ESTIMATES FOR THE ¯∂-NEUMANN PROBLEM AND

NONEXISTENCE OF LEVI-FLAT HYPERSURFACES IN ℂP^{n}

Jianguo Cao^{1}, Mei-Chi Shaw^{2} and Lihe Wang^{3}

Abstract. Let Ω be a pseudoconvex domain with C^{2}-smooth boundary in ℂP^{n}. We prove that the ¯∂-Neumann operator N exists for (p, q)-forms on Ω. Furthermore, there exists a t_{0} > 0 such that the operators N, ¯∂^{∗}N, ¯∂N and the Bergman projection are regular in the Sobolev space W^{t}(Ω̄) for t < t_{0}.

The boundary estimates above have applications in complex geometry. We use the estimates to prove the nonexistence of C^{2,α} real Levi-flat hypersurfaces in ℂP^{n}. We also show that there exist no non-zero L^{2}-holomorphic (p, 0)-forms on any pseudoconcave domain in ℂP^{n} with p > 0.

Introduction

One of the main results in this paper is the following:

Theorem 1. There exists no C^{2,α} real Levi-flat hypersurface M^{2n−1} in ℂP^{n}, where 0 < α < 1 and n ≥ 2.

Theorem 1 is inspired by the recent papers of Siu [Siu2,3] who proved that there exists no C^{8} Levi-flat hypersurface in ℂP^{n}, n ≥ 2. The required smoothness has been reduced to C^{4} by Iordan [Io]. Nonexistence of real analytic Levi-flat hypersurfaces in ℂP^{n} was obtained in Lins Neto [LNe] for n ≥ 3 and Ohsawa [Oh] for n = 2. Our proof of Theorem 1 follows arguments along the lines of [Siu2,3] who reduced the proof of Theorem 1 to the regularity of the tangential Cauchy-Riemann equations on M.

We derive new boundary regularity results for ¯∂-equation and the ¯∂-Neumann problem for domains in ℂP^{n}. Our Theorem 2 stated below can be applied to not only domains Ω with Levi-flat boundaries M but also all other weakly pseudoconvex C^{2} domains in ℂP^{n}, when compared with the earlier work mentioned above. The boundary regularity of the solutions is interesting itself.

To prove Theorem 1, Siu [Siu2,3] made the following observation: If there exists a C^{2}-smooth real Levi-flat hypersurface M = M^{2n−1} in ℂP^{n} and if ℂP^{n} has the standard Fubini-Study metric, then the curvature form iΘ̃^{N} of its complex normal line bundle N_{M,ℂP^{n}} is strictly positive definite on T^{(1,0)}(M) ⊕ T^{(0,1)}(M) by the Cartan-Chern-Gauss structure equation.

1Partially supported by NSF grant DMS-0102552. J. Cao is grateful to the University of Michigan for the hospitality. Research at MSRI is supported in part by NSF grant DMS-9810361.

2Partially supported by NSF grant DMS 01-00492.

3Partially supported by NSF grant DMS 01-00679.

Furthermore, one has Θ̃^{N} = dθ is an exact form, where θ is the connection form of the complex normal line bundle N_{M,ℂP^{n}}. If M is a Levi-flat hypersurface, it is locally foliated by complex manifolds of complex dimension n − 1. It is known that the restriction of Θ̃^{N} to each complex leaf of M, Θ_{b} = Θ̃^{N}|_{[T (M)]_{ℝ}∩J[T (M)]_{ℝ}} is a (1, 1)-form, where J is the complex structure of ℂP^{n} (see Proposition 1.2 below).

On the other hand, if one could show that Θ_{b} is ∂_{b} ¯∂_{b}-exact for some continuous real-valued function h (i.e., Θ_{b} = ∂_{b} ¯∂_{b}h on M), then iΘ_{b} is non-positive at the maximum point of h on M (cf. [Ca]). This contradicts the fact that iΘ̃^{N} is strictly positive definite on T^{(1,0)}(M) ⊕ T^{(0,1)}(M), and hence Theorem 1 would follow immediately.

Therefore, the proof of Theorem 1 is reduced to a problem of finding a continuous function u satisfying

(0.1)  i∂_{b} ¯∂_{b}u = f_{b} = f|_{[T (M)]_{ℝ}∩J[T (M)]_{ℝ}} on M,

under the condition that f = dθ is an exact real-valued (1, 1)-form when restricted to T^{(1,0)}(M) ⊕ T^{(0,1)}(M). Equation (0.1) corresponds to the classical ∂ ¯∂-Lelong equation

(0.2)  i∂ ¯∂ũ = f̃ in ℂP^{n}

where f̃ = dθ̃ is an exact real-valued (1, 1)-form. Using the fact that f̃ is an exact real-valued (1,1)-form to solve equation (0.2), it suffices to solve

(0.3)  ¯∂ũ = θ̃^{(0,1)} in ℂP^{n},

where θ̃^{(0,1)} is the (0, 1) part in θ̃, which is ¯∂-closed in ℂP^{n}. By the Hodge theory, there is no nontrivial harmonic (0, 1)-form in ℂP^{n}. It follows that any ¯∂-closed (0,1)-form θ̃^{(0,1)} on ℂP^{n} must be ¯∂-exact, and hence equation (0.3) can be solved easily (e.g., cf. [Zh]). For the proof of Theorem 1 and equation (0.1), one can similarly deduce that it suffices to solve

(0.4)  ¯∂_{b}u = θ_{b} in M,

where θ_{b} is a (0, 1)-form in M satisfying some compatibility condition. In [Siu2,3], existence and regularity of equation (0.4) on a Levi-flat boundary M in ℂP^{n} is studied. In this paper, we study the more general situation when M is the C^{2} boundary of any pseudoconvex domain Ω in ℂP^{n}.

When n > 2, the compatibility condition for equation (0.4) to be solvable is that

(0.5)  ¯∂_{b}θ_{b} = 0 on M.

When n = 2, (0.5) is satisfied trivially and the compatibility condition for equation (0.4) is substituted by the following moment condition:

(0.6)  ∫_{M} θ_{b} ∧ Ψ = 0,

where Ψ is any ¯∂_{b}-closed (2,0)-form on M (see (9.2.12a) of [CS, p216].) To show that such a compatibility condition holds for our case with n = 2, we also derived several Liouville type theorems which are of independent interest. In particular, we show that on any pseudoconcave domain Ω with C^{2} boundary in ℂP^{n}, there exist no non-zero L^{2}-integrable holomorphic (p, 0)-forms on Ω with p > 0, (cf. Proposition 4.5 below).

If one can extend a (0, 1)-form θ_{b} satisfying (0.5) or (0.6) from the boundary M = bΩ to a ¯∂-closed form θ̃ on the domain Ω, then (0.4) can be solved by restricting the solution ũ of the ¯∂-equation (0.3) to M = bΩ. To study the ¯∂-closed extension from M to Ω, one can formulate it as a ¯∂-Cauchy problem of finding solutions to the ¯∂-equation (0.3) with prescribed compact support. When Ω is strictly pseudoconvex with smooth boundary, the ¯∂-closed extension on Ω was pioneered in the work of Kohn and Rossi [KoR] using the boundary regularity of the ¯∂-Neumann operator on strongly pseudoconvex domains. When Ω ⊂ ℂ^{n} is only pseudoconvex, the ¯∂-closed extension for forms on M = bΩ to Ω can also be obtained if one can obtain the boundary regularity for the ¯∂-Neumann problem or weighted ¯∂-Neumann problem (see [Sh,BSh] or [CS, Chap.9]).

In our case, the connected real hypersurface M divides ℂP^{n} into two connected domains: Ω_{+} and Ω_{−}. Both Ω_{+} and Ω_{−} are pseudoconvex and have Levi-flat boundaries. Using the boundary regularity of the ¯∂-Neumann operator, a two-sided ¯∂-closed extension θ̃_{±} will be constructed for any form θ_{b} satisfying the compatibility condition (0.5) or (0.6) on the Levi-flat hypersurface M, where the ¯∂-closed extension θ̃ = θ̃_{±} is defined on the whole space ℂP^{n} = \overline{Ω_{+} ∪ Ω_{−}}.

We let N = N_{(p,q)} denote the ¯∂-Neumann operator on Ω, i.e., the inverse operator for the Laplace operator □_{(p,q)} acting on (p, q)-forms with the ¯∂-Neumann boundary condition on Ω. Let W_{(p,q)}^{s}(Ω) be the space of all (p, q)-forms whose coefficients are in the Sobolev space W^{s}(Ω). To prove Theorem 1, we first show that N = N_{(p,q)}|_{Ω} exists and that the operators N, ¯∂^{∗}N and ¯∂N are bounded on W^{s} for some s > 0, when Ω ⊂ ℂP^{n} is a pseudoconvex domain with C^{2} boundary.

To formulate our main regularity result for the ¯∂-Neumann operator N, we introduce a geometric invariant for any pseudoconvex domain Ω ⊂ ℂP^{n} as follows:

Definition 0.1. Let Ω be a pseudoconvex domain in a Kähler manifold with C^{2}- smooth boundary. Let δ(x) = δ_{bΩ}(x) = d(x, bΩ) be a distance function from the boundary. We call t_{0} = t_{0}(Ω) the order of plurisubharmonicity for the distance function δ_{bΩ} if

(0.7)  t_{0}(Ω) = sup{0 < ϵ ≤ 1|i∂ ¯∂(−δ^{ϵ}) ≥ 0 on Ω}.

When t_{0}(Ω) = 1, the condition implies that there exists a plurisubharmonic defining function on Ω. For a general smooth pseudoconvex domain in ℂ^{n}, such a plurisubharmonic defining function does not necessarily exist (cf. [DF2]). However, Diederich-Fornaess [DF1] showed that there exists a 0 < t_{0}(Ω) ≤ 1 for any pseudoconvex domain Ω in ℂ^{n} with C^{2}-smooth boundary (for a simple proof for pseudoconvex domains in ℂ^{n} with C^{3} boundary, see Range [R]). In ℂP^{n}, Ohsawa-Sibony [OS] showed that there exists 0 < t_{0}(Ω) ≤ 1 for any pseudoconvex domain Ω ⊂ ℂP^{n} with C^{2}-smooth boundary using results of Takeuchi [Ta] and [DF1].

Theorem 2. Let Ω be a pseudoconvex domain with C^{2}-smooth boundary in ℂP^{n} and let t_{0}(Ω) be the order of plurisubharmonicity for the distance function δ_{bΩ}. Then the ¯∂-Neumann operator N_{(p,q)} exists on L_{(p,q)}^{2}(Ω) where 0 ≤ p, q ≤ n. Furthermore, N, ¯∂N, ¯∂^{∗}N and the Bergman projection P are exact regular operators on W_{(p,q)}^{s}(Ω) for 0 < s < \frac{1}{2}t_{0} with respect to the W^{s}(Ω)-Sobolev norms.

The Bergman projection P is the orthogonal projection from L_{(p,q)}^{2}(Ω) to ker(¯∂). In general, P or the ¯∂-Neumann operator N is not necessarily a bounded operator from W_{(p,q)}^{s}(Ω) to W_{(p,q)}^{s}(Ω) for smooth pseudoconvex domains Ω (cf. [Ba]). In fact, Barrett [Ba] showed that, for any given β with 0 < β < 1, there is a pseudoconvex domain Ω_{β} (the so-called Diederich-Fornaess’ worm domain [DF2]) with C^{∞}-smooth boundary such that the Bergman projection P and the ¯∂-Neumann operator N on Ω_{β} are not bounded from W_{(0,1)}^{β}(Ω_{β}) to itself. Christ [Chr] showed that P does not map C^{∞}(Ω̄_{β}) into C^{∞}(Ω̄_{β}) for such a domain Ω_{β}. Therefore, one can only expect the boundary regularity of the ¯∂-Neumann operator N to be regular in W_{(p,q)}^{s}(Ω) with small s for general pseudoconvex domains.

When Ω ⊂⊂ ℂ^{n} with C^{∞}-smooth boundary and Ω has a plurisubharmonic defining function, Boas and Straube [BS] prove that the conclusion of Theorem 2 holds for any s > 0. This corresponds to the case when t_{0}(Ω) = 1. For the case t_{0}(Ω) < 1 and Ω ⊂ ℂ^{n}, Kohn [Ko2] and Berndtsson-Charpentier [BC] obtain the Sobolev regularity for the operators ¯∂^{∗}N and the Bergman projection P. Boas-Straube [BS1] show that regularity for the Bergman projection and the ¯∂-Neumann operator are equivalent for C^{∞}-smooth pseudoconvex domains in ℂ^{n} (and also for smooth pseudoconvex domains in complex manifolds with strongly plurisubharmonic functions in a neighborhood of the boundary). For pseudoconvex domains Ω ⊂ ℂP^{n}, such a smooth strictly plurisubharmonic function ϕ might not exist in a neighborhood of the boundary bΩ. Therefore, the proof of Theorem 2 has extra difficulties when compared to the case of domains in ℂ^{n}. We remark that our proof of the regularity for the Bergman projection and ¯∂^{∗}N is similar to the proof in [BC], but the regularity results of the ¯∂-Neumann operator N and ¯∂N in Theorem 2 are new, even for domains with C^{2}-smooth boundaries in ℂ^{n}.

The W^{s}(Ω)-regularity result for ¯∂N is sufficient and crucial in the proof of Theorem 1. The key observation is that the ¯∂_{b} equation (0.4), when restricted to each complex foliation leaf of M, is elliptic. Thus, any solution for (0.4) has Hölder regularity on each leaf from the elliptic theory. To prove continuity of the solution for equation (0.4), one only needs to prove continuity of the solution in the transversal direction, which can be proved by using the Besov norms and a finite difference scheme. Details are given in Section 5.

It should be pointed out that there exist non-smooth Levi-flat real hypersurfaces in ℂP^{2}. Recall that a C^{1} or Lipschitz real hypersurface M^{2n−1} is called Levi-flat in a complex manifold ℂP^{n} if ℂP^{n} \ M consists of two pseudoconvex domains. For example, let

M = {[z_{0}, z_{1}, z_{2}] ∈ ℂP^{2} | |z_{0}| = |z_{1}|},

where (z_{0}, z_{1}, z_{2}) are homogeneous coordinates in ℂP^{2}. Then M is Levi-flat since ℂP^{2} \ M consists of two pseudoconvex domains. The hypersurface M is smooth except at (0, 0, 1), where M is neither C^{1} nor Lipschitz (in the sense of a Lipschitz graph locally). It is still an open question if there exist Lipschitz or C^{2} Levi-flat hypersurfaces in ℂP^{n}.

In addition, the positive curvature condition (or the irreducible condition) is needed for the possible generalization of Theorem 1. One can inspect the example S^{1} × ℂP^{1}, which is a C^{∞}-smooth Levi-flat hypersurface in ℂP^{1} × ℂP^{1}. Clearly, ℂP^{1} × ℂP^{1} does not admit a Kähler metric with positive holomorphic bisectional curvature, (e.g., cf. [HSW]). Any compact Kähler manifold with positive holomorphic bisectional curvature must be biholomorphic to ℂP^{n} (e.g., cf. [Mok]). Although results in this paper are stated for domains in ℂP^{n}, the conclusions of these results remain to be true for corresponding domains in any compact Kähler manifold with positive holomorphic bisectional curvature as well.

