Existence and compactness for the ∂-Neumann operator on q-convex domains

1 23 Manuscripta Mathematica ISSN 0025-2611 manuscripta math. DOI 10.1007/s00229-014-0661-2 Existence and compactness for the $ ${\overline{\partial}}$$ ∂ ¯ -Neumann operator on q-convex domains Mau Hai Le, Quang Dieu Nguyen & Xuan Hong Nguyen 1 23 Your article is protected by copyright and all rights are held exclusively by Springer- Verlag Berlin Heidelberg. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. manuscripta math. © Springer-Verlag Berlin Heidelberg 2014 Mau Hai Le · Quang Dieu Nguyen · Xuan Hong Nguyen Existence and compactness for the ∂-Neumann operator on q-convex domains Received: 3 April 2012 / Revised: 11 December 2012 Abstract. The aim of this paper is to give a sufficient condition for existence and compact- ness of the ∂-Neumann operator Nq on L2(0,q)() in the case  is an arbitrary q-convex domain in Cn . 1. Introduction Let  be a domain in Cn . According to the fundamental work of Kohn and Hör- mander in the sixties, if  is bounded and pseudoconvex then for every 1 ≤ q ≤ n, the complex Laplacian q on square integrable (0, q) forms on  has a bounded inverse, denoted by Nq . This is the ∂-Neumann operator Nq . The most basic prop- erty of Nq is that if v is a ∂ closed (0, q + 1)-form, then u := ∂∗Nq+1v 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  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, Gansberger and Haslinger studied compactness estimates for the ∂-Neumann oper- ator 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 M. H. Le · Q. D. Nguyen(B)· X. H. Nguyen: Department of Mathematics, Hanoi National University of Education (Dai hoc Su Pham Ha Noi), 136 Xuan Thuy Street, Caugiay District, Hanoi, Vietnam. e-mail: dieu_vn@yahoo.com M. H. Le: e-mail: mauhai@fpt.vn X. H. Nguyen: e-mail: xuanhongdhsp@yahoo.com Mathematics Subject Classification (2010): Primary 32W05 DOI: 10.1007/s00229-014-0661-2 Author's personal copy M. H. Le et al. and the Laplacian ϕ at the infinite point and at boundary points of a domain  (Proposition 4.5 in [5]). The main step in Gansberger’s proof is to show that under the above assumption the embedding H10 (, ϕ,∇ϕ) ↪→ L2(, ϕ) is compact. The paper is organized as follows. In Sect. 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 (P ′q), a slight modification of the property (Pq) introduced earlier in the case of bounded domains by Catlin [2] and Straube [10,11]. Roughly speaking, we say that a (possibly unbounded) domain  has the property (P ′q) if  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 ∂. We should say that this notion is motivated from Theorem 1.3 in [5] in which the case where q = 1 and ∂ is smooth is studied. The main result of the section is Theorem 3.3 which gives sufficient conditions for domain  having the property (P ′q). The existence and compactness of the ∂-Neumann operator Nq on q-convex domains are discussed in Sect. 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  ⊂ Cn is a q-convex domain having property (P ′q) then there exists a bounded ∂-Neumann operator Nq on L2(0,q)() and Nq is compact. Our proof exploits the property (P ′ q) of  and some techniques from the book [11]. 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 ∈ C2() 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. 