Global Regularity for the 𝜕_𝑏-Equation on 𝐶𝑅 Manifolds of Arbitrary Codimension

Abstract

Let 𝑀 be a 𝒞^∞ compact 𝐶𝑅 manifold of 𝐶𝑅-codimension ℓ≥1 and 𝐶𝑅-dimension 𝑛−ℓ in a complex manifold 𝑋 of complex dimension 𝑛≥3. In this paper, assuming that 𝑀 satisfies condition 𝑌(𝑠) for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1, we prove an 𝐿²-existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for (0,𝑠)-forms on 𝑀.

1. Introduction and Basic Notations

The tangential Cauchy-Riemann complex (or 𝜕_𝑏-complex) was first introduced by Kohn and Rossi [1] for studying the holomorphic extension of 𝐶𝑅 functions from the boundary of a complex manifold. The closed range property is related to existence and regularity theorems for 𝜕_𝑏 and for 𝐶𝑅 manifolds to a reason of embedding. It is worth then to mention that the 𝜕_𝑏-operator has closed range in the 𝐿²-sense on boundaries of smooth bounded pseudoconvex domains in ℂ^𝑛 due to Shaw [2] for all 1≤𝑠<𝑛−2 and Boas and Shaw [3] for 𝑠=𝑛−2. Later, Kohn [4] obtained results analogue to those of [2, 3] on boundaries of smooth bounded pseudoconvex domains in a complex manifold. Nicoara [5] extended the results of Kohn [4] to compact, orientable, pseudoconvex 𝐶𝑅 manifold of real dimension 2𝑛−1, at least five, embedded in ℂ^𝑁, 𝑁≥𝑛, leading to global regularity for the 𝜕_𝑏-equation on such 𝐶𝑅 manifolds. The main tool in his proof is that of microlocalizations using a new type of weight functions called strongly 𝐶𝑅 plurisubharmonic functions (see also [6]).

In addition, Harrington and Raich [7] adapted the microlocal analysis used by Nicoara [5] to establish the closed range property for the 𝜕_𝑏-operator on 𝐶𝑅 manifold of hypersurface type satisfying weak 𝑌(𝑠) condition. More precisely, by using the weighted 𝜕-theory, they showed that the complex Green’s operator is continuous in the 𝐿²-Sobolev spaces 𝑊^𝑘, 𝑘∈ℕ, and they further obtained a global solution with 𝒞^∞-regularity for solutions of the 𝜕_𝑏-equation for (0,𝑠)-forms.

This paper is concerned with proving an 𝐿²-existence theorem for the 𝜕_𝑏-Neumann problem on a 𝒞^∞𝐶𝑅 compact manifold 𝑀 of real dimension 2𝑛−ℓ (ℓ≥1) that satisfies condition 𝑌(𝑠) for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1 in an 𝑛-dimensional complex manifold 𝑋 and with establishing the global regularity properties of the 𝜕_𝑏-equation. In particular, our 𝜕_𝑏-problem is set up in the usual 𝐿²-setting with no weights using our arguments in [8, 9]. Namely, via a partition of unity, we globalize first the local maximal 𝐿²-Sobolev estimates obtained by [10] for □_𝑏 and patching them together to obtain global ones on 𝑀. Further, we explore an 𝐿²-existence theorem for the 𝜕_𝑏-equation on 𝑀. These 𝐿² results allow us to prove that the complex Green operator 𝐺_𝑏 and the Szegö projection operators 𝑆_𝑠 are continuous in the Sobolev spaces 𝑊^𝑘_0,𝑠(𝑀) for some 𝑠 such that 1≤𝑠≤𝑛−ℓ−1 and 𝑘≥0. Furthermore, we obtain a global smooth solution for the 𝜕_𝑏-equation given smooth data on 𝑀. Before we proceed, we recall first some basic definitions and notations on 𝐶𝑅 manifolds.

Definition 1. Let 𝑀 be a 𝒞^∞-manifold of real dimension 2𝑛−ℓ. Then a 𝐶𝑅 structure on 𝑀 is given by a complex subbundle 𝑇^1,0(𝑀) of the complexified tangent bundle ℂ𝑇(𝑀)=𝑇(𝑀)⊗ℂ such that the following conditions are satisfied. (1)dim_ℂ𝑇_𝑧^1,0(𝑀)=𝑛−ℓ, where 𝑇_𝑧^1,0(𝑀) is the fiber at each 𝑧∈𝑀.(2)If we define 𝑇^0,1(𝑀)=𝑇^1,0(𝑀), then 𝑇^1,0(𝑀)∩𝑇^0,1(𝑀)={0}.(3)𝑇^1,0(𝑀) is involutive (or formally integrable); that is, if 𝐿₁ and 𝐿₂ are two smooth sections of 𝑇^1,0(𝑀), defined on an open subset 𝕌 of 𝑀, then so is their Lie bracket [𝐿₁,𝐿₂]=𝐿₁𝐿₂−𝐿₂𝐿₁, for every open subset 𝕌 of 𝑀. A 𝒞^∞ manifold 𝑀 endowed with this 𝐶𝑅 structure is called a 𝐶𝑅 manifold of 𝐶𝑅-dimension 𝑛−ℓ and 𝐶𝑅 codimension ℓ.