The plan of this paper is as follows: In Section 1 we derive some preliminary results on equidistant real hypersurfaces in ℂP^{n} for the proof of Theorems 1 and 2. In Section 2 we discuss the L^{2} existence theorems of ¯∂ and the ¯∂-Neumann problem on domains in ℂP^{n}. In Section 3 we prove that the ¯∂-Neumann operator on pseudoconvex domains with C^{2} boundary is regular in some Sobolev spaces as stated in Theorem 2. Using the existence and regularity of the ¯∂-Neumann operator, one can study the solution of the ¯∂-equation with prescribed support, including the case of forms with top degrees. We construct ¯∂-closed extensions from the boundary and prove the existence and regularity of equation (0.4) in Section 4. The proof of Theorem 1 is completed in Section 5.

1. Equidistant hypersurfaces and preliminaries for Theorems 1-2

In this section, we first study the geometry of equidistant real hypersurfaces which will be needed to prove Theorem 1. Afterwards we recall four preliminary results which are indispensable in the proof of Theorem 2.

Suppose that Ω is a bounded domain with C^{2}-smooth boundary bΩ in ℂP^{n}. Let bΩ_{s} = {x ∈ ℂP^{n}|d(x, bΩ) = s} and {bΩ_{s}}_{s<ϵ} be a family of equidistant real hypersurfaces from bΩ in ℂP^{n}.

We shall show that the connection and the curvature form of the complex line bundle of equidistant hypersurfaces {bΩ_{s}}_{s<ϵ} are C^{α}, as long as bΩ is a C^{2,α}-smooth hypersurface in ℂP^{n}. We will use the Cartan-Chern theory to prove this assertion as follows.

Observe that, if M = bΩ is a C^{2}-smooth connected hypersurface in ℂP^{n}, then M divides ℂP^{n} into two connected components Ω_{+} and Ω_{−}. Let ρ be a C^{2} defining function for M such that

(1.1)  ρ(x) = \{ \begin{matrix}−d(z, M), z ∈ Ω_{−}, \\ d(z, M), z ∈ Ω_{+}, \end{matrix}

where d(z, M) denotes the distance from z to M. The defining function ρ is called a signed distance function. Let U_{ϵ}(M) = {z ∈ ℂP^{n} | d(z, M) < ϵ} be a small tubular neighborhood of M. We consider the real two-dimensional normal plane bundle

N_{{∇ρ,J∇ρ}} = {(c_{1}∇ρ + c_{2}J∇ρ)|_{Q} | Q ∈ U_{ϵ}(M), c_{1}, c_{2} ∈ ℝ}

and the complex normal line bundle

N ˜∇ρ = N_{\{∇ρ⊗ℂ\}} = \{λ(∇ρ − \sqrt{−1}J(∇ρ))|_{Q} | Q ∈ U_{ϵ}(M), λ ∈ ℂ\}.

The Levi-Civita connection D of ℂP^{n} with the standard Fubini-Study metric ω satisfies

(1.2)  D_{ξ}(Jη) = J(D_{ξ}η)

since the metric ω is Kähler. We shall study the induced connections and curvature on N_{{∇ρ,J∇ρ}} and N_{{∇ρ⊗ℂ}}. There is also a hermitian connection D̃ on ℂT (ℂP^{n}) = T^{1,0}(ℂP^{n}) ⊕ T^{0,1}(ℂP^{n}). The two connections D and D̃ are related by

D̃_{ξ+\sqrt{−1}η}(Z + \sqrt{−1}W) = (D_{ξ}Z − D_{η}W) + \sqrt{−1}(D_{ξ}W + D_{η}Z)

for ξ, η, Z, W ∈ [T (ℂP^{n})]_{ℝ}.

Let us consider the corresponding real orthogonal decomposition

(1.3)  [T (ℂP^{n})]_{ℝ} = N_{{∇ρ,J∇ρ}} ⊕ N_{{∇ρ,J∇ρ}}^{⊥}.

For ξ ∈ [T (ℂP^{n})]_{ℝ} , ξ^{N} is the projection of ξ into the normal component N_{{∇ρ,J∇ρ}}. Similarly, we have

(1.4)  T^{1,0}(ℂP^{n}) = N_{{∇ρ⊗ℂ}} ⊕ N_{{∇ρ⊗ℂ}}^{⊥}.

For ξ̃ ∈ T^{1,0}(ℂP^{n}), ξ̃^{N} is the projection of ξ̃ into the complex normal component N_{{∇ρ⊗ℂ}}.

The map

u ↦ ũ = \frac{1}{\sqrt{2}}(u − \sqrt{−1}Ju)

is a linear isomorphism from T (ℂP^{n})_{ℝ} to T^{1,0}(ℂP^{n}). We define

(1.5)  D_{ξ}^{N}η = (D_{ξ}η)^{N},  η ∈ N_{{∇ρ,J(∇ρ)}}.

The induced curvature tensor is defined by

R^{N}(Z, W)η = −D_{Z}^{N}D_{W}^{N} η + D_{W}^{N} D_{Z}^{N}η + D_{[Z,W]}^{N}η,

for η ∈ N_{{∇ρ,J(∇ρ)}} and Z, W ∈ [T (ℂP^{n})]_{ℝ}. Similarly, we have the induced curvature tensor Θ̃^{N} for the complex line bundle N ˜∇ρ on U_{ϵ}.

Since ∥∇ρ∥^{2} = 1,

d(∥∇ρ∥^{2}) = 2⟨D∇ρ, ∇ρ⟩ = 0.

Thus D_{ξ}^{N}(∇ρ) has no component in the ∇ρ direction and

(1.6)  D_{ξ}^{N}(∇ρ) = β(ξ)J(∇ρ)

where β on U_{ϵ} is defined by

(1.7)  β(·) = Hess(ρ)(·, J(∇ρ)).

The 1-form β is the connection form for N_{{∇ρ,J(∇ρ)}}. Recall that from (1.2), D_{ξ}(J(∇ρ)) = JD_{ξ}(∇ρ) and ⟨∇ρ, J(∇ρ)⟩ ≡ 0. Thus we have

(1.8)  D_{ξ}^{N}(J(∇ρ)) = JD_{ξ}^{N}(∇ρ) = J(β(ξ)J(∇ρ)) = −β(ξ)∇ρ.

We shall study the connection form for N ˜∇ρ and its curvature form Θ̃^{N}.

Proposition 1.1. Let M be a C^{m} hypersurface in ℂP^{n} and let ρ be the signed distance function for M, m ≥ 2. Let ẽ_{n} = ˜∇ρ be the unit complex normal on a tubular neighborhood U_{ϵ}(M) for sufficiently small ϵ > 0. Then we have

(1.9)  D̃_{ξ}ẽ_{n} = θ(ξ)ẽ_{n}

where

(1.10)  θ(ξ) = \sqrt{−1}Hess(ρ)(ξ, J(∇ρ)) = \sqrt{−1}β(ξ).

Consequently, \sqrt{−1}θ is real-valued and the curvature form Θ̃^{N} of N ˜∇ρ is an exact 2-form on U_{ϵ}(M):

(1.11)  Θ̃^{N} = −dθ.

Furthermore, the connection θ and the curvature form Θ̃^{N} = −dθ above are of class C^{α} if M is a C^{2,α}-smooth hypersurface in ℂP^{n}.

Proof. The complex line bundle N ˜∇ρ is spanned by a global nowhere vanishing

section ˜∇ρ = \frac{1}{\sqrt{2}}(∇ρ − \sqrt{−1}J(∇ρ)), the complex normal vector on U_{ϵ}(M). A direct computation shows that, by (1.6) and (1.8) we have

\begin{aligned}D̃_{ξ}^{N} ẽ_{n} & = \frac{1}{\sqrt{2}}\{D̃_{ξ}^{N}∇ρ − \sqrt{−1}D̃_{ξ}^{N}(J(∇ρ))\} = \frac{1}{\sqrt{2}}\{β(ξ)J(∇ρ) + \sqrt{−1}β(ξ)(∇ρ)\} \\ & = \begin{matrix}√ & \overline{−1} \\ √ & \overline{2} \end{matrix}β(ξ)\{(∇ρ) − \sqrt{−1}J(∇ρ)\} = \sqrt{−1}β(ξ)ẽ_{n} = θ(ξ)ẽ_{n}, \end{aligned}

where θ = \sqrt{−1}β is the connection 1-form of the complex line bundle N ˜∇ρ. This proves (1.9). Using the Chern formula [Chern], the curvature form Θ̃^{N} is given by

Θ̃^{N} = −dθ + θ ∧ θ = −dθ = −\sqrt{−1}dβ

which proves (1.11).

By (1.10), θ is C^{m−2,α} if M is C^{m,α}. It remains to prove that the curvature form Θ̃^{N} of N ˜∇ρ is a 2-form with C^{α} coefficients on U_{ϵ}(M) if M is C^{2,α}-smooth. This can be proved by the generalized Gauss-Codazzi equation (the Cartan-Chern-Gauss structure equation). Let ẽ_{1}, ⋯, ẽ_{n} be an orthonormal local basis for T^{1,0}(U_{ϵ} ∩ V) such that ẽ_{n} = ˜∇ρ, where V is some neighborhood near a boundary point p ∈ M. Let θ̃ = (θ̃_{kl̄}) be the connection 1-form for ℂP^{n} defined by

D̃_{ξ}ẽ_{j} = ∑_{l=1}^{n} θ̃_{j,l̄}ẽ_{l}.

We remark that though T^{1,0}(ℂP^{n}) is a holomorphic vector bundle, our unitary frame {ẽ_{1}, ⋯, ẽ_{n}} is not necessarily holomorphic. Thus θ̃_{k,l̄} is not necessarily of (1, 0) type. Let R̃(X, Y) = −D̃_{X} D̃_{Y} + D̃_{Y} D̃_{X} + D̃_{[X,Y]} and

(1.12)  Θ̃_{k,l̄}(Z, \overline{W}) = ⟨R̃(Z, \overline{W})ẽ_{k}, ¯ ẽ_{l}⟩, Z, W ∈ T^{1,0}(V).

The matrix-valued 2-tensor Θ̃ is given by Chern’s formula for the curvature of complex vector bundles (cf. [Chern], [KN] or [Zh]),

(1.13)  Θ̃ = −dθ̃ + θ̃ ∧ θ̃.

In particular, we have

(1.14)  \begin{aligned}Θ̃_{n,n̄} & = ⟨R̃(·, ·)ẽ_{n}, ¯ ẽ_{n}⟩ = −dθ̃_{n,n̄} + ∑_{n=1}^{n} θ̃_{n,l̄} ∧ θ̃_{l,n̄} \\ & = Θ̃^{N} + ∑_{n=1}^{n−1} θ̃_{n,l̄} ∧ θ̃_{l,n̄}, \end{aligned}

where we have used the fact that θ̃_{n,n̄} = θ and (1.11). It remains to calculate θ̃_{n,l̄} and θ̃_{l,n̄}. For this purpose, we use the fact that

(1.15)  ⟨ẽ_{n}, ¯ ẽ_{l}⟩ = ⟨ẽ_{l}, ¯ ẽ_{n}⟩ = 0

where l = 1, ⋯, n − 1. Recall that for each ξ ∈ ℂT (V),

D̃_{ξ}ẽ_{n} = ∑_{l=1}^{m} θ̃_{n,l̄}(ξ)ẽ_{l} = ∑_{l=1}^{n}⟨D̃_{ξ}(ẽ_{n}), ¯ ẽ_{l}⟩ẽ_{l}.

Thus, we have

(1.16) \begin{aligned}θ̃_{n,l̄}(ξ) & = ⟨D̃_{ξ}(ẽ_{n}), ¯ ẽ_{l}⟩ = \frac{1}{2}⟨D̃_{ξ}(∇ρ − \sqrt{−1}J(∇ρ)), e_{l} + \sqrt{−1}Je_{l}⟩ \\ & = \frac{1}{2}⟨D̃_{ξ}(∇ρ), e_{l} + \sqrt{−1}Je_{l}⟩ − \frac{1}{2}⟨\sqrt{−1}JD̃_{ξ}(∇ρ), e_{l} + \sqrt{−1}Je_{l}⟩ \\ & = ⟨D̃_{ξ}(∇ρ), e_{l}⟩ + \sqrt{−1}⟨D̃_{ξ}(∇ρ), Je_{l}⟩ = \sqrt{2}Hess(ρ)(ξ, ẽ_{l}). \end{aligned}

Similarly, we have

(1.17)  \begin{aligned}θ̃_{l,n̄}(ξ) & = ⟨D̃_{ξ}(ẽ_{l}), ¯ ẽ_{n}⟩ = −⟨ẽ_{l}, D̃_{ξ}(¯ẽ_{n})⟩ \\ & = −⟨D̃ξ̄(ẽ_{n}), ¯ ẽ_{l}⟩ = −\sqrt{2}Hess(ρ)(ξ̄, ẽ_{l}). \end{aligned}

This implies that

(1.18)  θ̃_{l,n̄} = −¯θ̃_{n,l̄}, on V.

Thus

(1.19)  θ̃_{n,l̄} ∧ θ̃_{l,n̄} = −θ̃_{n,l̄} ∧ ¯ θ̃_{n,l̄}

is a skew-hermitian 2-form.

If ρ is C^{m,α}-smooth, then each ẽ_{k} is C^{m−1,α}-smooth for k = 1, ⋯ n. Thus Θ̃_{n,n̄} is C^{m−1,α}-smooth. It follows from (1.14) and (1.16)-(1.18) that Θ̃^{N} is a C^{m−2,α} smooth form, since Hess(ρ) is C^{m−2,α}. The proposition is proved. □

The C^{α}-regularity of θ and Θ̃^{N} will be used in the proof of Theorem 1, see Section 5 below. When M = bΩ is a Levi-flat real-hypersurface, the restriction Θ_{b} on M of the curvature form Θ̃^{N} above has some additional properties.

Proposition 1.2. Suppose that M is a C^{2}-smooth Levi-flat hypersurface in ℂP^{n} with the signed distance function ρ. Let ˜∇ρ be the unit complex normal on a tubular neighborhood U_{ϵ}(M) for sufficiently small ϵ > 0 and let Θ̃^{N} be the curvature tensor for the complex line bundle N ˜∇ρ. If J is the complex structure of ℂP^{n}, then Θ_{b} = Θ̃^{N}|_{[T (M)]_{ℝ}∩J[T (M)]_{ℝ}} is a (1, 1)-form. Furthermore, we have

(1.20)  \sqrt{−1}Θ̃^{N}(τ, τ̄) = \sqrt{−1}Θ_{b}(τ, τ̄) ≥ 2

where τ ∈ T^{1,0}(M) with |τ| = 1.

Proof. If M^{2n−1} is a Levi-flat real hypersurface, then by definition we have

(1.21)  Hess(ρ)( ¯Ỹ , X̃) = (i∂ ¯∂ρ)( ¯Ỹ , X̃) = 0

for any pair {X̃, Ỹ } ∈ T^{(1,0)}(M).

It follows from (1.16)-(1.17) and (1.21) that the connection forms {θ̃_{n,l̄}} are (1, 0)-forms on the Levi-flat hypersurface M = bΩ. Using (1.14) and (1.18), we have

(1.22)  Θ̃^{N} = Θ̃_{n,n̄} − ∑_{n=1}^{n−1} θ̃_{n,l̄} ∧ θ̃_{l,n̄} = Θ̃_{n,n̄} + ∑_{n=1}^{n−1} θ̃_{n,l̄} ∧ ¯ θ̃_{n,l̄}.

