (z)λi λj ≥ (|z|2 + 1)|λ|2 − 2( |zi zj ||λi λj |) ∂zi ∂zj 1≤i 0 small enough. The proof is complete. Proof of Proposition 3.6. Let φ be as in Definition 3.1 and let U0 U V Ω such that φ − 2|z|2 ∈ SHq (Ω \ U 0 ). Choose χ ∈ C0∞ (U ) such that 0 χ 1, χ ≡ 1 on U0 . Let m0 > 0 such that (1 − χ)φ + m0 |z|2 ∈ SH1 (V ). Choose r1 > 0 such that V D(0, r1 ) × . . . × D(0, r1 ), where D(0, r1 ) is the disc of radius r1 and center 0 ∈ C. By Lemma 3.7 there exists ψ ∈ C0∞ (Cw ) such that ψ + |w|2 ∈ SH1 (C), ψ − (m0 + 1)|w|2 ∈ SH1 (D(0, r1 )). Put n ∑ φ1 (z) := (1 − χ(z))φ(z) + ψ(zj ). j=1 For each j, we consider the canonical projection πj : Cn −→ C, z −→ zj . Since ψ + |w|2 ∈ SH1 (C) we have ψj (z) := ψ ◦ πj (z) + |zj |2 ∈ SH1 (Cn ). By the choice of φ1 we have, n n ∑ ∑ (φ1 − |z|2 )|Ω\U = φ + ψ ◦ πj − |z|2 = φ − 2|z|2 + ψj . j=1 j=1 It follows that φ1 (z) − |z|2 ∈ SHq (Ω\U ), (3.1) On the other hand, we have (φ1 − |z| )|V = (1 − χ)φ + 2 n ∑ ψ ◦ πj − |z|2 j=1 = ((1 − χ)φ + m0 |z| ) + 2 n ∑ (ψ ◦ πj − (m0 + 1)|zj |2 ). j=1 Thus by the choice of m0 and ψ we obtain φ1 (z) − |z|2 ∈ SHq (V ). (3.2) From (3.1) and (3.2) we get φ1 (z) − |z|2 ∈ SHq (Ω). (3.3) EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN . . . 15 Next we set C := ∥φ1 ∥Ω and φ1 + C . 2C 1 Then 0 φ 1 and by (3.3) we have φ − 2C |z|2 ∈ SHq (Ω). Now, we prove that for every M > 0 there exists ΩM Ω such that φ − M |z|2 ∈ SHq (Ω\ΩM ). Choose ΩM Ω such that V ΩM and φ − (2CM + 1)|z|2 ∈ SHq (Ω\ΩM ). We have 1 (φ − M |z|2 )|Ω\ΩM = (φ1 − 2CM |z|2 + C)|Ω\ΩM 2C n ∑ 1 = (φ + ψ ◦ πj − 2CM |z|2 + C) 2C j=1 φ := ∑ 1 (φ − (2CM + 1)|z|2 + ψj + C). 2C j=1 n = Hence, φ − M |z|2 ∈ SHq (Ω\ΩM ) because ψj ∈ P SH(Cn ), j = 1, . . . , n and φ − (2CM + 1)|z|2 ∈ SHq (Ω\ΩM ). Thus, φ satisfies Definition 3.1 and 1 φ − 2C |z|2 ∈ SHq (Ω). The proof is complete. The final result of this section relates the property (Pq′ ) of Ω and that of ∂Ω. Proposition 3.8. Let Ω be a bounded domain in Cn with smooth boundary. Assume that Ω satisfies property (Pq′ ) for 1 ≤ q ≤ n. Then ∂Ω satisfies property (Pq ). Proof. Since Ω is smoothly bounded, using a partition unity of ∂Ω, we can ∪m find a finite number of balls Bj , j = 1, 2, . . . , m such that ∂Ω j=1 Bj and Ω ∩ Bj is star-shaped for all j. For every j, being a strictly pseudoconvex domain, Bj has the property (Pq′ ). Moreover, since Ω has property (Pq′ ), we infer that so does Ω∩Bj . Since Ω∩Bj is star-shaped, using dilation, it is easy to show that ∂(Ω ∩ Bj ) has the property (Pq ). Because ∂Ω ∩ Bj ⊂ ∂(Ω ∩ Bj ) so ∂Ω ∩ Bj has property (Pq ). Therefore, Corollary 4.13 in [11] implies that ∂Ω also has property (Pq ). The proof is complete. 4. existence and compactness of the ∂-Neumann operator We start with a geometric sufficient condition for compactness of Nq which is a slight extension of an example given in [5]. Proposition 4.1. Let Ω be a domain in Cn , 0 ∈ Ω and 1 ≤ q ≤ n. Suppose that there exist constants r > 0, M > 0, a sequence of balls B(ξj , r) ⊂ Cq and a sequence of open sets Uj ⊂ Cn−q satisfying the following conditions. 