2. Preliminaries A complex-valued differential form u of type (0, q) on an open subset  ⊂ Cn can be expressed as u = ∑|J |=q ′u J dz J , where J are strictly increasing multi-indices with lengths q and {u J } are defined functions on . Let C∞(0,q)() be the space of complex-valued differential forms of class C∞ and of type (0, q) on . By C∞0 () we denote the space of C∞ functions with compact support in . We use L2(0,q)() to denote the space of (0, q)-forms on  with square-integrable coefficients. If u, v ∈ L2(0,q)(), the weighted L2-inner product and norms are defined by (u, v) = ∫  ∑ |J |=q ′ u J v J dλ2n and ‖u‖2 = (u, u). The ∂-operator on (0, q)-forms is given by ∂ ( ∑ |J |=q ′ u J dz J ) = ∑ |J |=q ′ n∑ j=1 ∂u J ∂z j dz j ∧ dz J , Author's personal copy Existence and compactness for the ∂-Neumann operator where ∑ ′ means that the sum is only taken over strictly increasing multi-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)(). So ∂ is a densely defined closed operator, and therefore has an adjoint operator from L2(0,q+1)() into L 2 (0,q)() denoted by ∂ ∗ . For u = ∑|J |=q+1 ′u J dz J ∈ dom(∂∗) one has ∂ ∗ u = − ∑ |K |=q ′ n∑ j=1 ∂u j K ∂z j dzK . 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 C∞0 (). 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 prob- lems in several complex variables. Next, we recall the definition of q-subharmonic functions which is an extension of plurisubharmonic functions (see [1,7,8]). Definition 2.1. Let  be a domain in Cn . An upper semicontinuous function u :  −→ [−∞,∞), u ≡ −∞ is called q-subharmonic if for every q-dimensional complex plane L in Cn , u|L is a subharmonic function on L ∩ . The set of all q-subharmonic functions on  is denoted by SHq(). The function u is called to be strictly q-subharmonic if for every U   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 plurisubharmonic- ity (resp. subharmonicity). We list below basic properties of q-subharmonic functions that will be used later on (see [7]). Proposition 2.3. Let  be an open set in Cn and let q is an integer with 1  q  n. Then we have. (a) If u ∈ SHq() then u ∈ SHr (), for every q  r  n. (b) If u, v ∈ SHq() and α, β > 0 then αu + βv ∈ SHq(). (c) If {u j }∞j=1 is a family of q-subharmonic functions, u = sup j u j < +∞ and u is upper semicontinuous then u is a q-subharmonic function. (d) If {u j }∞j=1 is a family of nonnegative q-subharmonic functions such that u =∑∞ j=1 u j < +∞ and u is upper semicontinuous then u is q-subharmonic. Author's personal copy M. H. Le et al. (e) If {u j }∞j=1 is a decreasing sequence of q-subharmonic functions then so is u = lim j→+∞ u j . (f) Let ρ ≥ 0 be a smooth radial function in Cn vanishing outside the unit ball and satisfying ∫ Cn ρdV = 1. For u ∈ SHq() we define uε(z) := (u ∗ ρε)(z) = ∫ B(0,ε) u(z − ξ)ρε(ξ)dλ2n(ξ), ∀z ∈ ε, where ρε(z) := 1ε2n ρ(z/ε) and ε = {z ∈  : d(z, ∂) > ε}. Then uε ∈ SHq(ε) ∩ C∞(ε) and uε ↓ u as ε ↓ 0. (g) Let u1, . . . , u p ∈ SHq() 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, . . . , u p) ∈ SHq(). 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 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  be a domain in Cn and let q be an integer with 1  q  n. Let u be a real valued C2-function defined on . Then the following are equivalent: (a) u is a q-subharmonic function . (b) i∂∂u ∧ (i∂∂|z|2)q−1  0 i.e., Hq(u)(z) ≥ 0 for every z ∈ . (c) For every (0, q)-form f = ∑|J |=q ′ f J dz J we have ∑ |K |=q−1 ′ n∑ j,k=1 ∂2u ∂z j∂zk f j K f kK  0. We also have the following simple result about smoothing q-subharmonic functions. Proposition 2.5. Let  be an open set in Cn and let u ∈ SHq() such that u − δ|idCn |2 ∈ SHq() for some δ > 0. Then for every ε > 0 we have uε − δ|idCn |2 ∈ SHq(ε), where ε := {z ∈  : d(z, ∂) > ε}. Proof. By Proposition 2.3 (f) we have (u − δ|idCn |2)ε ∈ SHq(ε). Since (u − δ|idCn |2)ε(z) = uε(z) − δ ∫ B(0,ε) |z − w|2ρε(w)dV (w) = uε(z) − δ|z|2 − δ ∫ B(0,ε) (2(z,−w) + |w|2)ρε(w)dV (w) = uε(z) − δ|z|2 − v(ε)(z), Author's personal copy Existence and compactness for the ∂-Neumann operator where v(ε)(z) := δ ∫ B(0,ε)(2(z,−w) + |w|2)ρε(w)dV (w) is a pluriharmonic function in Cn . Hence, uε − δ|idCn |2 = (u − δ|idCn |2)ε + v(ε) ∈ SHq(ε). This completes the proof. unionsq The following definition is an extension of pseudoconvexity. Definition 2.6. A domain  ⊂ Cn is said to be q-convex if there exists a q- subharmonic exhaustion function on . Moreover, a C2 smooth bounded domain  is called strictly q-convex if it admits a C2 smooth defining function which is strictly q-subharmonic on a neighbourhood of . 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 q-convex domain  can be written as an increasing union of bounded q-convex domains  j such that each  j has a smooth strictly q-convex exhaustion function. Using Sard theorem this result can be refined as follows. Proposition 2.7. Let  be a q-convex domain in Cn. Then  can be written as  = ∞⋃ j=1  j such that  j   j+1 and each  j is a strictly q-convex domain. 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  be a domain in Cn and p ∈ ∂. Assume that there exist r > 0, a sequence {p j } ⊂ , p j → p, and a sequence of q-dimensional complex subspaces L j satisfying the following conditions: (a) p j ∈ L j for every j ≥ 1, (b) ∂B(p, r) ∩ L j is contained in a fixed compact subset K of . Then  is not q-convex. Proof. Suppose that there exists a q-subharmonic exhaustion function u on . Since u|L j is subharmonic, by the maximum principle and the assupmtions (a) and (b) we get for every j ≥ 1 the inequality u(p j ) ≤ sup K u. By letting j → ∞ we arrive at a contradiction. unionsq Remark 2.9. (i) Let  := {z ∈ C3 : 1 < |z| < 3} and p = (1, 0, 0) ∈ ∂. Denote by L the hyperplane tangent to ∂ at the point p. Consider the sequences of points p j := (1 + 1/j, 0, 0), j ≥ 1 tending to p and hyper- planes L j := L + (1/j, 0, 0) passing through the points p j . Using the above result, we can see that  is not 2-convex. (ii) In [8], the following generalization of Levi convexity was introduced. We say that a bounded domain  in Cn with a C2 smooth defining function ρ is weakly q-convex if for every p ∈ ∂, for every (0, q)-form f = ∑|J |=q ′ f J dz J Author's personal copy M. H. Le et al. satisfying n∑ i=1 ∂ρ ∂zi (p) fi K = 0 ∀|K | = q − 1, we have ∑ |K |=q−1 ′ n∑ j,k=1 ∂2ρ ∂z j∂zk f j K f kK  0. 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  be a bounded q-convex domain in Cn, u = ∑|J |=q ′u J dz¯ J ∈ dom(∂¯) ∩ dom(∂¯∗) ⊂ L2(0,q)(). Then for all b ∈ C2(), b  0 the following holds ∑ |K |=q−1 ′ n∑ j,k=1 ∫  eb ∂2b ∂z j∂ z¯k u j K ukK dλ2n  ‖∂u‖2 + ‖∂¯∗u‖2. 3. The property (P′q) First we recall an important concept introduced and investigated by Catlin in [2] and 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 C2 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–Hörmander formula in [11], it is not hard to prove (see [2] for the case q = 1 and [11] for general q) that if  is a smoothly bounded pseudoconvex domain in Cn with the boundary ∂ 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  is unbounded. Definition 3.1. Let  be a domain in Cn . We say that  has the property (P ′q) if there exists a real valued, bounded C2 smooth function ϕ on  such that for every positive number M , we