Because the Fubini-Study metric ω is a Kähler metric, its curvature form Θ̃ is a matrix valued (1, 1)-form with respect to any unitary frame (cf. [Chern], [Zh]). Hence, Θ̃_{n,n̄} is a (1, 1)-form. Since {θ̃_{n,l̄}} are (1, 0)-forms on the Levi-flat hypersurface M = bΩ, we see that \sqrt{−1} ∑_{n=1}^{n−1} θ̃_{n,l̄} ∧ ¯ θ̃_{n,l̄} is a nonnegative (1, 1)-form. This fact together with (1.22) implies that Θ_{b} is a (1, 1)-form. Furthermore, it follows that, for τ ∈ T^{(1,0)}(M) with |τ| = 1,

\sqrt{−1}Θ̃^{N}(τ, τ̄) ≥ \sqrt{−1}Θ̃_{n,n̄}(τ, τ̄) ≥ 2,

where we used the fact that the Fubini-Study metric of ℂP^{n} has positive holomorphic bisectional curvature ≥ 2, (e.g., cf. [KN], [Zh]). This completes the proof. □

Let us now recall some preliminary results, which play important roles in the proof of Theorem 2. First of all, we need to recall the so-called ¯∂-Neumann boundary condition. The L^{2}-theory of the ¯∂-equation ¯∂u = v and the Laplace equation □u = f are related to the formal adjoint ϑ of ¯∂ on the space L_{(p,q)}^{2}(Ω). Let ∗ be the real Hodge star operator of the Riemannian metric and let ∗ be extended ℂ-linearly on the space of (p, q)-forms. Then

(1.23)  ϑu = −¯∗[¯∂(¯∗u)] = − ∗ ∂ ∗ u.

For u, v ∈ L^{2}(Ω), we use

(u, v̄)_{L^{2}(Ω)} = ∫_{Ω}⟨u, v̄⟩

to denote the inner product. For q > 0, using integration by parts on the domain Ω with C^{2}-smooth boundary bΩ, we have that u ∈ Dom(¯∂^{∗}|_{Ω}) ∩ C_{(p,q)}^{1}(\overline{Ω}) if and only if

(1.24)  (ϑu, v̄)_{L^{2}(Ω)} = (¯∂^{∗}u, v̄)_{L^{2}(Ω)} = (u, \overline{¯∂v})_{L^{2}(Ω)}

holds for all v ∈ L_{(p,q)}^{2}(Ω). The condition (1.24) is equivalent to the so-called ¯∂-Neumann boundary condition

(1.25.1)  u⌞_{( ¯∂ρ)_{#}}= 0 on bΩ,

where (¯∂ρ)_{#} = \frac{1}{\sqrt{2}}(∇ρ + \sqrt{−1}J∇ρ) is the dual of ¯∂ρ. In general, u ∈ Dom(¯∂^{∗}|_{Ω}) implies that (1.25.1) holds as currents.

Throughout this paper, we let

(1.25.2)  Dom(¯∂^{∗}) = Dom(¯∂^{∗}|_{Ω}) = {u ∈ L_{(p,q)}^{2}(Ω)|u⌞_{( ¯∂ρ)_{#}}= 0 on bΩ}

and

(1.25.3) Dom(□|_{Ω}) = {u ∈ L_{(p,q)}^{2}(Ω)|u⌞_{( ¯∂ρ)_{#}}= 0 and (¯∂u)⌞_{( ¯∂ρ)_{#}}= 0 on bΩ}.

There always exists the weak Hodge-Kodaira decomposition for any domain Ω:

(1.26)  L_{(p,q)}^{2}(Ω) = ker(□) ⊕_{L^{2}} \overline{Range(□|_{Ω})},

where \overline{Range(□|_{Ω})} denotes the closure of Range(□|_{Ω}) in L^{2}(Ω). A necessary condition for the existence of the ¯∂-Neumann operator on L_{(p,q)}^{2}(Ω) is the condition that the range of □ must be closed, i.e., \overline{Range(□)} = Range(□). To prove the first part of Theorem 2, by the definition of □ = ¯∂¯∂^{∗} + ¯∂^{∗} ¯∂, it is sufficient to show that both Range(¯∂) and Range(¯∂^{∗}) are closed subspaces of L_{(p,q)}^{2}(Ω). For this purpose (and for the convenience of the reader), we recall the following elementary but very useful fact in functional analysis (cf. [Hö2-3] or [CS, p60]):

Lemma 1.3. Let F : H_{1} → H_{2} be a linear, closed, densely defined operator between two Hilbert spaces. The following conditions on F are equivalent:

(1) Range(F) is closed;

(2) There is a constant C such that

(1.27.1)  ∥u∥_{1} ≤ C∥Fu∥_{2} for all u ∈ Dom(F) ∩ \overline{Range(F^{∗})};

(3) Range(F^{∗}) is closed;

(4) There is a constant C such that

(1.27.2)  ∥v∥_{2} ≤ C∥F^{∗}v∥_{2} for all v ∈ Dom(F^{∗}) ∩ \overline{Range(F)};

The best constants in (1.27.1)-(1.27.2) are the same.

Lemma 1.3 will be used in the proofs of Theorem 2.1, Theorem 2.6 and Theorems 3.4-3.5 below, which are parts of Theorem 2. By Lemma 1.3, it suffices to derive a priori estimates for the ¯∂ equation and the □ equation. We recall the curved version of the Morrey-Kohn-Hörmander formula, which is a Bochner type formula with a weight function e^{−ϕ}. The weighted function e^{−ϕ} induces a perturbed adjoint operator:

(1.28)  ϑ_{ϕ}u = ¯∂_{ϕ}^{∗}u = e^{ϕ}ϑ(e^{−ϕu}) = −e^{ϕ} ∗ ∂(e^{−ϕ} ∗ u).

Of course, one can formulate the weighted Laplace operator □_{ϕ} = (¯∂¯∂_{ϕ}^{∗} + ¯∂_{ϕ}^{∗} ¯∂) and the corresponding weighted ¯∂-Neumann operators N_{ϕ}.

Our strategy is to use the weighted ¯∂-Neumann operators N_{ϕ} to estimate the original ¯∂-Neumann operator N, see Sections 2-3 below. In order to estimate the operator ¯∂_{ϕ}^{∗}N_{ϕ}, we let L_{1}, ⋯, L_{n} be a local orthonormal frame for T^{(1,0)}(U), where U is a local neighborhood near some point in \overline{Ω}. For any u ∈ C_{(p,q)}(\overline{Ω}) and ϕ ∈ C^{2}(Ω), we define an operator

(1.29)  ⟨(i∂ ¯∂ϕ)u, \overline{u}⟩ = ∑_{j,k=1}^{n}(i∂ ¯∂ϕ)(L_{j}, \overline{L}_{k})⟨u⌞\overline{L}_{j}, \overline{u}⌞L_{k}⟩ on Ω.

Notice that ⟨(i∂ ¯∂ϕ)u, \overline{u}⟩ is independent of the choice of a local unitary basis. Similarly, for Q ∈ bΩ, we require that L_{n} = \sqrt{2}(∂ρ)_{#} and define

(1.30)  ⟨(i∂ ¯∂ρ)u, \overline{u}⟩ = ∑_{j,k=1}^{n−1}(i∂ ¯∂ρ)(L_{j}, \overline{L}_{k})⟨u⌞\overline{L}_{j}, \overline{u}⌞L_{k}⟩ on bΩ.

In addition, since ℂP^{n} has non-zero curvature, we set

(1.31)  ⟨Θu, \overline{u}⟩ = ∑_{j,k}^{n}⟨\overline{w}^{j} ∧ \{[R(L_{j}, \overline{L}_{k})u]⌞\overline{L}_{k}\}, \overline{u}⟩ on Ω,

where {w^{j}} is the dual of the frame {L_{k}} and R_{X,Y} = −D_{X}D_{Y} u+D_{Y} D_{X} +D_{[X,Y]} is the curvature operator on (p, q)-forms, see [Wu, Chapter 2]. Finally, we set

(1.32)  (u, \overline{v})_{ϕ} = ∫_{Ω}⟨u, \overline{v}⟩e^{−ϕ}.

Proposition 1.4. (Bochner-Hörmander-Kohn-Morrey formula) Let Ω be a compact domain Ω with C^{2}-smooth boundary bΩ and ρ(x) = −d(x, bΩ). Then, for any (p,q)-form u ∈ Dom(¯∂) ∩ Dom(¯∂^{∗}) with q ≥ 1 on Ω, we have

(1.33) ∥¯∂u∥_{ϕ}^{2} + ∥¯∂_{ϕ}^{∗}u∥_{ϕ}^{2} = ∥\overline{∇}u∥_{ϕ}^{2} + (Θu, \overline{u})_{ϕ} + ((i∂ ¯∂ϕ)u, \overline{u})_{ϕ} + ∫_{bΩ}⟨(i∂ ¯∂ρ)u, \overline{u}⟩e^{−ϕ},

where ∥\overline{∇}u∥_{ϕ}^{2} = ∫_{Ω} ∑_{j=1}^{n} |DL̄_{j}u|^{2}e^{−ϕ} and \{L_{1}, ..., L_{n}\} is a local unitary frame of T^{(1,0)}(Ω).

Proof. This formula is known (cf. [AV], [Hö2] [Dem1-2], [Siu1], [Wu]) for some special cases, although it has not been stated in the literature in the form (1.33). If u has compact support in the interior of Ω, the formula (1.33) was proved in [AV], Chapter 8 of [Dem2] and (2.12) of [Wu].

It remains to discuss the boundary term of (1.33). For the case ϕ = 0, the stated formula with the boundary term was proved in [Siu1]. To compute the boundary term, one notice that although the boundary terms in the proof of (1.33) involve the weight function, they have nothing to do with the Riemannian curvature R nor Θ. The boundary term had been computed in Hörmander [Hö2, Chapter 3] by combining the Morrey-Kohn technique on the boundary with non-trivial weight function. If one combines the results of [Hö2] with the interior formulae discussed above, one can prove that (1.33) holds for the general case with a weight function e^{−ϕ} and the curvature term. □

In later application, we will choose ϕ = −t log |ρ| in (1.33). Therefore, we need to discuss the term i∂ ¯∂ϕ with ϕ = −t log |ρ| and the curvature term ⟨Θu, ū⟩ in the right hand side of (1.33).

Proposition 1.5. Let u be a (p, q)-form of Ω ⊂ ℂP^{n} with q ≥ 1. Suppose that Θ is the curvature term defined in (1.31) with respect to the Fubini-Study metric. Then we have

(1.34)  \{ \begin{matrix}⟨Θu, \overline{u}⟩ & = 0, for any (n, q)-form u; \\ ⟨Θu, \overline{u}⟩ & ≥ 0, when p ≥ 1 and u is a (p, q)-form; \\ ⟨Θu, \overline{u}⟩ & = q(2n + 1)|u|^{2}, when u is a (0, q)-form. \end{matrix}

Proof. The assertion for (0, q)-forms and (n, q)-forms was computed in [Wu] and [Siu1]. For the curvature operator Θ acting on (1, 1)-forms u, [Siu1] showed that Θ is non-negative definite, but not positive definite, (see [Pe] as well). Henkin and Iordan [HI] extended this observation for all (p, q)-forms with p ≥ 1. In fact, Lemma 3.3 of [HI] and its proof showed that Θ acting on L_{(p,q)}^{2}(Ω) is a non-negative operator. Furthermore, Θ is positive definite on L_{(p,q)}^{2}(Ω) if and only if p = 0 and q ≥ 1. □

Proposition 1.6. If Ω is a pseudoconvex domain with C^{2}-smooth boundary bΩ in ℂP^{n} and ρ is the signed distance function, then

(1.35)  i∂ ¯∂(− log |ρ|)(X, \overline{X}) ≥ \frac{1}{2}|X|^{2}

for any X ∈ T^{(1,0)}(Ω).

Proposition 1.6 was proved in Takeuchi (e.g., cf. [Su], [Ta]). A new proof of Proposition 1.6 using comparison theorems in Riemannian geometry will appear in another paper in [CaS].

2. L^{2} theory for ¯∂ on pseudoconvex domains in ℂP^{n}

Let Ω be a bounded pseudoconvex domain in ℂP^{n}. We fix our Fubini-Study metric ω for ℂP^{n}. In this section, we will use preliminary results of Section 1 to study the ¯∂-Neumann operators N, ¯∂^{∗}N and ¯∂N acting on L_{(p,q)}^{2}(Ω). We first prove the L^{2} existence theorem of the ¯∂-Neumann operator for the easier case when p = 0.

Theorem 2.1. Let Ω be a pseudoconvex domain in ℂP^{n}, n ≥ 2. Then the ¯∂- Neumann operator N_{(0,q)} exists and is bounded on L_{(0,q)}^{2}(Ω), where 0 ≤ q ≤ n. For any f ∈ L_{(0,q)}^{2}(Ω),

(2.1)  \begin{aligned}f & = ¯∂¯∂^{∗}N_{(0,q)}f ⊕ ¯∂^{∗} ¯∂N_{(0,q)}f, 1 ≤ q ≤ n − 1. \\ f & = ¯∂^{∗} ¯∂N_{(0,0)}f ⊕ P f, q = 0, \end{aligned}

where P : L^{2}(Ω) → L^{2}(Ω) ∩ Ker(¯∂) is the Bergman projection with

(2.2)  P = I − ¯∂^{∗}N_{(0,1)} ¯∂

and

(2.3)  N_{(0,0)} = ϑN_{(0,1)}^{2} ¯∂.

Moreover, N, ¯∂^{∗}N and ¯∂N are bounded operators with respect to the L^{2}-norms.

Proof. We first assume that Ω is a pseudoconvex domain in ℂP^{n} with C^{2} boundary M. Let 1 ≤ q ≤ n and let ρ be a C^{2} defining function for Ω normalized by |dρ| = 1 on M. Choosing ϕ = 0 in Proposition 1.4 , we have for any (0, q)-form u ∈ Dom(¯∂) ∩ Dom(¯∂^{∗}),

(2.4.1) ∥¯∂u∥^{2} + ∥¯∂^{∗}u∥^{2} = ∥\overline{∇}u∥^{2} + ∫_{M}⟨(i∂ ¯∂ρ)u, ū⟩ + (Θu, ū) ≥ ∥\overline{∇}u∥^{2} + (Θu, ū),

where Θ is the Ricci form and \overline{∇} is the holomorphic gradient. Since the Ricci curvature of ℂP^{n} with the Fubini-Study metric is equal to 2n + 1, (cf. [KN] [Pe] or [Zh]), by (2.4.1) we have

(2.4.2) Q(u, ū) = ∥¯∂u∥^{2} + ∥¯∂^{∗}u∥^{2} ≥ q(2n + 1)∥u∥^{2}, u ∈ Dom(¯∂) ∩ Dom(¯∂^{∗}).

Consequently, we have

(2.4.3)  (□u, ū) = Q(u, ū) ≥ q(2n + 1)∥u∥^{2}, u ∈ Dom(□_{(0,q)}|_{Ω}),

for q ≥ 1.