16 MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN (a) The open sets Vj := Uj × B(ξj , r) are contained in Ω and pairwise disjoint. (b) λ2(n−q) (Uj ) = M for every j, where λ2n is the Lebesgue measure on C2n . Then Nq is not compact on L2(0,q) (Ω). Proof. We will modify the ideas given in p. 16 of [5] where the case q = n is essentially treated. Assume for the sake of seeking a contradiction that Nq is compact. Let θ be a C 2 smooth function with compact support in B(0, r) ⊂ Cq normalized so that ∥θ∥L2 (B(0,r)) = 1. For j ≥ 1, we let θj be the function with compact support in Vj and satisfies θj (z1 , . . . , zn ) = θ(z1 − ξj,1 , . . . , zq − ξj,q ), (zq+1 , . . . , zn ) ∈ Uj , where ξj = (ξj,1 , . . . , ξj,q ). Consider the following sequence in L2(0,q) (Ω) θˆj (z) := θj (z)dz1 ∧ · · · ∧ dzq . Using Fubini’s theorem and the assumption (b), it is easy to check that ✷q θˆj are uniformly bounded. Since Nq (✷θˆj ) = θˆj , we deduce that θˆj contains a convergent subsequence. Notice that the support of θˆj being contained in Vj are pairwise disjoint we infer that {θˆj }j≥1 is an orthogonal sequence in L2(0,n) (Ω). Applying again Fubini’s theorem and the assumption (b), we see that ∥θˆj ∥ = M . Putting all this together we obtain a contradiction. The proof is complete. It is pointed out in [5] that compactness of N1 is not invariance under biholomorphic maps. However, under additional assumptions on the map we have the following positive result. Proposition 4.2. Let Ω and Ω′ be domains in Cn and φ : Ω → Ω′ be a biholomorphic map satisfying 0 < inf |Jφ (z)| ≤ sup |Jφ (z)| < ∞. z∈Ω z∈Ω ′ Here Jφ is the Jacobian of φ. Suppose that both NqΩ and NqΩ exist. Then ′ NqΩ is compact if and only if so is NqΩ . Proof. Since the roles of Ω and Ω′ are similar, it is enough to prove that if ′ NqΩ is compact then so is NqΩ . By the proof of the implication (iv) → (i) given in Proposition 4.2 of [11], it suffices to prove the canonical solution ∗ ∗ Ω operators ∂ NqΩ and ∂ Nq+1 are compact. For simplicity of the exposition, ∗ we show only compactness of ∂ NqΩ . Using the assumptions on the Jacobian of φ, we see that the linear maps φ∗ : L2(0,q) (Ω′ ) → L2(0,q) (Ω) and (φ−1 )∗ : L2(0,q) (Ω) → L2(0,q) (Ω′ ) are bounded and preserve ∂-closed forms. Here φ∗ and (φ−1 )∗ are pull backs of φ and φ−1 , respectively. Next, let EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN . . . 17 ′ 2 M(0,q) (Ω′ ) := L2(0,q) (Ω) ∩ ker(∂) and PqΩ : L2(0,q) (Ω′ ) → M(0,q) (Ω′ ) be the orthogonal projection. We claim that the following equality holds on M(0,q) (Ω′ ) ′ ∗ ∗ ′ (Id − PqΩ ) ◦ (φ−1 )∗ ◦ ∂ NqΩ ◦ φ∗ = ∂ NqΩ . To see this, fix u ∈ M(0,q) (Ω′ ). Then φ∗ (u) ∈ L2(0,q) (Ω) ∩ ker(∂) and we have ( ) ( ) ∗ ∗ ∂ q (φ−1 )∗ ◦ ∂ NqΩ ◦ φ∗ (u) = (φ−1 )∗ ∂ q (∂ NqΩ (φ∗ (u)) = (φ−1 )∗ (φ∗ (u)) = (φ∗ )−1 (φ∗ (u)) = u. Here the second equality follows from the fact that if u ∈ ker(∂) then ∗ ∂ Nq u gives the solution f with L2 minimal norm of the equation ∂f = u (see Corollary 2.10 in [11]). This fact, combined with the property of the ∗ ′ ′ projection PqΩ also imply the desired claim. Thus ∂ NqΩ is compact. The proof is complete. Now we are ready to formulate the main result of this section. Theorem 4.3. Let Ω ⊂ Cn be a q-convex domain having the property (Pq′ ). Then there exists a bounded and compact ∂-Neumann Nq from L2(0,q) (Ω) to itself. The above theorem should be compared