Let 𝑀 be a generic 𝐶𝑅 manifold of real dimension 2𝑛−ℓ embedded in an 𝑛-dimensional complex manifold 𝑋. Such a manifold 𝑀 can be represented locally in the following form: for each 𝑧∈𝑀 there exists an open neighborhood 𝑈 of 𝑧 in 𝑋 such that 𝑀∩𝑈=𝜁∈𝑈∣𝜌₁(𝜁)=⋯=𝜌_ℓ(𝜁)=0, (1) where {𝜌_𝜈}_{𝜈=1,…,ℓ} are 𝒞^∞ real-valued functions on 𝑈 such that 𝜕𝜌₁(𝜁)∧⋯∧𝜕𝜌_ℓ(𝜁)≠0on𝑀∩𝑈. (2)

The complex subbundle which defines the induced 𝐶𝑅 structure on 𝑀 is given by 𝑇^1,0(𝑀)=𝑇^1,0(𝑋)∩ℂ𝑇(𝑀). Denote by 𝒞^∞_0,𝑠(𝑀) the space of (0,𝑠)-forms with 𝒞^∞-coefficients on 𝑀. The involution condition (3) of Definition 1 implies that there is a restriction of the de Rham exterior derivative 𝑑 to 𝒞^∞_0,𝑠(𝑀), which is defined by 𝜕_𝑏∶𝒞^∞_0,𝑠(𝑀)→𝒞^∞_0,𝑠+1(𝑀).

Let us equip 𝑋 with a Hermitian metric such that 𝑇^1,0(𝑋)⟂𝑇^0,1(𝑋) and consider on 𝑀 the induced metric, then 𝑇^1,0(𝑀)⟂𝑇^0,1(𝑀). Let 𝒟_0,𝑠(𝑀) be the space of (0,𝑠)-forms whose coefficients are 𝒞^∞ with compact support in 𝑀. We then can define a Hermitian inner product on 𝒟_0,𝑠(𝑀) by 𝜑,𝜓=_𝑀𝜑,𝜓_𝑧𝑑𝑣, (3) where 𝑑𝑣 is the volume element associated with the induced metric on 𝑀 and ⟨𝜑,𝜓⟩_𝑧 is the pointwise inner product induced on 𝒞^∞_0,𝑠(𝑀) by the metric on ℂ𝑇(𝑀) at each 𝑧∈𝑀. Let ‖𝜑‖²=(𝜑,𝜑) be the corresponding norm and 𝐿²_0,𝑠(𝑀) the 𝐿²-completion of 𝒟_0,𝑠(𝑀) with respect to this norm. Let 𝜕_𝑏∶𝐿²_0,𝑠(𝑀)→𝐿²_0,𝑠+1(𝑀) be the maximal closed extension of the original 𝜕_𝑏 on 𝒞^∞_0,𝑠(𝑀). A form 𝑢∈𝐿²_0,𝑠(𝑀) is in the domain of 𝜕_𝑏 if 𝜕_𝑏𝑢, defined in the sense of distributions, belongs to 𝐿²_0,𝑠+1(𝑀). In this way, 𝜕_𝑏 defines a linear, closed, densely defined operator. Let 𝜕^∗_𝑏∶𝐿²_0,𝑠+1(𝑀)→𝐿²_0,𝑠(𝑀) be the 𝐿²-Hilbert space adjoint of 𝜕_𝑏 such that (𝜑,𝜕_𝑏𝜓)=(𝜕^∗_𝑏𝜑,𝜓) for all 𝜓 in Dom(𝜕_𝑏) and 𝜑 in Dom(𝜕^∗_𝑏). The Kohn-Laplacian □_𝑏 is defined by □_𝑏=𝜕_𝑏𝜕^∗_𝑏+𝜕^∗_𝑏𝜕_𝑏∶Dom□_𝑏⟶𝐿²_0,𝑠(𝑀), (4) where Dom□_𝑏=𝜑∈Dom𝜕_𝑏∩Dom𝜕^⋆_𝑏⊂𝐿²_0,𝑠(𝑀)∣𝜕_𝑏𝜑∈Dom𝜕^⋆_𝑏,𝜕^⋆_𝑏𝜑∈Dom𝜕_𝑏. (5)

We recall that the Kohn-Laplacian □_𝑏 is not elliptic, so it has a characteristic set of dimension ℓ. Let 𝑁(𝑀) be the ℓ-dimensional bundle such that ℂ𝑇(𝑀)=𝑇^1,0(𝑀)⊕𝑇^0,1(𝑀)⊕𝑁(𝑀). (6) Let 𝑁^∗(𝑀) be the dual bundle of 𝑁(𝑀). Let 𝛾∈𝑁^∗(𝑀), then 𝛾 annihillates 𝑇^1,0(𝑀)⊕𝑇^0,1(𝑀). Thus 𝑁^∗(𝑀) is called the characteristic bundle. The Levi form of 𝑀 at a point 𝑧∈𝑀 is defined as the Hermitian form on 𝑇^1,0(𝑀) with values in 𝑁(𝑀) such that ℒ_𝑧𝐿₁,𝐿₂=𝑖𝜋_𝑧𝐿₁,𝐿_2𝑧,𝐿₁,𝐿₂∈𝑇^1,0(𝑀), (7) where 𝜋_𝑧 is the projection of ℂ𝑇_𝑧(𝑀) onto 𝑁_𝑧(𝑀).