It follows from (2.4.3) that Ker(□_{(0,q)}) = 0 and ∥N_{(0,q)}∥_{L^{2}} ≤ \frac{1}{q(2n+1)} for q ≥ 1. Moreover, by (2.4.2)-(2.4.3) and Lemma 1.3, one can show that the three operators ¯∂, ¯∂^{∗} and □ have closed ranges in L^{2}(Ω). Notice that if ¯∂f = 0 and if u = ¯∂^{∗}Nf, then u is the solution of ¯∂u = f with the smallest L^{2}-norm. Using this fact, by (2.4.2) and Lemma 1.3, one can further show that ¯∂^{∗}N is a bounded operator. Since ¯∂N = (N ¯∂^{∗})^{∗} = (¯∂^{∗}N)^{∗}, using Lemma 1.3 again, we see that the operator ¯∂N is bounded as well. This proves Theorem 2.1 for the case of q ≥ 1.

The existence of N_{(0,0)} also follows and one can prove (2.2) and (2.3) exactly as for domains in ℂ^{n}. The general case for nonsmooth domains follows from exhausting a pseudoconvex domain by C^{2} pseudoconvex domains and using (2.4) (See proofs of Theorem 4.4.1 and Theorem 4.4.3 in [CS] for details). □

When 0 < p ≤ n, the proof of the L^{2} existence of the ¯∂-Neumann operator is more involved. we will use Lemma 1.3 and Propositions 1.4-1.6 with non-trivial weight function e^{−ϕ} = |ρ|^{t}. Here is a preliminary result on the complex Hessian of the weight function.

Lemma 2.2. Let Ω ⊂⊂ ℂP^{n} be a pseudoconvex domain with C^{2}-smooth boundary bΩ and let δ(x) = d(x, bΩ) be the distance function to bΩ. Let t_{0} = t_{0}(Ω) be the order of plurisubharmonicity for the distance function δ defined by (0.7). Then, for any 0 < t < t_{0}, there exist C_{t} > 0 such that

(2.5)  i∂ ¯∂(−δ^{t}) ≥ C_{t}δ^{t} (ω + \frac{i∂δ ∧ ¯∂δ}{δ^{2}} ) ,

where ω is the Kähler form of the Fubini-Study metric on ℂP^{n}.

Proof. The existence of such t_{0} for pseudoconvex domains in ℂ^{n} with C^{2}-smooth boundary in ℂP^{n} is proved in Ohsawa-Sibony [OS]. Thus, by the inequality i∂ ¯∂(−δ^{t_{0}}) ≥ 0 with 0 < t_{0} ≤ 1, we obtain that

(2.6)  i\frac{∂ ¯∂(−δ)}{δ} + (1 − t_{0})\frac{i∂δ ∧ ¯∂δ}{δ^{2}} ≥ 0

15 Using Proposition 1.6, we have

(2.7)  i∂ ¯∂(− log δ) = i\frac{∂ ¯∂(−δ)}{δ} + \frac{i∂δ ∧ ¯∂δ}{δ^{2}} ≥ Cω

where C = \frac{1}{2}. Multiplying (2.6) by (1 − ϵ) and multiplying (2.7) by ϵ, and adding the two inequalities together (i.e., using the sum (1 − ϵ)(2.6) + ϵ(2.7)), we conclude that, for any 0 ≤ ϵ ≤ 1, the inequality

(2.8)  i∂ ¯∂(− log δ) = i\frac{∂ ¯∂(−δ)}{δ} + \frac{i∂δ ∧ ¯∂δ}{δ^{2}} ≥ Cϵω + (1 − ϵ)t_{0} \frac{i∂δ ∧ ¯∂δ}{δ^{2}}

holds. Hence, for any 0 < t < t_{0}, we choose ϵ_{t} such that (1 − ϵ_{t})t_{0} > t. Then

\begin{aligned}i∂ ¯∂(−δ^{t}) & = itδ^{t} (\frac{∂ ¯∂(−δ)}{δ} + (1 − t)\frac{∂δ ∧ ¯∂δ}{δ^{2}} ) \\ & ≥ C_{t}tδ^{t} (ω + \frac{i∂δ ∧ ¯∂δ}{δ^{2}} ) \end{aligned}

where C_{t} = min(\frac{1}{2}ϵ_{t}, (1 − ϵ_{t})t_{0} − t). The lemma is proved. □

We will use Hörmander’s weight function method to obtain the L^{2} existence for the ¯∂-equation with weights first. Let t be any real number and ϕ ∈ C^{2}(Ω). Let L^{2}(δ^{t}) denote the L^{2} space with respect to the weight function e^{−tϕ} = δ^{t} and

∥f∥_{(t)}^{2} = ∫_{Ω} |f|^{2}e^{−tϕ} = ∫_{Ω} δ^{t}|f|^{2}.

We use ¯∂_{t}^{∗} to denote the adjoint of ¯∂ with respect to the weighted space.

We use the norm ∥.∥_{(t)}^{2} since it is equivalent to the Sobolev norm on a sub-space of W^{− \frac{t}{2}} (Ω), (see [CS p348] and [MS1-2]). In fact, we have

(2.9)  ∥ũ∥_{(t)}^{2} ≃ ∥ũ∥_{W^{− \frac{t}{2}} (Ω)}

when ũ satisfies an elliptic equation such as ¯∂ ⊕ ϑ.

In order to use Proposition 1.4 to derive the desired estimates for the operator ¯∂^{∗}N, we need to introduce the following two asymmetric weighted norms. These new norms will be used to obtain more refined estimates than those obtained in [Hö2].

For any (p, q)-form f on Ω, we decompose f into complex normal and tangential parts by setting

\{ \begin{matrix}f^{ν} & = (f⌞_{( ¯∂δ)_{#}}) ∧ ¯∂δ \\ f^{τ} & = f − f^{ν}. \end{matrix}

The above decomposition is well-defined for any (p, q)-form f supported in [Ω \ Cut_{Ω}(bΩ)], where Cut_{Ω}(bΩ) is the cut-loci of bΩ in Ω, see [Cha], [CE] or [Pe]. It is well-known that the cut-loci Cut_{Ω}(bΩ) of bΩ has real dimension ≤ (2n − 1) and hence has zero measure in Ω, (e.g., [Cha], [CE, p90ff], [Pe]). In summary, the above decomposition exists almost everywhere in Ω.

We define the asymmetric weighted norm

(2.10.1)  |f|_{A}^{2} = |f^{τ} |^{2} + \frac{|f^{ν}|^{2}}{|δ|^{2}} .

We also define the dual norm

(2.10.2)  |f|_{A′}^{2} = |f^{τ} |^{2} + |f^{ν}|^{2}|δ|^{2}.

For any t > 0, let L_{A}^{2}(δ^{t}) and L_{A′}^{2}(δ^{t}) denote the weighted L^{2} spaces on (p, q)-forms defined by the norm

(2.10.3)  |||u|||_{(t)}^{2} = ∫_{Ω} δ^{t}|f|_{A}^{2} = ∫_{Ω} δ^{t}(|f^{τ} |^{2} + \frac{|f^{ν}|^{2}}{|δ|^{2}} )

and

(2.10.4)  |||u|||_{(t)′}^{2} = ∫_{Ω} δ^{t}|f|_{A′}^{2} = ∫_{Ω} δ^{t}|(f^{τ} |^{2} + |f^{ν}|^{2}|δ|^{2}).

We may assume that Diam(ℂP^{n}) ≤ 1 up to a factor \frac{π}{2} . It is obvious that

|||u|||_{(t)′}^{2} ≤∥ u ∥_{(t)}^{2}≤ |||u|||_{(t)}^{2},

if Diam(Ω) ≤ 1.

The following is a preliminary estimate for the operator ¯∂^{∗}N, because u = ¯∂^{∗}Nf is the solution of ¯∂u = f with the least L^{2}-norm.

Proposition 2.3. Let Ω ⊂⊂ ℂP^{n} be a pseudoconvex domain with C^{2}-smooth boundary bΩ. Let t_{0} = t_{0}(Ω) be the order of plurisubharmonicity for the distance function δ defined by (0.7). For any 0 < t < t_{0} and any (p, q)-form f ∈ L_{A′}^{2}(δ^{t}), where 0 ≤ p ≤ n and 1 ≤ q ≤ n, such that ¯∂f = 0 in Ω, there exists u ∈ L_{(p,q−1)}^{2}(δ^{t}) satisfying ¯∂u = f and

(2.11)  ∥u∥_{(t)}^{2} ≤ \frac{1}{C_{0}t}|||f|||_{(t)′}^{2}.

Proof. By Proposition 1.6, we have that ϕ = − log δ is strictly plurisubharmonic and i∂ ¯∂ϕ ≥ \frac{1}{2}ω, where ω is the Kähler form of ℂP^{n} with the Fubini-Study metric.

It is known that Dom(¯∂_{t}^{∗}) = Dom(¯∂^{∗}), (e.g., [CS, Chapter 4]). Using Hörmander’s weighted estimates (cf. Proposition 1.4) and Proposition 1.5, we have the following formula: for any (p, q)-form g ∈ Dom(¯∂) ∩ Dom(¯∂_{t}^{∗}),

(2.12.1)  ∥¯∂g∥_{(t)}^{2} + ∥¯∂_{t}^{∗}g∥_{(t)}^{2} ≥ t((i∂ ¯∂)ϕg, ḡ)_{(t)}.

From (2.8), the positive (1,1)-form i∂ ¯∂ϕ induces a pointwise norm on (p, q) forms

(2.12.2)  |g|_{i∂ ¯∂ϕ}^{2} = ⟨i(∂ ¯∂ϕ)g, ḡ⟩ ≥ C_{0}|g|_{A}^{2},

where C_{0} = min (\frac{1}{2}, t_{0}) with t_{0} = t_{0}(Ω) as in (2.5). Thus, we have

(2.13)  ∥¯∂g∥_{(t)}^{2} + ∥¯∂_{t}^{∗}g∥_{(t)}^{2} ≥ C_{0}t|||g|||_{(t)}.

Let (i∂ ¯∂ϕ)′ denote the dual norm for (p, q)-forms induced by i∂ ¯∂ϕ. Using (2.13) and the same argument as in Demailly [Dem1,2], we conclude that for any f ∈ L_{A′}^{2}(δ^{t}), there exist u ∈ L^{2}(δ^{t}) satisfying ¯∂u = f and

(2.14)  ∫_{Ω} |u|^{2}δ^{t} ≤ ∫_{Ω} |¯∂u|_{(i∂ ¯∂ϕ)′}^{2}δ^{t} ≤ \frac{1}{C_{0}t} ∫_{Ω} |¯∂u|_{A′}^{2}δ^{t}.

This completes the proof. □

Notice that the terms in (2.14) have an extra weight factor δ^{t}. We need to remove this factor δ^{t} in order to obtain the desired L^{2}-estimate for all p ≥ 0. The next two propositions have already been obtained in Berndtsson-Charpentier [BC] (see also [HI]). We include the proof here for the sake of completeness.

Proposition 2.4. Let Ω ⊂⊂ ℂP^{n} be a pseudoconvex domain with C^{2} boundary bΩ. For any f ∈ L_{(p,q)}^{2}(Ω), where 0 ≤ p ≤ n and 1 ≤ q ≤ n, such that ¯∂f = 0 in Ω, there exists u ∈ L_{(p,q−1)}^{2}(Ω) satisfying ¯∂u = f with ∫_{Ω} |u|^{2} ≤ Ĉ ∫_{Ω} |f|^{2}.

Proof. By Proposition 2.3, for any t > 0 with 0 < t < t_{0}/4, there exists u ∈ L_{(p,q−1)}^{2}(δ^{t}) satisfying ¯∂u = f, such that u is perpendicular to Ker(¯∂) in L^{2}(δ^{t}) and u satisfies inequality (2.11).

Consider v = uδ^{−t}. Then v ∈ L^{2}(δ^{2t}) and v ⊥ Ker(¯∂) in L^{2}(δ^{2t}). It follows from (2.14) that the following holds:

(2.15)  ∫_{Ω} |u|^{2} = ∫_{Ω} |v|^{2}δ^{2t} ≤ \frac{1}{2C_{0}t} ∫_{Ω} |¯∂v|_{A′}^{2} δ^{2t}.

By the definitions of the norm |.|_{A′} and ϕ = − log δ (cf. (2.10.2)), we have

(2.16)  \begin{aligned}|¯∂v|_{A′}^{2} e^{−2tϕ} & ≤ 2|¯∂u|_{A′}^{2} + 2t^{2}|¯∂ϕ|_{A′}^{2} |u|^{2} \\ & ≤ C(|¯∂u|^{2} + 2t^{2}|u|^{2}). \end{aligned}

Choosing t sufficiently small and substituting (2.16) into (2.15), one obtains

(2.17)  ∫_{Ω} |u|^{2} ≤ Ĉ ∫_{Ω} |¯∂u|^{2},

where Ĉ = \frac{C}{2C_{0}t(1−2Ct^{2})}. This proves the theorem. □

By (2.9), the Sobolev norm ∥u∥_{W \frac{t}{2} ; (p,q)(Ω)} of the solution u is related to the weighted

norm ∥u∥_{(−t)}.

Proposition 2.5. Let Ω ⊂⊂ ℂP^{n} be a pseudoconvex domain with C^{2} boundary bΩ. Let t_{0} = t_{0}(Ω) be the order of plurisubharmonicity for the distance function δ defined by (0.7). For any 0 ≤ t < t_{0}, f ∈ L_{A′}^{2}(δ^{−t}), where 0 ≤ p ≤ n and 1 ≤ q ≤ n, such that ¯∂f = 0 in Ω, then u = ¯∂^{∗}Nf is in L_{(p,q−1)}^{2}(δ^{−t}) with ¯∂u = f and

∥u∥_{(−t)}^{2} ≤ C̃_{t}|||f|||_{(−t)′}^{2},

where the weighted norms are given by (2.10.1)-(2.10.4).

Proof. Let v = uδ^{−t}. We have that ¯∂_{t}^{∗}v = 0 if q > 1 and v ⊥ Ker(¯∂) in L^{2}(δ^{t}) for q = 0. This implies that v = ¯∂_{t}^{∗}N_{t} ¯∂v. By (2.8) (with ϵ = 0), we have that

(2.18)  |\frac{¯∂δ}{δ} ∧ u|_{(i∂ ¯∂ϕ)′}^{2} ≤ \frac{1}{t_{0}} |\frac{¯∂δ}{δ} ∧ u|_{A′}^{2} ≤ \frac{1}{t_{0}} |u|^{2}.

Thus for any ϵ > 0, by Proposition 2.3 (the proof of (2.14)), we have

(2.19) \begin{aligned}∫_{Ω} |u|^{2}δ^{−t} & = ∫_{Ω} |v|^{2}δ^{t} ≤ \frac{1}{C_{0}t} ∫_{Ω} |¯∂v|_{(i∂ ¯∂ϕ)′}^{2}δ^{t} \\ & ≤ (1 + \frac{1}{ϵ})\frac{1}{C_{0}t} ∫_{Ω} |¯∂u|_{(i∂ ¯∂ϕ)′}^{2}δ^{−t} + (1 + ϵ)\frac{t}{t_{0}} ∫_{Ω} |\frac{¯∂δ}{δ} ∧ u|_{A′}^{2}δ^{−t} \\ & ≤ (1 + \frac{1}{ϵ})\frac{1}{C_{0}t} ∫_{Ω} |¯∂u|_{A′}^{2}δ^{−t} + (1 + ϵ)\frac{t}{t_{0}} ∫_{Ω} |u|^{2}δ^{−t}. \end{aligned}

Choosing ϵ sufficiently small such that (1 + ϵ)\frac{t}{t_{0}} = r < 1 in (2.19), we conclude that

(2.20)  ∫_{Ω} |u|^{2}δ^{−t} ≤ \frac{(1 + \frac{1}{ϵ})\frac{1}{C_{0}t}}{1 − r} ∫_{Ω} |¯∂u|_{A′}^{2}δ^{−t} ≤ C̃_{t}|||f|||_{(−t)′}^{2}.