to Theorem 1.3 and Lemma 6.4 in [5]. More precisely, it is proved there that if the unbounded domain Ω is smooth, pseudoconvex and locally B1 -regular and if there exists a bounded function λ ∈ SH1 (Ω)∩C 2 (Ω) such that the lowest eigenvalues of the complex Hessian of λ tends to ∞ then N1 is compact. Notice that we do not assume smoothness of ∂Ω. The original proof of the above theorem relies heavily on the technique of weighted norms and is quite complicated. We are grateful to the referee for showing us a simpler and more direct proof that we will now present. Proof. We split the proof into two parts. (a) Existence of Nq (as a bounded operator on L2(0,q) (Ω)). According to Proposition 3.6, we can take a bounded function φ ∈ C 2 (Ω) such that φ(z) − ε|z|2 ∈ SHq (Ω) for some fixed ε > 0. We may assume that ∗ 0 φ 1. Note that the functions in dom(∂) ∩ dom(∂ ) with compact ∗ support in Ω are dense in dom(∂) ∩ dom(∂ ). Hence, it suffices to show an estimate ¯ 2 + ∥∂ ∗ u∥2 ), ∥u∥2 C(∥∂u∥ (4.1) for such forms with C independent of u. By density, (4.1) then holds on ∗ dom(∂) ∩ dom(∂ ), and the existence of Nq follows by standard arguments (see e.g. p. 27 in [11] or p. 78 in [3]). 18 MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN Let u as above and choose a large ball B such that the support of u is contained in Ω ∩ B. By applying Proposition 2.10 on the bounded q-convex domain Ω ∩ B with b = φ − 1 we have ∫ n ∑ ∑ ∂ 2φ −1 2 ′ e qε∥u∥ eφ−1 ujK ukK dλ2n ∂zj ∂z k j,k=1 |K|=q−1 ∑ Ω∩B ′ n ∑ ∫ e φ−1 |K|=q−1 j,k=1 Ω ∂ 2φ ujK ukK dλ2n ∂zj ∂z k ∗ ∥∂u∥2 + ∥∂ u∥2 . Hence ∥u∥2 e ∗ ∗ (∥∂u∥2 + ∥∂ u∥2 ) = C(∥∂u∥2 + ∥∂ u∥2 ), qε with C = qεe and the desired conclusion follows. (b) Compactness of Nq . From the property (Pq′ ) it follows that for M > 0 there exists KM Ω such that φ(z)−M |z|2 ∈ SHq (Ω\KM ) where as above ∗ we assume that 0 φ 1. Assume that u ∈ dom(∂) ∩ dom(∂ ). Again by applying Proposition 2.10 we infer that ∫ n ∑ ∑ ∂ 2φ ′ −1 2 eφ−1 e qM ∥u∥Ω\KM ujK ukK dλ2n ∂z j ∂z k j,k=1 |K|=q−1 Ω\KM ∗ 2 ∥∂u∥ + ∥∂ u∥ . 2 Thus we get C e ∗ (∥∂u∥2 + ∥∂ u∥2 ), C = . M q ∥u∥2Ω\KM Next, choose a bounded domain with smooth boundary VM with KM ⊂ ∗ VM Ω. We prove that the restriction map RM : dom(∂) ∩ dom(∂ ) −→ L2(0,q) (VM ) : u → u|VM with the graph norm on the first space is compact. Indeed, using the estimate (2.85) in [11] for s = 0, t = 0, V = VM and U = Ω, we get ∗ ∥u∥2W 1 (V ) (0,q) M C1 (∥∂u∥2 + ∥∂ u∥2 + ∥u∥2 ), where u ∈ W 1 (Ω). Combining with (4.1) we obtain ∥u∥2W 1 (0,q) ∗ (VM ) D(∥∂u∥2 + ∥∂ u∥2 ), with D independent of u. Hence Rellich’s Lemma in [3] implies that the ∗ restriction map RM : dom(∂) ∩ dom(∂ ) −→ L2(0,q) (VM ) is compact and the ∗ desired conclusion follows. On the other hand, for u ∈ dom(∂) ∩ dom(∂ ), EXISTENCE AND COMPACTNESS FOR THE ∂-NEUMANN . . . 