The Levi form of 𝑀 at a point 𝑧∈𝑀 in the direction 𝛾∈𝑁^∗(𝑀) is the scalar Hermitian form denoted ℒ_𝑧(𝛾) and is given by ℒ_𝑧𝛾=ℒ_𝑧𝐿₁,𝐿₂𝐿,𝛾=𝑖₁,𝐿₂,𝛾_𝑧,𝐿₁,𝐿₂∈𝑇^1,0(𝑀). (8)

Definition 2 (see [10, Definition 1.2]). A 𝐶𝑅 manifold 𝑀 of real dimension 2𝑛−ℓ and codimension ℓ≥1 in a complex manifold of complex dimension 𝑛 is said to satisfy condition 𝑍(𝑠), 1≤𝑠≤𝑛−ℓ−1, at a point 𝑧∈𝑀 in the direction 𝛾∈𝑁^∗(𝑀) if the Levi form ℒ_𝑧(𝛾) has at least 𝑛−ℓ−𝑠+1 positive eigenvalues or at least 𝑠+1 negative eigenvalues. 𝑀 is said to satisfy condition 𝑌(𝑠) at 𝑧∈𝑀 if it satisfies condition 𝑍(𝑠) for all directions 𝛾∈𝑁^∗_𝑧(𝑀).

Note that in the hypersurface case, that is, ℓ=1, the condition 𝑌(𝑠) defined above is equivalent to the classical 𝑌(𝑠) condition of Kohn for hypersurfaces (see, e.g., [11] for more details). In particular, if the 𝐶𝑅 structure is strictly pseudoconvex; that is, the Levi form of 𝑀 is positive or negative definite, condition 𝑌(𝑠) holds for all 1≤𝑠≤𝑛−2.

2. 𝐿²-Existence Theory for 𝝏_𝑏

Let 𝑀 be a 𝒞^∞ generic 𝐶𝑅 manifold of real dimension 2𝑛−ℓ and codimension ℓ≥1 in a complex manifold 𝑋 of complex dimension 𝑛. For each point 𝑝₀∈𝑀, there is then a neighborhood 𝑈 of 𝑝₀ in 𝑋 and a local orthonormal basis consisting of smooth vector fields 𝐿₁,…,𝐿_𝑛−ℓ for 𝑇^1,0(𝑈) (see, e.g., [12, Section 7.2; Theorem 3]). The collection of vector fields {𝐿₁,…,𝐿_𝑛−ℓ} forms a local orthonormal basis for 𝑇^0,1(𝑈). Let 𝑇₁,…,𝑇_ℓ be real vector fields on 𝑈 such that the set {𝐿₁,…,𝐿_𝑛−ℓ,𝐿₁,…,𝐿_𝑛−ℓ,𝑇₁,…,𝑇_ℓ} forms a local orthonormal basis for ℂ𝑇(𝑈). Denote by {𝜔¹,…,𝜔^𝑛−ℓ, 𝜔¹,…,𝜔^𝑛−ℓ, 𝛾₁,…,𝛾_ℓ} the basis for ℂ𝑇^⋆(𝑈) dual to {𝐿₁,…,𝐿_𝑛−ℓ,𝑇₁,…,𝑇_ℓ}. In terms of this basis, an element 𝜑 in 𝒞^∞_0,𝑠(𝑈) can be uniquely expressed as a sum: 𝜑=_|𝐼|=𝑠𝜑_𝐼𝜔^𝐼, (9) where 𝐼=(𝑖₁,𝑖₂,…,𝑖_𝑠) is an 𝑠-tuple of integers with 1≤𝑖₁<⋯<𝑖_𝑠≤𝑛−ℓ and 𝜔^𝐼=𝜔^𝑖1∧⋯∧𝜔^𝑖𝑠.