have ϕ(z) − M |z|2 ∈ SHq(\KM ) for some compact subset KM of . Author's personal copy Existence and compactness for the ∂-Neumann operator Remark 3.2. (i) The function ϕ is not assumed to be q-subharmonic on the whole . We will prove, however, that ϕ can be chosen to have this additional prop- erty. (ii) If  has the property (P ′q) then for every complex space L of dimension q, ∩ L is quasibounded (in L) i.e.,  ∩ L contains only a finite number of 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(z j , r) contained in  with |z j | → ∞. By passing to a subsequence, we can find a sequence m j of real numbers such that m j → +∞ and ϕ(z)− m j |z|2 ∈ SHn(B(z j , 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 θ j (z) := θ(z − z j ). By Stoke’s theorem we have ∫ B(z j ,r/2) i∂∂ϕ ∧ (i∂∂|z|2)n−1 ≤ ∫ B(z j ,r) θ j i∂∂ϕ ∧ (i∂∂|z|2)n−1 = ∫ B(z j ,r) iϕ∂∂θ j ∧ (i∂∂|z|2)n−1 ≤ C‖ϕ‖λ2n(B(z j , 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 m j ≤ C ′‖ϕ‖ for every j ≥ 1. By letting j → ∞ we get a contradiction. It is easy to see that finite intersection of domains possessing the (P ′q) property still has this property. The main result of the section provides a substantial class of domains satisfying the property (P ′q). More precisely, we have Theorem 3.3. Let  be an open set in Cn with ∂ locally has the property (Pq). Assume that there exist negative q-subharmonic functions ρ, ρ˜ on  satisfying the following conditions. (a) ρ ∈ C2(), ρ(z) − |z|2 ∈ SHq(). (b) lim|z|→∞ Hq (ρ)(z)1+ρ(z)2 = ∞. (c) ρ˜ is strictly q-subharmonic on  and satisfies lim z→ξ ρ˜(z) = 0 for every ξ ∈ ∂. Then  has the property (P ′q). 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  be a bounded domain in Cn with ∂ has the property (Pq). Assume that there exists a negative strictly q-subharmonic exhaustion function ϕ of . Then for every real valued continuous function f on ∂ the function P B f,(z) := sup{u(z) : u ∈ SHq(), lim sup x→ξ u(x) ≤ f (ξ) ∀ξ ∈ ∂} belongs to SHq() ∩ C() and P B f,|∂ = f. Author's personal copy M. H. Le et al. Proof. Since ∂ has the property (Pq), by Proposition 4.10 in [11], there exists a sequence f j of continuous q−subharmonic functions on neighborhoods of ∂ such that f j converges uniformly to f on ∂. By Proposition 2.3 (f), after taking convolution with a smoothing kernel we may achieve that f j is q−subharmonic and C2 smooth near ∂ for every j . Now we fix j ≥ 1 and choose a real valued C2 smooth function θ j on Cn with compact support such that θ j = 1 on a small neighborhood of ∂. Since ϕ is strictly q-subharmonic on , by taking a constant M j > 0 large enough the function Fj (z) := M jϕ(z) + θ j (z) f j (z) will belong to SHq() ∩ C() and satisfies Fj |∂ = f j . So the function P B f j ,(z) := sup{u(z) : u ∈ SHq(), lim sup x→ξ u(x) ≤ f j (ξ) ∀ξ ∈ ∂} satisfies lim inf z→ξ P B f j ,(z) ≥ limz→ξ Fj (z) = f j (ξ) ∀ξ ∈ ∂. On the other hand, since ϕ is a negative subharmonic exhaustion function of , by a well known result in potential theory we know that  is regular with respect to the Dirichlet problem for Laplacian. So we can find a real valued continuous function Hj on  which is harmonic on  and satisfies Hj = f j on ∂. By the maximum principle for subharmonic functions we obtain P B f j , ≤ Hj on . Therefore lim sup z→ξ P B f j ,(z) ≤ lim z→ξ Hj (z) = f j (ξ) ∀ξ ∈ ∂. Summing up, we have proved that lim z→ξ P B f j , = f j (ξ) ∀ξ ∈ ∂. Now we apply Lemma 1 in [12] to conclude that P B f j , is in fact continuous on . 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