Proposition 2.5 is proved. □

We arrive at the main result of this section.

Theorem 2.6. Let Ω ⊂⊂ ℂP^{n} be a pseudoconvex domain with C^{2}-smooth boundary bΩ. Then □_{(p,q)} has closed range and the ¯∂-Neumann operator N_{(p,q)} : L_{(p,q)}^{2}(Ω) → L_{(p,q)}^{2}(Ω) exists for every p, q such that 0 ≤ p ≤ n, 0 ≤ q ≤ n. Moreover, for any f ∈ L_{(p,q)}^{2}(Ω), we have

\begin{aligned}f & = ¯∂¯∂^{∗}N_{(p,q)}f ⊕ ¯∂^{∗} ¯∂N_{(0,q)}f, 1 ≤ q ≤ n − 1. \\ f & = ¯∂^{∗} ¯∂N_{(p,0)}f ⊕ P f, q = 0, \end{aligned}

where P is the projection from L_{(p,0)}^{2}(Ω) onto L_{(p,0)}^{2}(Ω) ∩ Ker(¯∂) and

N_{(p,0)} = ¯∂^{∗}N_{(p,1)}^{2} ¯∂.

In addition, ¯∂N, ¯∂^{∗}N and N are bounded linear operators on L_{(p,q)}^{2}(Ω).

Proof. The L^{2}-existence theorem for the ¯∂-Neumann operator N on Ω follows from the L^{2}-existence of the solution u for the ¯∂-equation, which is proved in Proposition 2.4. The proof of Theorem 2.6 follows from Lemma 1.3 using Proposition 2.4 and (2.17) (see e.g., proofs of Theorem 4.4.1 and Theorem 4.4.3 in [CS] or [Hö3].) □

We remark that when p = 0, Theorem 2.6 holds for pseudoconvex domains not necessarily with C^{2} boundary from Theorem 2.1.

3. Sobolev estimates for the ¯∂-Neumann operator on pseudoconvex domains in ℂP^{n}

In this section we prove that the ¯∂-Neumann operator N is a bounded linear operator on Sobolev spaces W_{(p,q)}^{t}(Ω) for small t > 0. In order to derive some a priori estimates for the ¯∂-equation, we need a variant of the Bochner-Kodaira-Morrey-Kohn-Hörmander formula (cf. Proposition 1.4) as follows:

Proposition 3.1. Let Ω be a domain in ℂP^{n} with C^{2}-smooth boundary bΩ. Let δ(x) = d(x, bΩ) be the distance function and λ = −(δ)^{t} for some t > 0. For any f ∈ C_{(p,q)}^{1}(\overline{Ω}) ∩ Dom(¯∂^{∗}), where 0 ≤ p ≤ n and 1 ≤ q ≤ n, we have

(3.1.1)  \begin{aligned}∫_{Ω}(−λ)|¯∂f|^{2} & + ∫_{Ω}(−λ)|ϑf|^{2} + 2ℜ(ϑf, \overline{f⌞_{( ¯∂λ)_{#}}}) \\ & = ∫_{Ω}(−λ) (|\overline{∇}f|^{2} + ⟨Θf, f̄⟩ + ⟨(i∂ ¯∂λ)f, f̄⟩) \end{aligned}

where Θ and \overline{∇} are the same as in Proposition 1.4 and ℜ(h) is the real part of h.

Proof. The proof follows from the same calculation as in Hörmander [Hö2]. There is no boundary term since t > 0. Recall that ϑ = ¯∂^{∗}. A simple calculation shows that

(3.1.2)  ϑ_{ϕ}f = ¯∂_{ϕ}^{∗}f = ¯∂^{∗}f − f⌞_{( ¯∂ϕ)_{#}}

holds for any (p, q)-form u with q ≥ 1. On the other hand, by expanding the term δ^{t}i∂ ¯∂(− log δ^{t}) we have

(3.1.3)  δ^{t}i∂ ¯∂(− log δ^{t}) = ti∂ ¯∂(−δ^{t}) + t^{2} \frac{i∂δ ∧ ¯∂δ}{δ^{2}} .

Substituting (3.1.2)-(3.1.3) into (1.33) of Proposition 1.4 with ϕ = −t log δ, we obtain (3.1.1). □

Using lemma 2.2 and Proposition 3.1, we have the following estimates:

Proposition 3.2. Let Ω be a domain in ℂP^{n} with C^{2} boundary bΩ. Let δ be the distance function from bΩ. Let 0 < t < 1 such that λ = −(δ)^{t} satisfies condition (2.5). For any f ∈ L_{(p,q)}^{2}(Ω) ∩ Dom(¯∂) ∩ Dom(¯∂^{∗}), where 0 ≤ p ≤ n and 1 ≤ q ≤ n, we have

(3.2)  \begin{matrix}∫_{Ω}(−λ)|¯∂f|^{2} + ∫_{Ω}(−λ)|ϑf|^{2} \\ ≥ Ĉ_{t} (∫_{Ω}(−λ)|\overline{∇}f|^{2} + ∫_{Ω}(−λ)|f|^{2} + ∫_{Ω}(−λ)\frac{|f⌞_{( ¯∂ρ)_{#}}|^{2}}{|ρ|^{2}} ) , \end{matrix}

where Ĉ_{t} is a constant number independent of f.

Proof. Let f ∈ C_{(p,q)}^{1}(\overline{Ω}) ∩ Dom(¯∂^{∗}). By (2.5), we have

(3.3)  ⟨(i∂ ¯∂λ)f, f̄⟩ ≥ C_{1} (∫_{Ω}(−λ)|f|^{2} + ∫_{Ω}(−λ)\frac{|f⌞_{( ¯∂ρ)_{#}}|^{2}}{|ρ|^{2}} ) .

Also, we have for any ϵ > 0,

(3.4)  |2ℜ(−ϑf, \overline{f⌞_{( ¯∂λ)_{#}}})| ≤ \frac{t}{ϵ} ∫_{Ω}(−λ)|ϑf|^{2} + tϵ ∫_{Ω}(−λ)\frac{|f⌞_{( ¯∂ρ)_{#}}|^{2}}{|ρ|^{2}} .

Choosing ϵ sufficiently small, (3.2) follows from (3.1.1), (3.3) and Proposition 1.5. This proves the proposition for C^{1}-smooth forms.

Since C_{(p,q)}^{1}(\overline{Ω}) ∩ Dom(¯∂^{∗}) is dense in Dom(¯∂) ∩ Dom(¯∂^{∗}) in the graph norm (cf. Lemma 4.3.2 in [CS]), we have that C_{(p,q)}^{1}(\overline{Ω}) ∩ Dom(¯∂^{∗}) is also dense in the graph norm with weights defined by (∫_{Ω}(−λ)|¯∂f|^{2} +∫_{Ω}(−λ)|ϑf|^{2} +∫_{Ω}(−λ)|f|^{2})^{\frac{1}{2}} from the Dominated Convergence Theorem. The proposition is proved by approximation of C^{1}-smooth forms. □

Recall that u = ¯∂^{∗}Nf is a solution of ¯∂u = f with the least L^{2}-norm. By the estimate above, we have the following:

Corollary 3.3. Let Ω ⊂⊂ ℂP^{n} and t be the same as in Proposition 3.2. Then

(3.5)  ∥ ¯∂^{∗}N_{(p,q)}f ∥_{(t)}≤ C ∥ f ∥_{(t)}, f ∈ Ker(¯∂), 2 ≤ q ≤ n.

(3.6)  ∥ ¯∂N_{(p,q)}f ∥_{(t)}≤ C ∥ f ∥_{(t)}, f ∈ Ker(¯∂^{∗}), 0 ≤ q ≤ n − 1.

where C depends only on t.

Proof. Notice that N = □^{−1} on the range of □. It follows that Nf ∈ Dom(□). Hence, by (1.25.3), we see that ¯∂^{∗}Nf is in Dom(¯∂) ∩ Dom(¯∂^{∗}). If q ≥ 2, for any f ∈ L_{(p,q)}^{2}(Ω), ¯∂^{∗}Nf is a (p, q − 1)-form. Thus for f ∈ Ker(¯∂), one gets from (3.2) that

\begin{matrix}∫_{Ω}(−λ)|f|^{2} = ∫_{Ω}(−λ)|¯∂¯∂^{∗}Nf|^{2} + ∫_{Ω}(−λ)|¯∂^{∗} ¯∂^{∗}Nf|^{2} \\ ≥ Ĉ_{t} ∫_{Ω}(−λ)|¯∂^{∗}Nf|^{2}, \end{matrix}

where we used the following facts: f = (¯∂¯∂^{∗} + ¯∂^{∗} ¯∂)Nf, ¯∂^{∗} ¯∂^{∗}f = 0 and ¯∂Nf = N ¯∂f = 0 with f ∈ ker(¯∂). This proves inequality (3.5) by choosing C = \frac{1}{Ĉ_{t}}.

Similarly, for any 0 ≤ q ≤ n − 1, ¯∂Nf is in Dom(¯∂) ∩ Dom(¯∂^{∗}). Substituting ¯∂N_{(p,q)}f into (3.2), for f ∈ Ker(¯∂^{∗}), we have

\begin{matrix}∫_{Ω}(−λ)|f|^{2} = ∫_{Ω}(−λ)|¯∂¯∂Nf|^{2} + ∫_{Ω}(−λ)|¯∂^{∗} ¯∂Nf|^{2} \\ ≥ Ĉ_{t} ∫_{Ω}(−λ)|¯∂Nf|^{2}. \end{matrix}

This proves the corollary by choosing C = \frac{1}{Ĉ_{t}}. □

Let W_{(p,q)}^{s}(Ω) be the Sobolev space with −\frac{1}{2} < s < \frac{1}{2} and let ∥ ∥_{s(Ω)} denote its norm. For any u in Dom(¯∂) ∩ Dom(¯∂^{∗}), we have u ∈ W^{1}(Ω, loc). This implies that u satisfies an elliptic system and u ∈ W^{s}(Ω) for −\frac{1}{2} < s < \frac{1}{2} if and only if

(3.7)  ∥u∥_{(−2s)}^{2} = ∫_{Ω} δ^{−2s}|u|^{2} < ∞.

For a proof of this, see Theorem C.4 in the Appendix in [CS].

Using (3.7) or (2.9) and the estimates above, we are ready to prove the Sobolev boundary regularity result for the (generalized) Bergman operator P : L_{(p,q)}^{2}(Ω) → L_{(p,q)}^{2}(Ω) ∩ ker(¯∂). From Theorem 2.6, we have P = I − ¯∂^{∗}N_{(p,q+1)} ¯∂.

Theorem 3.4. Let Ω ⊂⊂ ℂP^{n} be a pseudoconvex domain with C^{2}-smooth boundary. Let 0 < t < t_{0} and t_{0} = t_{0}(Ω) be given by (0.7). Then the Bergman projection operator P is bounded from W(p,q)^{\frac{t}{2}}(Ω) to W(p,q)^{\frac{t}{2}}(Ω), where 0 ≤ p ≤ n and 0 ≤ q ≤ n − 1.

Proof. By Proposition 2.5, we have that ¯∂^{∗}N is bounded on Ker(¯∂) with estimates

(3.8)  ∥ ¯∂^{∗}N_{(p,q)}f ∥_{(−t)}≤ C|||f|||_{(−t)′}, f ∈ Ker(¯∂), 1 ≤ q ≤ n − 1.

Let P_{t} denote the Bergman projection with respect to the weighted space L^{2}(δ^{t}). For any f, g ∈ L_{(p,0)}^{2}(Ω) with ¯∂g = 0, we have

(P f, g) = (f, g) = (δ^{−t}f, g)_{(t)} = (P_{t}δ^{−t}f, g)_{(t)} = (δ^{t}P_{t}δ^{−t}f, g).

This implies that

(3.9)  \begin{aligned}P & = P^{2} = P δ^{t}P_{t}δ^{−t} \\ & = (I − ¯∂^{∗}N ¯∂)δ^{t}P_{t}δ^{−t} \\ & = δ^{t}P_{t}δ^{−t} − ¯∂^{∗}N(¯∂(δ^{t}) ∧ P_{t}δ^{−t}) \end{aligned}

since ¯∂P_{t} = 0. For any f ∈ L_{(p,q)}^{2}(Ω),

(3.10)  ∥δ^{t}P_{t}δ^{−t}f∥_{(−t)}^{2} ≤ ∥P_{t}δ^{−t}f∥_{(t)}^{2} ≤ ∥δ^{−t}f∥_{(t)}^{2} = ∥f∥_{(−t)}^{2}.

By (3.8), we have

(3.11)  \begin{aligned}∥ ¯∂^{∗}N(¯∂(δ^{t}) ∧ P_{t}δ^{−t}f) ∥_{(−t)}^{2} & ≤ C|||¯∂(δ^{t}) ∧ P_{t}δ^{−t}f|||_{(−t)′}^{2} \\ & ≤ C ∥ δ^{\frac{t}{2}} P_{t}δ^{−t}f ∥^{2}= C ∥ P_{t}δ^{−t}f ∥_{(t)}^{2} \\ & ≤ C ∥ δ^{−t}f ∥_{(t)}^{2}= C ∥ f ∥_{(−t)}^{2} . \end{aligned}

From (3.9)-(3.11), we get

(3.12)  ∥ P f ∥_{(−t)}≤ C ∥ f ∥_{(−t)} .

Note that W^{\frac{t}{2}} ⊂ L^{2}(δ^{−t}) by (3.7) or (2.9). From (3.10), we get

(3.13.1)  ∥ P f ∥_{(−t)}≤ C ∥ f ∥_{(−t)}≤ C_{1} ∥ f ∥_{\frac{t}{2}} .

Using (3.13.1) and (3.7) or (2.9), one obtains that the Bergman projection satisfies

(3.13.2)  ∥ P f ∥_{\frac{t}{2}} ≤ C_{2} ∥ f ∥_{\frac{t}{2}} .

Theorem 3.4 is proved. □

Let us now study the Sobolev regularities for other operators: N, ¯∂N and ¯∂^{∗}N.

Theorem 3.5. Let Ω ⊂⊂ ℂP^{n} be a pseudoconvex domain with C^{2}-smooth boundary bΩ. Let t_{0} = t_{0}(Ω) be the order of plurisubharmonicity for the distance function δ defined by (0.7). For any 0 < t < t_{0}, the ¯∂-Neumann operator N is bounded from W(p,q)^{\frac{t}{2}}(Ω) to W(p,q)^{\frac{t}{2}}(Ω), where 0 ≤ p ≤ n and 0 ≤ q ≤ n − 1. We also have the following estimates: for any f ∈ W_{(p,q)}^{s}(Ω),

(3.14)  ∥ Nf ∥_{\frac{t}{2} (Ω)}≤ 2C^{2} ∥ f ∥_{\frac{t}{2} (Ω)},

(3.15)  ∥ ¯∂^{∗}Nf ∥_{\frac{t}{2} (Ω)}≤ C ∥ f ∥_{\frac{t}{2} (Ω)},

(3.16)  ∥ ¯∂Nf ∥_{\frac{t}{2} (Ω)}≤ C ∥ f ∥_{\frac{t}{2} (Ω)},

where C depends only on t.