We then have 𝜕_𝑏𝜑=_|𝐼|=𝑠^𝑛−ℓ_𝑗=1𝐿_𝑗𝜑_𝐼𝜔^𝑗∧𝜔^𝐼+⋯=_{|𝐽|=𝑠+1𝑗,𝐼}𝜀_𝐽^𝑗𝐼𝐿_𝑗𝜑_𝐼𝜔^𝐽+⋯, (10) where 𝜀_𝐽^𝑗𝐼 is zero if 𝑗∪{𝐼}≠𝐽 as sets and is the sign of the permutation that reorders 𝑗𝐼 as 𝐽 if 𝑗∪{𝐼}=𝐽, and the ⋯ stands for terms of order zero. Using integration by parts, we obtain 𝜕^∗_𝑏𝜑=−_|𝐼|=𝑠^𝑛−ℓ_𝑗=1𝐿_𝑗𝜑_𝑗𝐼𝜔^𝐼+⋯=−_{|𝐾|=𝑠−1𝑗,𝐼}𝜀_𝐾^𝑗𝐼𝐿_𝑗𝜑_𝐼𝜔^𝐾+⋯. (11) For 𝜑 in 𝒞^∞_0,𝑠(𝑈), the subspace of smooth (0,𝑠)-forms on 𝑈 that can be extended smoothly up to and including the boundary, we set 𝜑²_ℒ(𝑈)=^𝑛−ℓ_𝑗=1𝐿_𝑗(𝜑)²+𝜑²,𝜑²_ℒ(𝑈)=^𝑛−ℓ_𝑗=1𝐿_𝑗(𝜑)²+𝜑². (12) If we further assume that 𝑀 satisfies condition 𝑌(𝑠) for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1, for each 𝑝₀∈𝑀, we can find a constant 𝐶=𝐶(𝑝₀)>0 such that 𝜑²_ℒ(𝑈)+𝜑²_ℒ(𝑈)≤𝐶𝜕_𝑏𝜑²+𝜕^∗_𝑏𝜑²+𝜑² (13) uniformly for all 𝜑∈𝒟_0,𝑠(𝑈) (see, e.g., [10]).

Set 𝐿_𝑗=𝑋_2𝑗−1+𝑖𝑋_2𝑗; 𝑗=1,…,𝑛−ℓ. The condition 𝑌(𝑠) implies that the real vector 𝑋₁,…,𝑋_2𝑛−2ℓ and their commutators of length at most two span the tangent space at each point in 𝑈. Thus 𝑋₁,…,𝑋_2𝑛−2ℓ satisfy Hörmander’s finite rank condition of order two. It follows then from [13, Theorem A] (see also [14]) that there is a positive constant 𝐶=𝐶(𝑈) satisfying the following 1/2-subelliptic estimate: 𝜑²_(𝑈)1/2≤𝐶^2𝑛−2ℓ_𝑖=1𝑋_𝑖𝜑²+𝜑²,𝜑∈𝒟_0,𝑠(𝑈). (14)

Here and always ‖⋅‖_𝑘(𝑈) denotes the 𝐿² Sobolev space 𝑘-norm, ‖⋅‖_−𝑘 is the norm of its dual space, and ‖⋅‖ is the usual 𝐿²-norm. We may omit the subscript 𝑈 from the norm notation when there is no danger of confusion.

Combining the above 1/2-subelliptic estimate with (13), as in [10], we get the following theorem.

Theorem 3. Let 𝑀 be a 𝒞^∞𝐶𝑅 manifold of real dimension 2𝑛−ℓ and codimension ℓ≥1 in a complex manifold 𝑋 of complex dimension 𝑛. Suppose that 𝑀 satisfies condition 𝑌(𝑠) for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1. For each point 𝑝₀∈𝑀, there is then an open neighborhood 𝑈 on which the Kohn Laplacian □_𝑏 satisfies the 1/2-subelliptic estimate 𝜑_1/2(𝑈)≤𝐶𝜕_𝑏𝜑²+𝜕^∗_𝑏𝜑²+𝜑² (15) uniformly for all 𝜑 in 𝒟_0,𝑠(𝑈).
In addition, if 𝑀 is compact, the estimate (15) holds uniformly on 𝑀 for all 𝜑 in 𝒞^∞_0,𝑠(𝑀).

Theorem 4 (see [10]). Let 𝑀 be given as in Theorem 3 and 𝜙 the unique solution of the equation (□_𝑏+𝐼𝑑)𝜙=𝑓 for 𝑓∈𝐿²_0,𝑠(𝑀), where 𝐼𝑑 is the identity operator. Let 𝑈⊂⊂𝑀 be a relatively compact subset of 𝑀. If the restriction of 𝑓 to 𝑈 is in 𝒞^∞_0,𝑠(𝑈), the restriction of 𝜙 to 𝑈 is then in 𝒞^∞_0,𝑠(𝑈). In addition, suppose that 𝜂 and 𝜂₁ are two cut-off functions supported in 𝑈 such that 𝜂=1 on the support of 𝜂₁; then if the restriction of 𝑓 to 𝑈 is in the 𝐿²-Sobolev space 𝑊^𝑘_0,𝑠(𝑈) for some nonnegative integer 𝑘, the restriction of 𝜂₁𝜙 to 𝑈 is in 𝑊^𝑘+1_0,𝑠(𝑈) and there is a constant 𝐶_𝑘>0 (independent of 𝑓) such that 𝜂₁𝜙_𝑘+1(𝑈)≤𝐶_𝑘𝜂𝑓_𝑘(𝑈)+𝑓. (16)

Patching the above local estimates, we obtain the following global one.

Theorem 5. Let 𝑀 be a 𝒞^∞ compact 𝐶𝑅 manifold of real dimension 2𝑛−ℓ and codimension ℓ≥1 in an 𝑛-dimensional complex manifold 𝑋. Suppose that 𝑀 satisfies condition 𝑌(𝑠) for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1. Let 𝜙∈Dom(□_𝑏) such that (□_𝑏+𝐼𝑑)𝜙=𝑓 for 𝑓 in 𝑊^𝑘_0,𝑠(𝑀), 𝑘≥0, then 𝜙 is in 𝑊^𝑘+1_0,𝑠(𝑀) and there exists a constant 𝐶_𝑘>0 (independent of 𝑓) such that 𝜙_𝑘+1(𝑀)≤𝐶_𝑘𝑓_𝑘(𝑀). (17)

Using Theorem 5 and following an induction argument on 𝑘, we get the following result.