19 after rescaling, we have ∥u∥ ∥u∥Ω\KM + ∥u∥VM 1 ∗ (∥∂u∥ + ∥∂ u∥) + ∥RM u∥. M It now follows from [11], Lemma 4.3, part (ii) that the embedding dom(∂) ∩ ∗ dom(∂ ) −→ L2(0,q) (Ω) is compact and, hence, by Proposition 4.2 in [11] this is equivalent with compactness of Nq . This completes the proof. References [1] H. Ahn and N. Q. Dieu, The Donnelly-Fefferman theorem on q-pseudoconvex domains, Osaka J. Math., 46(2009), 599 - 610. [2] D.W. Catlin, Global regularity of the ∂-Neumann problem. In Complex analysis of several variables(Madison, 1982). Proc. Symp. Pure Math., 41(1984), 39-49. [3] S.C. Chen and M.C.Shaw, Partial Differential Equations in Several Complex Variables, Studied in Advanced Mathematics, Vol. 19, Amer. Math. Soc, 2001. [4] J. Demailly, Complex Analytic and Differential Geometry, http://www-fourier.ujfgrenoble.fr/ demailly/manuscripts/agbook.pdf [5] K. Gansberger, On the weighted ∂-Neumann problem on unbounded domains, arXiv:0912.0841. [6] K. Gansberger and F. Haslinger, Compactness estimates for the ∂-Neumann problem in weighted L2 -spaces on Cn , to appear in Proccedings of the Conference on Complex Analysis 2008, in honour of Linda Rothschild, arXiv:0903.1783. [7] L. M. Hai, N. Q. Dieu and N. X. Hong, L2 -Approximation of differential forms by ∂-closed ones on smooth hypersurfaces, J. Math. Anal. Appl., 383(2011), 379-390. [8] L. H. Ho, ∂-problem on weakly q-convex domains, Math. Ann., 290(1991), 3–18. [9] N. Sibony, Une classes des domaines pseudoconvexe, Duke Math. J., 55(1987), 299319. [10] E. J. Straube, Plurisubharmonic functions and subellipticity of the ∂-Neumann problem on non-smooth domains, Math.Res. Lett. 4(1997), 459 - 467. [11] E. J. Straube, Lectures on the L2 -Sobolev Theory of the ∂-Neumann Problem, ESI Lectures in Mathematics and Physics, European Mathematical Society, Z¨ urich, 2010. [12] J. Walsh, Continuity of envelopes of plurisubharmonic functions, J. Math. Mech., 18(1968), 143–148. Department of Mathematics, Hanoi National University of Education (Dai hoc Su Pham Ha Noi), 136 Xuan Thuy Street, Caugiay District, Ha Noi, Viet Nam E-mail address: mauhai@fpt.vn, dieu vn@yahoo.com, xuanhongdhsp@yahoo.com [...]... obtain ∥u∥2W 1 (0,q) ∗ (VM ) D(∥∂u∥2 + ∂ u∥2 ), with D independent of u Hence Rellich’s Lemma in [3] implies that the ∗ restriction map RM : dom (∂) ∩ dom (∂ ) −→ L2(0,q) (VM ) is compact and the ∗ desired conclusion follows On the other hand, for u ∈ dom (∂) ∩ dom (∂ ), EXISTENCE AND COMPACTNESS FOR THE ∂- NEUMANN 19 after rescaling, we have ∥u∥ ∥u∥Ω\KM + ∥u∥VM 1 ∗ (∥∂u∥ + ∂ u∥) + ∥RM u∥ M It now follows... easy to see that v is continuous on C and C 2 smooth outside the origin On the other hand, from the choice of h we check easily that the functions v + ε|z|2 and v − M |z|2 are subharmonic on 0 < |z| < b′ and 0 < |z|, respectively Since these functions are continuous at the origin, they are indeed subharmonic on |z| < b′ and C, respectively Finally, we obtain the desired function u by setting u(z) =... and NqΩ exist Then ′ NqΩ is compact if and only if so is NqΩ Proof Since the roles of Ω and Ω′ are similar, it is enough to prove that if ′ NqΩ is compact then so is NqΩ By the proof of the implication (iv) → (i) given in Proposition 4.2 of [11], it suffices to prove the canonical solution ∗ ∗ Ω operators ∂ NqΩ and ∂ Nq+1 are compact For simplicity of the exposition, ∗ we show only compactness of ∂. .. numbers and f be a real valued C 2 smooth function on the interval (c2 , d2 ) Then the function u(z) := f (|z|2 ) is subharmonic on the annulus c < |z| < d if and only if f ′ (x) + xf ′′ (x) ≥ 0, ∀c2 < x < d2 For the proof of the lemma, we