Proof. We already proved that the Bergman projection P = I − ¯∂^{∗}N ¯∂ are bounded on W^{\frac{t}{2}} . It is easy to verify that, if F^{∗} is the adjoint map of F with respect to the L^{2}-norm, then

∥Fu ∥_{W ^{\frac{t}{2}} (Ω)}= sup_{v∈L^{2}}\{\frac{(Fu, v̄)_{L^{2}}}{∥v ∥_{W^{− \frac{t}{2}}}} \}

= sup_{v∈L^{2}}\{\frac{(u, F^{∗}v̄)_{L^{2}}}{∥v ∥_{W^{− \frac{t}{2}}}} \} ≤∥ F^{∗} ∥_{W^{− \frac{t}{2}} (Ω)} ∥u∥_{W ^{\frac{t}{2}} (Ω)},

(3.17)

(compare with Lemma 1.3).

From the proof of Theorem 3.4, we have that the canonical solution satisfies (3.8). Since the Bergman projection P is self-adjoint on L^{2}(Ω), by Theorem 3.4 and (3.17), we see that the Bergman projection P is bounded on W_{(p,q)}^{± \frac{t}{2}}(Ω).

Inequality (3.15) follows easily from Theorem 3.4 and the boundedness of the Bergman projection on W^{\frac{t}{2}} since ¯∂^{∗}Nf = ¯∂^{∗}NP f.

Let P̃ = ¯∂^{∗}N ¯∂ be the projection operator into Ker¯∂^{∗}. Then P = I − ¯∂^{∗}N ¯∂ = I − P̃. It follows that ¯∂Nf = ¯∂NP̃f. Since the Bergman projection is self-adjoint, both P and P̃ satisfy

(3.18)  ∥ P f ∥_{(t)} + ∥ P̃f ∥_{(t)}≤ C_{3} ∥ f ∥_{(t)} .

Thus from Corollary 3.3, we have that both ¯∂N and ¯∂^{∗}N are bounded with estimates

(3.19)  ∥ ¯∂Nf ∥_{(t)}=∥ ¯∂NP̃f ∥_{(t)}≤ C_{4} ∥ P̃f ∥_{(t)}≤ C_{4}C_{3} ∥ f ∥_{(t)}, q ≥ 0

(3.20)  ∥ ¯∂^{∗}Nf ∥_{(t)}=∥ ¯∂^{∗}NP f ∥_{(t)}≤ C_{4} ∥ P f ∥_{(t)}≤ C_{4}C_{3} ∥ f ∥_{(t)}, q ≥ 2.

In fact (3.20) also holds for q = 1. This follows from the formula ¯∂^{∗}Nf = ¯∂_{t}^{∗}N_{t}f − P_{t} ¯∂_{t}^{∗}N_{t}f for any f ∈ Ker(¯∂). Thus for any f ∈ L_{(p,1)}^{2}(Ω), by (2.14) and (2.10.4) we have

(3.21) \begin{aligned}∥ ¯∂^{∗}Nf ∥_{(t)} & =∥ ¯∂^{∗}NP f ∥_{(t)} \\ & =∥ ¯∂_{t}^{∗}N_{t}P f − P_{t} ¯∂_{t}^{∗}N_{t}P f ∥_{(t)}≤ C_{5} ∥ P f ∥_{(t)}≤ C_{5}C_{3} ∥ f ∥_{(t)} . \end{aligned}

Notice that (¯∂N)^{∗} = N ¯∂^{∗} = ¯∂^{∗}N and (¯∂^{∗}N)^{∗} = N ¯∂ = ¯∂N. By (3.17)-(3.21), we obtain that both ¯∂N and ¯∂^{∗}N are bounded operators on W_{(p,q)}^{± \frac{t}{2}}(Ω). Thus, (3.15)-(3.16) are true, by choosing C = max{C_{3}C_{4}, C_{3}C_{5}}.

It remains to verify (3.14). By an observation of Range, we have

(3.22)  N = ¯∂¯∂^{∗}N^{2} + ¯∂^{∗} ¯∂N^{2} = ¯∂N ¯∂^{∗}N + ¯∂^{∗}N ¯∂N.

It follows from (3.15)-(3.16) and (3.22) that

(3.23)  ∥ Nf ∥_{(t)}≤ 2C^{2} ∥ f ∥_{(t)}, q ≥ 0.

By (3.23) and (3.17), we conclude that

(3.24)  ∥ Nf ∥_{(−t)}≤ 2C^{2} ∥ f ∥_{(−t)}, q ≥ 1,

since N is self-adjoint. Thus (3.14) now follows from (3.24) and (3.7) or (2.9). □

Theorem 2 is a direct consequence of Theorems 3.4-3.5.

4. ¯∂-closed extension from pseudoconvex boundaries in ℂP^{n}

In this section we study the extension of ¯∂_{b}-closed forms from the boundary of a pseudoconvex domain in ℂP^{n}. This is equivalent to solving the ¯∂-Cauchy problem on pseudoconvex domains, which is the dual of the ¯∂-Neumann problem.

Recall that if ¯∂_{b}v^{(0,1)} = 0 on bΩ and if w^{(0,1)} is an arbitrary extension of v^{(0,1)} on Ω, then we can correct w^{(0,1)} to be a ¯∂-closed extension ṽ^{(0,1)} on Ω by setting

(4.0)  ṽ^{(0,1)} = w^{(0,1)} + ⋆¯∂N_{(n,n−2)}⋆(¯∂w^{(0,1)}) on Ω,

where ⋆ = ¯∗ : L_{(p,q)}^{2}(\overline{Ω}) is the Hodge star operator defined by

(4.1)  ⟨ϕ, ψ̄⟩dV = ϕ ∧ ⋆ψ̄.

Notice that ⋆ = ¯∗ satisfies ⋆(λu) = λ̄(⋆u) for any complex number λ ∈ ℂ.

There are two issues in the application of the formula (4.0) above. The first one is that we require ¯∂w^{(0,1)} ∈ L^{2} when we apply Theorem 3.5 to the (n, n − 2)-form [⋆(¯∂w)]. Therefore, in some results stated below, we require that w ∈ W^{1}(Ω) or equivalently v = w|_{bΩ} ∈ W^{\frac{1}{2}} (bΩ), (cf. Proposition 4.3).

The second difficulty occurs when [⋆(¯∂w)] is an (n, 0)-form, i.e., n = 2. It is easy to see that, for a (p, q)-form v on M with q < n − 1, the ¯∂-closed extension ṽ on ℂP^{n} exists only if ¯∂_{b}v = 0. However, for all (p, n − 1)-forms v on M, the equation ¯∂_{b}v = 0 always holds. There is another necessary condition on a (p, n − 1)-form v so that its ¯∂-closed extension on Ω exists, see (4.9) (or equivalently (4.3)) below. In order to show that (4.9) holds for any (p, n−1)-form v on the Levi-flat hypersurface M, we derive a new Liouville type theorem for a pseudoconcave boundary M = bΩ, see Proposition 4.5 below.

To find a ¯∂-closed extension ṽ on ℂP^{n} for v with ¯∂_{b}v = 0 on bΩ, it suffices to solve the inhomogeneous equation ¯∂u = f with compact support Supp(u) ⊂ \overline{Ω}.

Proposition 4.1. Let Ω be a pseudoconvex domain with C^{2} boundary in ℂP^{n}, n ≥ 2. For every f ∈ L_{(p,q)}^{2}(ℂP^{n}), where 0 ≤ p ≤ n and 1 ≤ q ≤ n − 1, with ¯∂f = 0 in the distribution sense in ℂP^{n} and f supported in \overline{Ω}, one can find u ∈ L_{(p,q−1)}^{2}(ℂP^{n}) such that ¯∂u = f in the distribution sense in ℂP^{n} with u supported in \overline{Ω} and

∫_{Ω} | u |^{2} dV ≤ C ∫_{Ω} | f |^{2} dV

for some C > 0.

Proof. From Theorem 2.6, the ¯∂-Neumann operator of degree (n − p, n − q) in Ω, denoted by N_{(n−p,n−q)}, exists. Let ⋆ : L_{(p,q)}^{2}(Ω) be the Hodge star operator defined as in (4.1) above. We define

(4.2)  u = −⋆¯∂N_{(n−p,n−q)}⋆f = −¯∗¯∂N_{(n−p,n−q)}\overline{∗f},

then u ∈ L_{(p,q−1)}^{2}(Ω). By the proof of Corollary 3.3, we can verify that ⋆u = ¯∂N(⋆f) ∈ Dom(¯∂^{∗}), because N = □^{−1} on Range(□) and Range(N) ⊂ Dom(□). Since (⋆u) ∈ Dom(¯∂^{∗}), it follows from (1.25.2) that u|_{T^{(0,1)}(bΩ)} = 0. Extending u to ℂP^{n} by defining u = 0 in ℂP^{n} \ Ω, we obtain that u has support in \overline{Ω}. It follows from a theorem of Kohn-Rossi [KoR] (see also Theorem 9.1.2 in Chen-Shaw [CS]) that ¯∂u = f in the distribution sense in ℂP^{n}. □

In order to solve the ¯∂-equation with compact support when q = n, there is another compatibility condition (see (4.3) below) and we have the following result:

Proposition 4.2. Let Ω be a pseudoconvex domain with C^{2}-smooth boundary in ℂP^{n}, n ≥ 2. For any f ∈ L_{(p,n)}^{2}(ℂP^{n}), 0 ≤ p ≤ n, such that f is supported in \overline{Ω} and

(4.3)  ∫_{Ω} f ∧ g = 0  for every g ∈ L_{(n−p,0)}^{2}(Ω) ∩ Ker(¯∂),

one can find u ∈ L_{(p,n−1)}^{2}(ℂP^{n}) such that ¯∂u = f in the distribution sense in ℂP^{n} with u supported in \overline{Ω} and

∫_{Ω} | u |^{2} dV ≤ C ∫_{Ω} | f |^{2} dV,

for some C > 0. Proof. Using Theorem 2.6, the ¯∂-Neumann operator N_{(p,0)} exists for any 0 ≤ p ≤ n and we have

(4.4)  N_{(p,0)} = ¯∂^{∗}N_{(p,1)}^{2} ¯∂.

The Bergman projection operator P_{(p,0)} is given by

(4.5)  ¯∂^{∗} ¯∂N_{(p,0)} = I − P_{(p,0)}.

We define u by

(4.6)  u = − ⋆ ¯∂N_{(n−p,0)}⋆f.

(4.7)  \begin{aligned}¯∂u & = (−1)^{p+n}⋆¯∂^{∗} ¯∂N_{(n−p,0)} ⋆ f \\ & = f − (−1)^{p+n}⋆P_{(n−p,0)} ⋆ f. \end{aligned}

From (4.3), we get for any g ∈ L_{(n−p,0)}^{2}(Ω) ∩ Ker(¯∂),

(⋆f, ḡ) = (−1)^{p+n}\overline{∫_{Ω} g ∧ f} = 0.

Thus P_{(n−p,0)}(⋆f) = 0 and ¯∂u = f in Ω. Using ⋆u ∈ Dom(¯∂^{∗}) and extending u to be zero outside Ω, then ¯∂u = f in ℂP^{n} in the distribution sense. □

Let us now summarize the necessary and sufficient condition on f ∈ W(p,q)^{\frac{1}{2}}(bΩ) to have a ¯∂-closed extension F on Ω. Proposition 4.3. Let Ω ⊂⊂ ℂP^{n} be a pseudoconvex domain with C^{2} boundary M = bΩ. Let f ∈ W(p,q)^{\frac{1}{2}}(bΩ), where 0 ≤ p ≤ n, 1 ≤ q ≤ n − 1. We assume that

(4.8)  ¯∂_{b}f = 0,  if 1 ≤ q < n − 1,

and

(4.9)  ∫_{M} f ∧ ψ = 0 for every ψ ∈ L_{(n−p,0)}^{2}(\overline{Ω}) ∩ Ker(¯∂), if q = n − 1.

Then there exists F ∈ L_{(p,q−1)}^{2}(Ω) such that F = f on bΩ and ¯∂F = 0 in Ω.

We remark that any ψ ∈ L_{(n−p,0)}^{2}(Ω) ∩ Ker(¯∂) has boundary values on bΩ in the W^{− \frac{1}{2}} (bΩ) space (see e.g. Lemma 2.1 in Michel-Shaw [MS2]). Thus the pairing (4.9) is well-defined in the sense of currents.

Proof. Since f ∈ W(p,q)^{\frac{1}{2}}(bΩ) is a form, one can extend f = Σ_{|I|=p,|J|=q}f_{I,J̄} componentwise to Ω such that each component f_{I,J̄} is in W^{1}(Ω). For detailed construction of such an extension, see e.g. Lemma 9.3.3 in [CS]. Let f̃ be an arbitrary extension of f with f̃ ∈ W_{(p,q)}^{1}(Ω)

We first assume that q + 1 < n. From (4.8), we can require that f_{1} = ¯∂_{b}f̃ = 0 in M. If we extend f_{1} to be zero outside Ω, we get ¯∂f_{1} = 0 in ℂP^{n} in the distribution sense. we set v_{0} = −⋆¯∂N_{(n−p,n−q−1)}⋆f_{1}. From Proposition 4.1 and its proof, we have v_{0} ∈ L_{(p,q)}^{2}(Ω). We set v_{0} = 0 outside Ω. Then ¯∂v_{0} = f_{1} in the sense of distributions on ℂP^{n} with supp(v_{0}) ⊂ \overline{Ω}.

Setting F = f̃− v_{0} in Ω, we have F = f on bΩ and ¯∂F = 0 in Ω. This proves the proposition for q < n − 1.

When q = n−1 and f satisfies (4.9), we let f_{1} = ¯∂f̃. Then for any ψ ∈ C_{(n,0)}^{1}(\overline{Ω}) with ¯∂ψ = 0, we have

(4.10)  ∫_{Ω} f_{1} ∧ ψ = ∫_{Ω} ¯∂f̃∧ ψ = ∫_{bΩ} f ∧ ψ = 0.

When ψ ∈ L_{(n,0)}^{2}(Ω)∩ker(¯∂), (4.10) follows from approximating ψ by smooth forms. Let us now apply Proposition 4.2 to f_{1}, since f_{1} satisfies (4.3) by (4.10). Setting v_{0} = − ⋆ ¯∂N_{(n−p,n−q−1)}⋆f_{1} and F = f̃− v_{0} as above, by the proof of Proposition 4.2, we can conclude the proof for the case of q = n − 1 as well. □

Recall that the domain Ω_{+} is pseudoconcave if and only if its complement Ω_{−} = ℂP^{n} \ \overline{Ω}_{+} is pseudoconvex. In order to show that the condition (4.9) automatically holds on the Levi-flat boundary M = bΩ_{±}, we observe that M is the boundary of a pseudoconcave domain as well. Indeed, if Ω_{+} ∪ Ω_{−} = ℂP^{n} \ M and if M is Levi-flat, then Ω_{±} is both pseudoconvex and pseudoconcave.