Proposition 6. Let 𝑀 be given as in Theorem 5. Then the Kohn Laplacian □_𝑏 is hypoelliptic. Moreover, if □_𝑏𝜙=𝑓 for 𝑓 in 𝑊^𝑘_0,𝑠(𝑀), 𝑘≥0, then 𝜙 is in 𝑊^𝑘+1_0,𝑠(𝑀) and there is a constant 𝐶_𝑘>0 (independent of 𝑓) such that 𝜙²_(𝑀)𝑘+1≤𝐶_𝑘𝑓²_𝑘(𝑀)+𝜙². (18)

Let ℋ^𝑏_0,𝑠(𝑀)=𝛼∈Dom𝜕_𝑏∩Dom𝜕^⋆_𝑏⊂𝐿²_0,𝑠(𝑀)∣𝜕_𝑏𝛼=𝜕^⋆_𝑏𝛼=0 (19) be the closed subspace of 𝐿²_0,𝑠(𝑀) consisting of harmonic forms and ^⟂ℋ^𝑏_0,𝑠(𝑀)=𝛼∈𝐿²_0,𝑠(𝑀)∣𝛼,𝜙=0∀𝜙∈ℋ^𝑏_0,𝑠(𝑀). (20) The main 𝐿²-result is the following theorem.

Theorem 7. Let 𝑀 be a 𝒞^∞ compact 𝐶𝑅 manifold of real dimension 2𝑛−ℓ and codimension ℓ≥1 in an 𝑛-dimensional complex manifold 𝑋. Suppose that 𝑀 satisfies condition 𝑌(𝑠) for some 𝑠 such that 1≤𝑠≤𝑛−ℓ−1. Then the following holds. (1)The space of harmonic (0,𝑠)-forms ℋ^𝑏_0,𝑠(𝑀) is of finite dimensional.(2)The operators 𝜕_𝑏∶𝐿²_0,𝑠(𝑀)→𝐿²_0,𝑠+1(𝑀), 𝜕^⋆_𝑏∶𝐿²_0,𝑠+1(𝑀)→𝐿²_0,𝑠(𝑀), and □_𝑏=𝜕_𝑏𝜕^∗_𝑏+𝜕^∗_𝑏𝜕_𝑏∶Dom(□_𝑏)→𝐿²_0,𝑠(𝑀) have closed ranges.(3)The complex Green operator 𝐺_𝑏∶𝐿²_0,𝑠(𝑀)→Dom(□_𝑏) exists and is a compact operator in 𝐿²_0,𝑠(𝑀).(4)For any 𝑓 in 𝐿²_0,𝑠(𝑀), we have 𝑓=𝜕_𝑏𝜕^⋆_𝑏𝐺_𝑏𝑓+𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓+𝐻^𝑏_0,𝑠𝑓, (21) where 𝐻^𝑏_0,𝑠 is the orthogonal projection of 𝐿²_0,𝑠(𝑀) onto ℋ^𝑏_0,𝑠(𝑀).(5)𝐺_𝑏𝐻^𝑏_0,𝑠=𝐻^𝑏_0,𝑠𝐺_𝑏=0. 𝐺_𝑏□_𝑏=□_𝑏𝐺_𝑏=𝐼𝑑−𝐻^𝑏_0,𝑠 on Dom(□_𝑏).(6)If 𝐺_𝑏 is defined on 𝐿²_0,𝑠+1(𝑀) (resp., 𝐿²_0,𝑠−1(𝑀)), 𝜕_𝑏𝐺_𝑏=𝐺_𝑏𝜕_𝑏 on Dom(𝜕_𝑏) (resp., 𝜕^⋆_𝑏𝐺_𝑏=𝐺_𝑏𝜕^⋆_𝑏 on Dom(𝜕^⋆_𝑏)).(7)If 𝑓 is in 𝐿²_0,𝑠(𝑀) such that 𝜕_𝑏𝑓=0 and 𝑓⟂ℋ^𝑏_0,𝑠(𝑀), then 𝑓=𝜕_𝑏𝜕^⋆_𝑏𝐺_𝑏𝑓 and 𝑢=𝜕^⋆_𝑏𝐺_𝑏𝑓 is the unique solution to the equation 𝜕_𝑏𝑢=𝑓 which is orthogonal to Ker(𝜕_𝑏) and satisfies ‖𝑢‖²≤𝐶‖𝑓‖².(8)𝐺_𝑏(𝒞^∞_0,𝑠(𝑀))⊆𝒞^∞_0,𝑠(𝑀), and for each 𝑘∈ℝ there is a positive constant 𝐶_𝑠 such that the estimate ‖𝐺_𝑏𝑓‖_𝑘+1≤𝐶_𝑠‖𝑓‖_𝑘 holds uniformly for all 𝑓 in 𝒞^∞_0,𝑠(𝑀).