fix a < a′ < b′ < b Next, we pick a continuous function h on [0, b′2 ] such that h > −ε on [0, b′2 ], h ≥ M on [0, a′2 ] and h(b′2 ) = 0 We define the following functions... strictly q-subharmonic on Ω We will prove that θ satisfies the condition given in Definition 3.1 For this, we note that the following inequalities holds ( ) 1 Hq (ρ)(z) Hq (ρ)(z) i ∂ ∧(i ∂| z|2 )q−1 ≥ (i ∂| z|2 )q ≥ (i ∂| z|2 )q 2 1 − ρ(z) (1 − ρ2 (z)) 2(1 + ρ(z)2 ) So using assumption (b) we infer that for every M > 0 there exists rM > 0 1 − M |z|2 is q-subharmonic on Ω \ B(0, rM ) Now using the such that... Using the assumptions on the Jacobian of φ, we see that the linear maps φ∗ : L2(0,q) (Ω′ ) → L2(0,q) (Ω) and (φ−1 )∗ : L2(0,q) (Ω) → L2(0,q) (Ω′ ) are bounded and preserve ∂- closed forms Here φ∗ and (φ−1 )∗ are pull backs of φ and φ−1 , respectively Next, let EXISTENCE AND COMPACTNESS FOR THE ∂- NEUMANN 17 ′ 2 M(0,q) (Ω′ ) := L2(0,q) (Ω) ∩ ker (∂) and PqΩ : L2(0,q) (Ω′ ) → M(0,q) (Ω′ ) be the orthogonal... strictly q-subharmonic function ρ˜ So by Theorem 3.3 Ω has the property (Pq′ ) Finally, the domain Ω is q-convex since it admits the q-subharmonic exhaustion function ψ(z) := −1/˜ ρ(z) + |z|2 The following proposition is useful for the proof of the main result Proposition 3.6 Let Ω be an open set in Cn and assume that Ω satisfies the property (Pq′ ) Then the function φ in Definition 3.1 can be chosen... solution f with L2 minimal norm of the equation ∂f = u (see Corollary 2.10 in [11]) This fact, combined with the property of the ∗ ′ ′ projection PqΩ also imply the desired claim Thus ∂ NqΩ is compact The proof is complete Now we are ready to formulate the main result of this section Theorem 4.3 Let Ω ⊂ Cn be a q-convex domain having the property (Pq′ ) Then there exists a bounded and compact ∂- Neumann. .. NGUYEN AND XUAN HONG NGUYEN Let u as above and choose a large ball B such that the support of u is contained in Ω ∩ B By applying Proposition 2.10 on the bounded q-convex domain Ω ∩ B with b = φ − 1 we have ∫ n ∑ ∑ ∂ 2φ −1 2 ′ e qε∥u∥ eφ−1 ujK ukK dλ2n ∂zj ∂z k j,k=1 |K|=q−1 ∑ Ω∩B ′ n ∑ ∫ e φ−1 |K|=q−1 j,k=1 Ω ∂ 2φ ujK ukK dλ2n ∂zj ∂z k ∗ ∥∂u∥2 + ∂ u∥2 Hence ∥u∥2 e ∗ ∗ (∥∂u∥2 + ∂ u∥2 ) = C(∥∂u∥2 + ∂. .. Bj ) so ∂ ∩ Bj has property (Pq ) Therefore, Corollary 4.13 in [11] implies that ∂ also has property (Pq ) The proof is complete 4 existence and compactness of the ∂- Neumann operator We start with a geometric sufficient condition for compactness of Nq which is a slight extension of an example given in [5] Proposition 4.1 Let Ω be a domain in Cn , 0 ∈ Ω and 1 ≤ q ≤ n Suppose that there exist constants ... restriction map RM : dom (∂) ∩ dom (∂ ) −→ L2(0,q) (VM ) is compact and the ∗ desired conclusion follows On the other hand, for u ∈ dom (∂) ∩ dom (∂ ), EXISTENCE AND COMPACTNESS FOR THE ∂- NEUMANN. .. sufficient conditions for domain Ω having the property (Pq′ ) The existence and compactness of the ∂- Neumann operator Nq on q-convex domains are discussed in Section We start the section with a... ρ˜j (z) i ∂ ∧(i ∂| z| ) ≥ i ∂ ρ ˜ ∧(i ∂| z| ) e ≥ (i ∂| z| )e j 2j EXISTENCE AND COMPACTNESS FOR THE ∂- NEUMANN 11 Observe that lim ρ˜j (z) = for every j such that j > |ξ| + 1, so the above z→ξ