Notice that, by a theorem of Chern-Lashof, any compact domain Ω̄ with C^{2}- smooth boundary bΩ in ℂ^{n} must have a point Q_{0} in bΩ with strictly positive principle curvature. To see this fact, one considers the Gauss map from bΩ to the unit sphere S^{2n−1} ⊂ ℂ^{n}. Therefore, this theorem of Chern-Lashof implies that there is no compact pseudoconcave domain \overline{Ω} in ℂ^{n}. However, the complex projective space ℂP^{n} contains a lot of pseudoconcave domains.

The classical Liouville Theorem states that if f : ℂ^{n} → ℂ is a bounded holomorphic function on ℂ^{n} then f must be constant. We will extend the classical Liouville Theorem of functions to the (p, q)-forms on pseudoconcave domains Ω^{+} as follows.

First we study the one-sided ¯∂-closed extension from a pseudoconcave domain Ω_{+} to the whole space ℂP^{n} by using (4.8).

Proposition 4.4. Let Ω_{+} ⊂ ℂP^{n} be a pseudoconcave domain with C^{2}-smooth boundary. Suppose that f ∈ W_{(p,q)}^{1}(\overline{Ω}_{+}) with ¯∂f = 0, where 0 ≤ p ≤ n and 1 ≤ q < n − 1. Then there exists a ¯∂-closed form F ∈ L_{(p,q)}^{2}(ℂP^{n}) such that F is a ¯∂-closed extension of f.

Proof. Let M = bΩ_{+} be the C^{2}-smooth pseudoconcave boundary in ℂP^{n}. We consider the so-called Fermi map (cf. [Cha]) along M as follows:

\begin{aligned}h : & M × ℝ → ℂP^{n} \\ & (Q_{0}, t) → Exp_{Q_{0}}(t∇ρ), \end{aligned} (4.11)

where Exp is the exponential map of ℂP^{n} and ρ(x) = −d(x, bΩ_{−}) for x ∈ Ω_{−}. It is well-known that, for sufficiently small ϵ > 0, the Fermi-map h|_{M×[−ϵ,ϵ]} is an embedding map.

Let us now choose an extension w ∈ L_{(p,q)}^{2}(ℂP^{n}) of f as follows: for 0 ≥ t ≥ −ϵ, let

(4.12)  w(h(Q_{0}, t)) = f(h(Q_{0}, −t)).

Thus, w is well-defined on U_{ϵ}(M). We further extend w to [Ω_{−} \ U\frac{ϵ}{2} (M)] so that w|_{\overline{Ω}_{+}} = f and w ∈ W_{(p,q)}^{1}(ℂP^{n}).

Recall that Ω_{−} is pseudoconvex by the assumption. Applying Proposition 4.1 to the equation

¯∂u = ¯∂w on Ω_{−},

we get a solution u with support in \overline{Ω}_{−} by choosing

(4.13) u = −⋆¯∂N_{(n−p,n−q−1)}(⋆¯∂w) on Ω_{−}.

Inspired by (4.0), we let

(4.14) F_{−} = w − u = w + ⋆¯∂N_{(n−p,n−q−1)}(⋆¯∂w) on Ω_{−},

where u is given by (4.13). A direct computation shows that

(4.15)  ¯∂u = ¯∂w − (−1)^{p+q} ⋆ P_{(n−p,n−q)} ⋆ (¯∂w),

where P_{(n−p,n−q)} : L_{(p,q)}^{2}(Ω_{−}) → Ker(¯∂) is the Bergman projection.

When w is a (p, q)-form with q < n − 1, ¯∂w ∈ Ker(¯∂) and hence P (⋆(¯∂w)) = 0. By (4.15) and P (⋆(¯∂w)) = 0, we have that ¯∂u = ¯∂w and ¯∂F_{−} = 0.

By the proof of Proposition 4.1, we know that u has compact support in \overline{Ω}_{−}. Define

(4.16)  F = \{ \begin{matrix}f, x ∈ Ω̄_{+}, \\ F_{−}, x ∈ Ω_{−}. \end{matrix}

Then F ∈ L_{(p,q)}^{2}(ℂP^{n}), F = f on Ω̄_{+} and ¯∂F = 0 in ℂP^{n} in the distribution sense. This proves the Proposition. □

In order to show that (4.9) holds, it is sufficient to derive a new Liouville type Theorem, i.e., to show that L_{(n−p,0)}^{2}(Ω) ∩ Ker(¯∂) = 0 for a pseudoconcave domain and n − p > 0.

Proposition 4.5. Let Ω_{+} ⊂⊂ ℂP^{n} be a pseudoconcave domain with C^{2}-smooth boundary bΩ_{+}. Then L_{(p,0)}^{2}(Ω_{+}) ∩ Ker(¯∂) = {0} for every 0 < p ≤ n; and L_{(0,0)}^{2}(Ω_{+}) ∩ Ker(¯∂) = ℂ.

Proof. We will extend f to be a ¯∂-closed form in ℂP^{n} in a similar way as in the proof of Proposition 4.4. If f ∈ L_{(p,0)}^{2}(Ω_{+}) ∩ Ker(¯∂), then there exists a ¯∂-closed extension F ∈ L^{2} on the whole ℂP^{n}. To see this, we extend f to be f̃ in ℂP^{n} such that f_{1} ≡ ¯∂f̃ ∈ L_{(p,1)}^{2}(ℂP^{n}) with support f_{1} ⊂ \overline{Ω}_{−}, where Ω_{−} = ℂP^{n} \ \overline{Ω}_{+}.

Such an L^{2}-extension f̃ can be carried out, because bΩ is a C^{2}-smooth real hypersurface. Here is the detail for the construction of f̃ and f_{1}. Let h : bΩ × [−ϵ, ϵ] → ℂP^{n} be the Fermi-map given by (4.11). For sufficiently small ϵ, h is an embedding map which gives rise to the so-called Fermi coordinate along bΩ. We now define f̃(h(Q, t)) = f(h(Q, −t)) for h(Q, t) ∈ Ω_{−} and |t| < ϵ/2. Outside the annuli neighborhood h(bΩ × [−ϵ, ϵ]), we require f̃ to be C^{2}-smooth. Therefore, the (p, 1)-form f_{1} ≡ ¯∂f̃ ∈ L_{(p,1)}^{2}(ℂP^{n}) with support f_{1} ⊂ \overline{Ω}_{−}.

Using Propositions 4.1-4.4, we see that there exists a solution v_{0} ∈ L_{(p,0)}^{2}(Ω_{−}) such that ¯∂v_{0} = f_{1} in ℂP^{n} if we extend v_{0} to be zero on Ω_{+}. Thus, the extension form F = f̃− v_{0} is ¯∂-closed in the whole space ℂP^{n}. We observe that L_{(p,0)}^{2}(ℂP^{n}) ∩ Ker(¯∂) = {0} for all 0 < p ≤ n; and L_{(0,0)}^{2}(ℂP^{n}) ∩ Ker(¯∂) = ℂ. Hence, f = 0 when p > 0, and f must be a constant when it is a L^{2}-holomorphic function on Ω_{+}. This proves Proposition 4.5. □

Some related results similar to Propositions 4.4-4.5 were obtained by Henkin-Iordan [HI] earlier via a different argument. Let us now apply our results above to a C^{2}-smooth Levi-flat hypersurface M if it exists.

Corollary 4.6. Let M ⊂ ℂP^{n} be a C^{2}-smooth Levi-flat hypersurface. For any f ∈ W(p,n−1)^{\frac{1}{2}}(M), where 0 ≤ p < n, there exists a ¯∂-closed form F ∈ L_{(p,n−1)}^{2}(ℂP^{n}) such that F is a ¯∂-closed extension of f.

Proof. Let M be a compact connected C^{2}-smooth real hypersurface in ℂP^{n}. It is well-known that f ∈ W(p,q)^{\frac{1}{2}}(M) if and only if there exists an extension f̃ of f with f̃ ∈ W_{(p,q)}^{1}(ℂP^{n}).

Since ℂP^{n} is simply-connected, ℂP^{n} \ M = Ω_{+} ∪ Ω_{−} has exactly two connected components: Ω_{+} and Ω_{−}. Since Ω_{±} is pseudoconvex, we let

(4.17)  F_{±} = f̃+ ⋆[¯∂N_{(n−p,0)}|_{Ω_{±}}](⋆¯∂f̃)  on Ω_{±}

and v_{±} = − ⋆ ¯∂N_{(n−p,0)}(⋆¯∂f̃). By Theorem 2.6, we have F_{±} ∈ L_{(p,n−1)}^{2}(ℂP^{n}).

Since M is Levi-flat, we see that both Ω_{+} and Ω_{−} are pseudoconcave as well. By Proposition 4.5, the form f_{1} = ¯∂f̃ satisfies (4.3) and (4.9) on each of Ω_{±}. By the proofs of Propositions 4.1-4.5, we conclude that F is the ¯∂-closed extension of f on the whole ℂP^{n}. □

We are now ready to study the inhomogeneous equation (0.4).

Corollary 4.7. Let M ⊂ ℂP^{n} be a C^{2}-smooth Levi-flat hypersurface. Let f ∈ W(p,q)^{\frac{1}{2}}(M), where 0 ≤ p ≤ n, 1 ≤ q ≤ n − 1 and p ≠ q. We assume that f is ¯∂_{b}-closed in M if q < n − 1 and p < n − 1 if q = n − 1. Then there exists

u ∈ W(p,q−1)^{\frac{1}{2}}(M) such that ¯∂_{b}u = f in M.

Proof. From Proposition 4.4 (for q < n − 1) and Corollary 4.6 (for q = n − 1), there exists a ¯∂-closed F ∈ L_{(p,q)}^{2}(ℂP^{n}) such that F = f on M.

From our assumption p ≠ q, we have that H^{p,q}(ℂP^{n}) = {0}. This implies that there exists an inverse 𝔾, the Green operator, for □ on ℂP^{n} such that □𝔾 = I on L_{(p,q)}^{2}(ℂP^{n}) and the the Hodge decomposition theorem holds for (p, q)-forms on ℂP^{n}. For any ¯∂-closed (p, q)-form F, we have

(4.18)  F = ¯∂¯∂^{∗}𝔾F + ¯∂^{∗} ¯∂𝔾F = ¯∂¯∂^{∗}𝔾F + ¯∂^{∗}𝔾¯∂F = ¯∂¯∂^{∗}𝔾F.

Thus F = ¯∂ũ where ũ = ¯∂^{∗}𝔾F is the canonical solution on ℂP^{n}. Using the interior regularity for ¯∂, we see that ũ ∈ W^{1}(ℂP^{n}). Restricting ũ to M and denoting the restriction by u, we have u ∈ W^{\frac{1}{2}} (M) and ¯∂_{b}u = f.

5. Proof of Theorem 1

We will use results of Section 4 to prove Theorem 1. Let M be a C^{2,α} Levi-flat hypersurface in ℂP^{n}, n ≥ 2 and let ρ = −δ be the signed distance function for M. We first discuss solvability with regularity for the ¯∂_{b} equation in the Sobolev spaces on M.

As we pointed out in Section 0, the proof of Theorem 1 is reduced to a problem of finding a continuous solution u of

(5.1)  i∂_{b} ¯∂_{b}u = f_{b} = f|_{[T (M)]_{ℝ}∩J[T (M)]_{ℝ}} on M^{2n−1}

on M under the assumption that f = dv is an exact form and f_{b} is a real-valued Hölder continuous (1, 1)-form. When f_{b} is a (1, 1)-form and v^{(0,1)} is part of v, we have ¯∂_{b}v^{(0,1)} = 0 and ∂_{b}v^{(1,0)} = 0.

There is the corresponding classical Lelong equation

(5.2)  i∂ ¯∂ũ = f̃ on ℂP^{n}

where f̃ = dṽ is an exact real-valued (1, 1)-form on ℂP^{n}. Since f̃ is of type (1, 1), we have ¯∂ṽ^{(0,1)} = 0 and ∂ṽ^{(1,0)} = 0 where ṽ = ṽ^{(0,1)} + ṽ^{(1,0)} on ℂP^{n}. The ∂ ¯∂- exact Poincaré Lemma states that if 𝔾_{(0,1)} is the Green’s operator of the Laplace operator □_{(0,1)} acting on (0, 1)-forms of ℂP^{n}, then ũ = 2ℜ{¯∂^{∗}𝔾_{(0,1)}ṽ^{(0,1)}} on ℂP^{n} is a solution of (5.2), where ℜ{ϕ} denotes the real part of ϕ (cf. [Zh]).

Hence, to solve our equation (5.1) it is sufficient to extend the ¯∂_{b}-closed 1-form v^{(0,1)} of M to be a global ¯∂-closed form ṽ^{(0,1)} on the whole space ℂP^{n}. Recall that if w_{±}^{(0,1)} is an arbitrary extension of v^{(0,1)} from M = bΩ_{±} to Ω_{±}, the ¯∂-closed extension ṽ_{±}^{(0,1)} of v^{(0,1)} is given by

ṽ_{±}^{(0,1)} = w_{±}^{(0,1)} + [¯∗(¯∂N_{(n,n−2)}|_{Ω_{±}})]¯∗(¯∂w_{±}^{(0,1)}).

We would also like to explain why the W^{s}(Ω)-regularity result of Theorem 2 is good enough for the proof of Theorem 1. When the Levi-flat hypersurface M is C^{2,α}-smooth, we can choose w = θ to be the connection form for the complex line bundle of equidistant hypersurfaces in a neighborhood U_{ϵ}(M) of M. It was shown in Section 1 that both w and dw (i.e., θ and the curvature Θ^{N} = dθ) are C^{0,α}-smooth in U_{ϵ}(M). If Theorem 2 holds, the ¯∂-closed extension ṽ^{(0,1)} of θ^{(0,1)} is in W_{(0,1)}^{s}(ℂP^{n}). Using elliptic theory on ℂP^{n}, we obtain that the solution h̃ = ũ to (5.2) is in W_{(0,0)}^{1+s}(ℂP^{n}), see [GT]. Applying the trace theorem in Sobolev spaces to

h̃, we conclude that h = h̃|_{M} is in W^{\frac{1}{2} +s}(M). By the classical elliptic theory on the holomorphic leaves of M, we already know that h is C^{∞} in each holomorphic leaf of M. With some extra efforts, we can show that h is Hölder continuous in remaining directions of [T (M)]_{ℝ}, whenever h ∈ W^{\frac{1}{2} +s}(M) and i∂_{b} ¯∂_{b}h = Θ_{b} ∈ C^{0,α}(M) for some s, α > 0, see Lemmas 5.1-5.2 below.

Lemma 5.1. Let M ⊂ ℂP^{n} be a C^{2}-smooth Levi-flat hypersurface and let f ∈ W(p,q)^{\frac{1}{2}}(M), where 0 ≤ p ≤ n, 1 ≤ q ≤ n − 1 and p ≠ q. Suppose that ϵ_{0} = \frac{1}{2} min\{t_{0}(Ω_{+}), t_{0}(Ω_{−})\} and t_{0}(Ω) is the order of plurisubharmonicity of Ω given by

Definition 0.1. We further assume that f ∈ W_{(p,q)}^{\frac{1}{2} +ϵ}(M) is ¯∂_{b}-closed in M if q < n−1

and p < n − 1 if q = n − 1. Then for any 0 ≤ ϵ < ϵ_{0}, there exists u ∈ W_{(p,q−1)}^{\frac{1}{2} +ϵ}(M) satisfying ¯∂_{b}u = f in M.