Proof. Since 𝑀 is compact, via a partition of unity, the estimate (15) holds globally on 𝑀. Suppose that 𝑓_𝑘 is a sequence in Dom(𝜕_𝑏)∩Dom(𝜕^∗_𝑏)∩𝐿²_0,𝑠(𝑀) such that ‖𝑓_𝑘‖ is bounded, 𝜕_𝑏𝑓_𝑘→0 in the 𝐿²_0,𝑠+1(𝑀)-norm and 𝜕^∗_𝑏𝑓_𝑘→0 in the 𝐿²_0,𝑠−1(𝑀)-norm as 𝑘→∞. Thus, we have ‖𝑓_𝑘‖_1/2(𝑀)≤𝑐 for some constant 𝑐. By Rellich’s Lemma, the inclusion map 𝑖_𝑀∶𝑊^1/2_0,𝑠(𝑀)→𝐿²_0,𝑠(𝑀) is compact; we can then extract a subsequence of 𝑓_𝑘 which converges in 𝐿²_0,𝑠(𝑀). Then the hypotheses of Theorem 1.1.3 in Hörmander [15] are satisfied which implies that ℋ^𝑏_0,𝑠(𝑀) is finite dimensional and the estimate 𝑓²≤𝐶𝜕_𝑏𝑓²+𝜕^∗_𝑏𝑓² (22) holds for every 𝑓 in Dom(𝜕_𝑏)∩Dom(𝜕^∗_𝑏) with 𝑓⟂ℋ^𝑏_0,𝑠(𝑀).
By Theorem  1.1.2 in [15], we then conclude that the operators 𝜕_𝑏∶𝐿²_0,𝑠(𝑀)→𝐿²_0,𝑠+1(𝑀) and 𝜕^⋆_𝑏∶𝐿²_0,𝑠(𝑀)→𝐿²_0,𝑠−1(𝑀) have closed ranges. We obtain also from (22) that 𝑓≤𝐶□_𝑏𝑓,𝑓∈Dom□_𝑏,𝑓⟂ℋ^𝑏_0,𝑠(𝑀). (23) This estimate implies that □_𝑏 is one-to-one and in view of Theorem  1.1.1 in [15] that the range of □_𝑏 is closed. It forces, since □_𝑏 is self-adjoint, the strong Hodge decomposition: 𝐿²_0,𝑠(𝑀)=Range□_𝑏⊕ℋ^𝑏_0,𝑠(𝑀)=𝜕_𝑏𝜕^∗_𝑏Dom□_𝑏⊕𝜕^∗_𝑏𝜕_𝑏Dom□_𝑏⊕ℋ^𝑏_0,𝑠(𝑀). (24) Thus □_𝑏∶Dom(□_𝑏)→^⟂ℋ^𝑏_0,𝑠(𝑀) is one-to-one and onto. This implies the existence of the complex Green operator 𝐺_𝑏∶𝐿²_0,𝑠(𝑀)→Dom(□_𝑏) as a unique operator that inverts □_𝑏 on ^⟂ℋ^𝑏_0,𝑠(𝑀). The operator 𝐺_𝑏 is defined as follows: if 𝑓 is in Range(□_𝑏), we define 𝐺_𝑏𝑓=𝜙, where 𝜙 is the unique solution of □_𝑏𝜙=𝑓 with 𝑓⟂ℋ^𝑏_0,𝑠(𝑀). 𝐺_𝑏 is extended to the whole 𝐿²_0,𝑠(𝑀) space by setting 𝐺_𝑏=0 on ℋ^𝑏_0,𝑠(𝑀). The boundedness of 𝐺_𝑏 in 𝐿²_0,𝑠(𝑀) follows from (23).
To show that 𝐺_𝑏 is compact in 𝐿²_0,𝑠(𝑀), it suffices to show compactness on ^⟂ℋ^𝑏_0,𝑠(𝑀) (since 𝐺_𝑏≡0 on ℋ^𝑏_0,𝑠(𝑀)). When 𝑓⟂ℋ^𝑏_0,𝑠(𝑀) (and hence 𝐺_𝑏𝑓⟂ℋ^𝑏_0,𝑠(𝑀)), the integration by parts, Cauchy-Schwarz inequality (|(𝑢,𝑣)|≤‖𝑢‖‖𝑣‖), and (23) imply 𝜕_𝑏𝐺_𝑏𝑓²+𝜕^∗_𝑏𝐺_𝑏𝑓²=𝜕_𝑏𝐺_𝑏𝑓,𝜕_𝑏𝐺_𝑏𝑓+𝜕^∗_𝑏𝐺_𝑏𝑓,𝜕^∗_𝑏𝐺_𝑏𝑓=𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝐺_𝑏𝑓+𝜕_𝑏𝜕^∗_𝑏𝐺_𝑏𝑓,𝐺_𝑏𝑓=𝑓,𝐺_𝑏𝑓≤𝑓𝐺_𝑏𝑓≤𝐶𝑓². (25) By applying (15) to 𝐺_𝑏𝑓 and using (23), we get 𝐺_𝑏𝑓²_(𝑀)1/2≤𝐶𝜕_𝑏𝐺_𝑏𝑓²+𝜕^∗_𝑏𝐺_𝑏𝑓²+𝐺_𝑏𝑓²≤𝐾𝑓², (26) where 𝐾 is a positive constant. Thus the compactness of 𝐺_𝑏 in 𝐿²_0,𝑠(𝑀) follows from Rellich’s Lemma.
The assertions in (5) follow immediately from the definition of 𝐺_𝑏. For assertion (6), if 𝑓∈Dom(𝜕_𝑏) and 𝐺_𝑏 is also defined on 𝐿²_0,𝑠+1(𝑀), by (21) and the first assertion of (5), we have 𝐺_𝑏𝜕_𝑏𝑓=𝐺_𝑏𝜕_𝑏𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓=𝐺_𝑏𝜕_𝑏𝜕^⋆_𝑏+𝜕^⋆_𝑏𝜕_𝑏𝜕_𝑏𝐺_𝑏𝑓=𝐺_𝑏□_𝑏𝜕_𝑏𝐺_𝑏𝑓=𝜕_𝑏𝐺_𝑏𝑓. (27) A similar equation holds for 𝜕^⋆_𝑏. Assertions (1)–(6) have been established.
To show assertion (7), if 𝑓⟂ℋ^𝑏_0,𝑠(𝑀) and 𝜕_𝑏𝑓=0, then 𝜕_𝑏𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓=0 as well (from (21)). Consequently, ‖𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓‖²=(𝜕_𝑏𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝜕_𝑏𝐺_𝑏𝑓)=0, since 𝜕_𝑏𝐺_𝑏𝑓∈Dom(𝜕^⋆_𝑏), and hence 𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓=0. Thus 𝑓=𝜕_𝑏(𝜕^⋆_𝑏𝐺_𝑏𝑓) and 𝑢=𝜕^⋆_𝑏𝐺_𝑏𝑓 is orthogonal to Ker(𝜕_𝑏). Following assertion (3) and the fact that 𝐺_𝑏 is bounded, 𝑢 satisfies the following 𝐿²-estimate: ‖𝑢‖²=𝜕^⋆_𝑏𝐺_𝑏𝑓²=𝜕^⋆_𝑏𝐺_𝑏𝑓,𝜕^⋆_𝑏𝐺_𝑏𝑓=𝜕_𝑏𝜕^⋆_𝑏𝐺_𝑏𝑓,𝐺_𝑏𝑓=𝜕_𝑏𝜕^⋆_𝑏+𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝐺_𝑏𝑓=𝑓,𝐺_𝑏𝑓≤𝑓𝐺_𝑏𝑓≤𝐶𝑓². (28) Finally, we show assertion (8); if 𝑓∈𝒞^∞_0,𝑠(𝑀), then 𝑓−𝐻^𝑏_0,𝑠𝑓∈𝒞^∞_0,𝑠(𝑀) and, since 𝑀 is compact, 𝑓∈Dom(□_𝑏). On other hand, from assertion (5), □_𝑏𝐺_𝑏𝑓=𝑓−𝐻^𝑏_0,𝑠𝑓. Since □_𝑏 is hypoelliptic, by Proposition 6, 𝐺_𝑏𝑓∈𝒞^∞_0,𝑠(𝑀).
Again Proposition 6 implies 𝐺_𝑏𝑓_𝑘+1(𝑀)≤𝐶_𝑘□_𝑏𝐺_𝑏𝑓_𝑘(𝑀)+𝐺_𝑏𝑓≤𝐶_𝑘𝑓_𝑘(𝑀)+𝐻^𝑏_0,𝑠𝑓_𝑘(𝑀)+(const.𝑓𝑓)≤𝐶_𝑘(𝑀). (29) Here we have used the fact that ℋ^𝑏_0,𝑠(𝑀) is of finite dimension to conclude the estimate 𝐻^𝑏_0,𝑠𝑓_𝑘(𝑀)≤𝐶_𝑘𝐻^𝑏_0,𝑠𝑓≤𝐶_𝑘𝑓_𝑘(𝑀) (30) for some constant 𝐶_𝑘. The theorem is proved.