Proof. Let ℂP^{n} \ M = Ω_{+} ∪ Ω_{−}. Let t_{0}^{+} and t_{0}^{−} be the orders of plurisubharmonicity for the distance functions associated with Ω_{+} and Ω_{−} respectively defined in Definition 0.1.

We use the same notion as in Section 4. We can construct a ¯∂-closed form F ∈ W_{(p,q)}^{ϵ}(ℂP^{n}) such that F |_{M} = f. This is proved by Theorem 3.5 and (4.14)- (4.18) as follows.

Since f ∈ W_{(p,q)}^{\frac{1}{2} +ϵ}(M), there exists an extension f̃ ∈ W^{1+ϵ}(ℂP^{n}) of f by the Trace Theorem (cf. Appendix of [CS]). As in the proof of Corollary 4.6, we let

F_{±} = f̃+ ⋆[¯∂N_{(n−p,n−q−1)}|_{Ω_{±}}](⋆¯∂f̃) on Ω_{±}.

This implies that F ∈ W^{ϵ}(ℂP^{n}) by Theorem 3.5.

Next, we solve the equation ¯∂ũ = F on the whole manifold ℂP^{n}. For the same reason as in the proof of Corollary 4.7, we conclude that ũ ∈ W^{1+ϵ} with ¯∂ũ = F. This gives that u = ũ|_{M} is in W_{(p,q−1)}^{\frac{1}{2} +ϵ}(M) by the Trace Theorem again. □

We need two more preliminary results for the proof of Theorem 1. We denote by N 1,0 ; ˜∇ρ the complex line bundle and its associated curvature form by Θ̃^{N} as in Section 1. From Propositions 1.1-1.2, Θ_{b} is a (1, 1)-form on the Levi-flat hypersurface M. As in Section 1, we let β be the real 1-form such that

Θ̃^{N} = −dθ = \sqrt{−1}dβ on U_{ϵ}(M),

where

(5.3)  β(·) = Hess(ρ)(·, J(∇ρ)).

Write

β = β^{1,0} + β^{0,1}

where β^{1,0} and β^{0,1} are the (1,0) and (0,1) components of β.

Lemma 5.2. Let M be a compact C^{2,α} Levi-flat hypersurface in ℂP^{n} and β^{0,1} be the (0, 1)-form defined in (5.3). Then there exists a function u ∈ C^{ϵ}(M) such that

(5.4)  ¯∂_{b}u = β^{0,1} in M

for sufficiently small ϵ < \frac{1}{4} min\{α, t_{0}(Ω_{±})\}, where t_{0}(Ω_{±}) is the plurisubharmonicity of Ω_{±} and Ω_{+} ∪ Ω_{−} = ℂP^{n}.

Proof. Since ρ is of class C^{2,α}, the 1-form β is obtained from the Hessian of ρ, hence, is in C^{α}. The Levi-flat hypersurface M is locally foliated by complex manifolds {Σ_{t}}, where V = ∪_{|t|<μ}Σ_{t} is an open subset of M^{2n−1}. Since each leaf Σ_{t} is a complex hypersurface, it follows from the Chern formula that the curvature tensor Θ_{b} = Θ̃^{N}|_{Σ_{t}} is a (1, 1)-form (see Proposition 1.2). Thus from (1.11) and type consideration, we get for n > 2,

(5.5)  ¯∂_{b}β^{0,1} = 0 in M.

When n = 2, β^{0,1} satisfies the compatibility condition (4.9) by Proposition 4.5.

We claim that there exists a solution u of (5.4) such that u ∈ W^{\frac{1}{2} +ϵ}(M) for some ϵ > 0. If α ≥ \frac{1}{2}, we can use Lemma 5.1 directly. If α < \frac{1}{2}, we note that the Cartan-Chern-Gauss structure equation holds in a tubular neighborhood U = U_{ϵ}(M) of M. Thus, β^{0,1} ∈ C^{α}(U) ⊂ W^{α}(U). Using the last assertion of Proposition 1.1, ¯∂β^{0,1} is in C^{α}(U) since it is the (0, 2)-component of the curvature form Θ̃^{N}, which is in C^{α}. Thus β^{0,1} has an extension to ℂP^{n} with ¯∂β^{0,1} in W_{(0,2)}^{α}(Ω_{+}) and W_{(0,2)}^{α}(Ω_{−}) respectively.

Repeating the same arguments as in the proof of Proposition 4.4 for q < n − 1 and Corollary 4.6 for q = n − 1, we obtain a ¯∂-closed extension F ∈ L^{2} of β^{0,1} on the whole space ℂP^{n} with F = β^{0,1} on M. Also from the boundary regularity of the ¯∂-Neumann operator proved in Theorem 3.5, the extension F is in W^{ϵ}(ℂP^{n}) with ϵ < min{α, ϵ_{0}}, where ϵ_{0} is defined in Lemma 5.1.

Using the same arguments as in the proof of Corollary 4.7, we can prove that there exists ũ ∈ W^{1+ϵ}(ℂP^{n}) satisfying ¯∂ũ = F. Therefore, if u = ũ|_{M}, then we have u ∈ W^{\frac{1}{2} +ϵ}(M) satisfying (5.4).

It remains to show that u is Hölder continuous in M. This follows from the following version of the Sobolev embedding theorem. We note that, on each leaf Σ_{t}, u satisfies an elliptic equation. Thus we already have that u is smooth on each leaf Σ_{t}, because (∂ ¯∂u)|_{Σ_{t}} = Θ_{b} and Θ_{b} is analytic on each holomorphic leaf Σ_{t}. It remains only to show that u is Hölder continuous in the transversal direction \frac{∂}{∂t}. To do this we need to parametrize our hypersurface M locally.

Let (z′, g(z′, t)) denote the leaf L_{t} where g(z′, t) is holomorphic in z′ ∈ 𝔹^{n−1} ⊂ ℂ^{n−1} and C^{2}-smooth in t. We can parametrize M locally as a graph of a function η + g, by setting

Ψ(z′, t) = (z′, η(t) + g(z′, t)),

where z′ ∈ ℂ^{n}, 0 ≤ |t| < μ and η(t) is a C^{1,α} function in t with η(0) = 0 and η′(0) = 1. Clearly, Ψ : 𝔹^{n−1} × (−μ, μ) → M is a local coordinate map of M, where 𝔹^{n−1} ⊂ ℂ^{n−1} is an open set of ℂ^{n−1}. Using a result of Barrett-Fornaess (see [BaF]), the foliation is actually C^{2,α}.

It is easy to see that the push forwards L̄_{i} = Ψ_{∗}(\frac{∂}{∂z̄_{i}}) of \frac{∂}{∂z̄_{i}}, i = 1, ⋯, n − 1, are the tangential Cauchy-Riemann equations for M. Let

(5.6)  T = Ψ_{∗}(\frac{∂}{∂t}).

Since

(5.7)  [\frac{∂}{∂t}, \frac{∂}{∂z̄_{i}} ] = 0, [\frac{∂}{∂t}, \frac{∂}{∂z_{i}} ] = 0,

we have

[T, L̄_{i}] = 0, [T, L_{i}] = 0.

Thus the tangential Cauchy-Riemann equations ¯∂_{b} are just the Cauchy-Riemann equations on Σ_{t} and they commute with T. When u is restricted to each leaf Σ_{t}, u satisfies an elliptic system in coordinates z′ and

(5.8)  ∂_{z′} ¯∂_{z′}u = Θ_{b}  on Σ_{t},

where Θ_{b} = Θ̃^{N}|_{[T (M)]_{ℝ}∩J[T (M)]_{ℝ}} = Θ̃^{N}|_{Σ_{t}} is a (1, 1)-form and J is the complex structure of ℂP^{n}, see Proposition 1.2.

From the classic Schauder theorem (cf. [GT]) for elliptic systems on ℂ^{n−1}, we get that u ∈ C^{2,α}(Σ_{t}) for each t since Θ̃^{N} is in C^{α}. Furthermore, we have that there exists a constant C_{1} independent of t such that

(5.9)  |u|_{C^{2,α}(Σ_{t})} ≤ C_{1}(|Θ̃^{N}|_{C^{α}(Σ_{t})} + ∥u∥_{L^{2}(Σ_{t})}),

where C_{1} depends on V and Ψ, but C_{1} is independent of t, (because the local foliation is C^{2}-smooth and the equation (5.8) is uniformly elliptic on Σ_{t} ⊂ V independent of t).

Recall that V = ∪_{|t|<μ}Σ_{t} ⊂ M. From the Sobolev trace theorem, a function u ∈ W^{\frac{1}{2} +ϵ}(M) has L^{2}-trace on each leaf. Therefore, there exists C_{2} > 0 independent of t such that

(5.10)  ∥u∥_{L^{2}(Σ_{t})} ≤ C_{2}∥u∥_{W ^{\frac{1}{2} +ϵ}(M)}.

Combining (5.9) and (5.10), we get

(5.11)  |u|_{L^{∞}(V)} ≤ sup_{|t|<μ}|u|_{C^{2,α}(Σ_{t})} ≤ C_{3}.

Thus we have already proved that u is bounded.

It remains to show that u is Hölder continuous in t as well. To do this we differentiate the equation (5.8) in t with order 0 < ϵ < 1.

Let D_{t,h}^{ϵ} denote the finite difference

D_{t,h}^{ϵ}u = \frac{u(z′, t + h) − u(z′, t)}{|h|^{ϵ}}

and let | |_{s} be the Besov norm given by

|u|_{s} = sup_{0<|h|<η_{0}} \frac{∥u(x + h) − u(x)∥}{|h|^{s}}.

The Besov norm is weaker than the Sobolev norm ∥u∥_{s}. It is easier to use the Besov norm than the Sobolev norm (see e.g. Hörmander [Hö4]) and we have

(5.12)  ∥u∥_{s′} ≤ |u|_{s} ≤ ∥u∥_{s}

for any s′ < s. Let 0 < ϵ′ < ϵ. We claim that

(5.13)  sup_{0<|h|<η_{0}}|D_{t,h}^{ϵ′}u|_{\frac{1}{2} +ϵ−ϵ′} ≤ C|u|_{\frac{1}{2} +ϵ}.

35 Assuming (5.13) for the moment, we get from (5.12) that D_{t,h}^{ϵ′}u ∈ W^{s′} for any \frac{1}{2} < s′ < \frac{1}{2} + ϵ − ϵ′. Note that from (5.7), the operators D_{t,h}^{ϵ′} and ∂_{b} ¯∂_{b} commute. Thus we have

(5.14)  ∂_{b} ¯∂_{b}D_{t,h}^{ϵ′}u = D_{t,h}^{ϵ′}∂_{b} ¯∂_{b}u = D_{t,h}^{ϵ′}Θ̃^{N} ∈ C^{α−ϵ′}(U).

Applying the classical Schauder estimates to the equation (5.14) and repeating the above arguments used to obtain (5.11), we have

(5.15)  D_{t,h}^{ϵ′}u ∈ L^{∞}(U).

This implies that u ∈ C^{ϵ′}(U). To finish the proof of the lemma, it remains to prove the claim (5.13). □

Lemma 5.3. Let u be a function with |u|_{s+ϵ′} < ∞ as above. Then

sup_{0<|h|<η_{0}}|D_{t,h}^{ϵ′}u|_{s} ≤ C|u|_{s+ϵ′}.

Proof. We identify h = (0, ⋯, 0, h). We have

(5.16)  \frac{∥D_{t,h}^{ϵ′}u(x + h̃) − D_{t,h}^{ϵ′}u(x)∥}{|h̃|^{s}} ; = \frac{∥(u(x + h̃ + h) − u(x + h̃)) − (u(x + h) − u(x))∥}{|h|^{ϵ′}|h̃|^{s}} ; ≤ \frac{(|u(x + h̃)|_{s+ϵ′} + |u|_{s+ϵ′})|h|^{s+ϵ′}}{|h|^{ϵ′}|h̃|^{s}} ; ≤ \frac{2|u|_{s+ϵ′}|h|^{s+ϵ′}}{|h|^{ϵ′}|h̃|^{s}} .

Similarly, we have

(5.17)  \frac{∥D_{t,h}^{ϵ′}u(x + h̃) − D_{t,h}^{ϵ′}u(x)∥}{|h̃|^{s}} ; = \frac{∥(u(x + h̃ + h) − u(x + h)) − (u(x + h̃) − u(x))∥}{|h|^{ϵ′}|h̃|^{s}} ; ≤ \frac{(|u(x + h)|_{s+ϵ′} + |u|_{s+ϵ′})|h̃|^{s+ϵ′}}{|h|^{ϵ′}|h̃|^{s}} ; ≤ \frac{2|u|_{s+ϵ′}|h̃|^{s+ϵ′}}{|h|^{ϵ′}|h̃|^{s}} .

By (5.16) and (5.17), we conclude that

sup_{0<|h|<η_{0}}|D_{t,h}^{ϵ′}u|_{s} = sup_{0<|h|<η_{0}} sup_{0<|h̃|<η_{0}} \frac{∥D_{t,h}^{ϵ′}u(x + h̃) − D_{t,h}^{ϵ′}u(x)∥}{|h̃|^{s}}

≤ C sup_{0<|h|+|h̃|<2η_{0}} min(|h|^{s+ϵ′}, |h̃|^{s+ϵ′})\frac{|u|_{s+ϵ′}}{|h|^{s′}|h̃|^{s}}

≤ C|u|_{s+ϵ′}.

This proves the lemma. □

Proof of Theorem 1. Let M be a Levi-flat hypersurface of class C^{2,α} in ℂP^{n}, n ≥ 2. Let u be obtained in Lemma 5.2 and h = 2Imu, where Imu is the imaginary part of u. From Lemma 5.2, the function h is Hölder continuous on M. Since h is real-valued and M is compact, there exists a point Q_{0} such that h assumes a global maximum at Q_{0}. On the other hand, we have

(5.18)  \begin{aligned}Θ_{b} & = Θ̃^{N}|_{[T (M)]_{ℝ}∩J[T (M)]_{ℝ}} \\ & = (−dβ)|_{[T (M)]_{ℝ}∩J[T (M)]_{ℝ}} = −∂_{b}β^{0,1} − ¯∂_{b}β^{1,0} \\ & = −∂_{b} ¯∂_{b}u − ¯∂_{b}∂_{b}ū \\ & = ∂_{b} ¯∂_{b}h \end{aligned}

on M. From Proposition 1.2, the curvature form iΘ_{b} is a positive (1, 1)-form on M. Although h is only C^{0,ϵ}- continuous on M, its Hessian and i∂_{b} ¯∂_{b}h can be computed by using the barrier functions, (e.g., see [Ca]). From (5.18) and Proposition 1.2, h is a strictly plurisubharmonic function when restricted to each leaf Σ_{t} of M. Hence, at the global maximum point Q_{0} ∈ M of h, we obtain a strictly plurisubharmonic function h on that particular leaf Σ_{0} containing Q_{0} as an interior maximum. This contradicts the maximum principle (cf. [Ca]). Thus there does not exist any Leviflat hypersurface of class C^{2,α} in ℂP^{n}. Theorem 1 is proved. □

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Jianguo Cao, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 USA

E-mail address: cao.7@nd.edu

Mei-Chi Shaw, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 USA

E-mail address: mei-chi.shaw.1@nd.edu

Lihe Wang, Department of Mathematics, University of Iowa, Iowa City, IA 52242 USA

E-mail address: lwang@math.uiowa.edu