3. Sobolev Space Estimates

In this section, we prove that the complex Green operator 𝐺_𝑏, the canonical solution operators 𝜕_𝑏𝐺_𝑏 and 𝜕^∗_𝑏𝐺_𝑏, and the Szegö projection 𝑆_𝑠 operators enjoy some regularity properties in the 𝐿²-Sobolev spaces 𝑊^𝑘_0,𝑠(𝑀), 𝑘≥0, for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1. Furthermore, we obtain a global regularity for the solutions of the 𝜕_𝑏-equation. By the same way for bounded pseudoconvex domains, a differential operator is said to be exactly regular if it maps all 𝐿²-Sobolev spaces 𝑊^𝑘_0,𝑠(𝑀) (𝑘≥0) to themselves and globally regular if it maps the space 𝒞^∞_0,𝑠(𝑀) continuously to itself.

3.1. Continuity of the Complex Green Operator

We prove first the continuity of the complex Green operator 𝐺_𝑏 on 𝑊^k_0,s(𝑀), 𝑘≥0.

Theorem 8. Let 𝑀 be a 𝒞^∞ compact 𝐶𝑅 manifold of real dimension 2𝑛−ℓ and codimension ℓ≥1 in an 𝑛-dimensional complex manifold 𝑋. Suppose that 𝑀 satisfies condition 𝑌(𝑠) for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1. Then the complex Green operator 𝐺_𝑏 is continuous on the Sobolev space 𝑊^𝑘_0,𝑠(𝑀), 𝑘≥0; that is, there is a constant 𝐶=𝐶(𝑘)>0 such that 𝐺_𝑏𝑓_𝑘(𝑀)≤𝐶𝑓_𝑘(𝑀),𝑓∈𝑊^𝑘_0,𝑠(𝑀). (31)

Proof. We consider the special case when 𝑘=0,1,2,3,…. Indeed the general case is then derived by means of interpolation of linear operators. Since 𝑀 is compact, it is easy to show that 𝒞^∞_0,𝑠(𝑀) is a dense subspace in 𝑊^𝑘_0,𝑠(𝑀). Further, by Theorem 7 (8), we have 𝐺_𝑏𝑓∈𝒞^∞_0,𝑠(𝑀) for 𝑓∈𝒞^∞_0,𝑠(𝑀). Thus it suffices to establish (31) for 𝑓∈𝒞^∞_0,𝑠(𝑀). For 𝑘=0, (31) follows from (23).
For each 𝑘≥0, let Λ^𝑘(𝜉) be a pseudodifferential operator of order 𝑘 with symbol (1+|𝜉|²)^𝑘/2. Let 𝑈 be an open neighborhood of 𝜁 in 𝑀 and let 𝜂 and 𝜂₁ be two cutoff functions with supports in 𝑈 such that 𝜂=1 on supp 𝜂₁; then 𝜂Λ^𝑘𝜂₁𝑓∈𝒟_0,𝑠(𝑈) whenever 𝑓∈𝒟_0,𝑠(𝑈).
Recall that the compactness of 𝐺_𝑏 in 𝐿²_0,𝑠(𝑈) is equivalent to the compactness estimate: for every 𝜖>0 there is a constant 𝐶(𝜖)>0 such that for every 𝜑∈Dom(𝜕_𝑏)∩Dom(𝜕^∗_𝑏)𝜑²≤𝜖𝑄_𝑏𝜑,𝜑+𝐶(𝜖)𝜑²_−1(𝑈), (32) where 𝑄_𝑏(𝜑,𝜑)=(𝜕_𝑏𝜑,𝜕_𝑏𝜑)+(𝜕^∗_𝑏𝜑,𝜕^∗_𝑏𝜑). For this estimate and further results on the compactness of the complex Green operator see, e.g., [16–19].
Applying (32) for 𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓, we obtain 𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓²≤𝜖𝑄_𝑏𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓+𝐶(𝜖)𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓²_−1(𝑈). (33) We sometimes use 𝐴 for 𝜂Λ^𝑘𝜂₁ and 𝐴^∗ for its formal adjoint, which is also a tangential operator of order 𝑘. We estimate the first term on the right hand side in (33), it is a standard consequence of [20, Corollary 3.1] (or [11, Lemma 2.4.2]) that 𝑄_𝑏𝐴𝐺_𝑏𝑓,𝐴𝐺_𝑏𝑓=Re𝑄_𝑏𝐺_𝑏𝑓,𝐴^∗𝐴𝐺_𝑏𝑓+𝒪𝐷𝐺_𝑏𝑓²_{(𝑈)𝑘−1}≤Re𝑄_𝑏𝐺_𝑏𝑓,𝐴^∗𝐴𝐺_𝑏𝑓+𝐶𝐺_𝑏𝑓²_𝑘(𝑈). (34) Here we have used the fact that the tangential derivative 𝐷^𝛼 of order |𝛼|=𝜆 satisfies the tangential Sobolev estimate |||𝐷^𝛼𝑓|||_𝑟≤‖𝑓‖_𝑟+𝜆.
Taking 𝑣=𝐴^∗𝐴𝑓 in the form 𝑄_𝑏(𝐺_𝑏𝑢,𝑣)=(𝑢,𝑣), we get 𝑄_𝑏𝐴𝐺_𝑏𝑓,𝐴𝐺_𝑏𝑓≤Re𝑓,𝐴^∗𝐴𝐺_𝑏𝑓+𝐶𝐺_𝑏𝑓²_𝑘(𝑈)≤𝑓,𝐴^∗𝐴𝐺_𝑏𝑓+𝐶𝐺_𝑏𝑓²_𝑘(𝑈). (35) The Cauchy-Schwarz inequality implies 𝑄_𝑏𝐴𝐺_𝑏𝑓,𝐴𝐺_𝑏𝑓≤𝐴𝑓𝐴𝐺_𝑏𝑓+𝐶𝐺_𝑏𝑓²_𝑘(𝑈). (36) Inequality (33) becomes 𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓²≤𝜖𝑓_𝑘(𝑈)𝐺_𝑏𝑓_𝑘(𝑈)+𝐶(𝜖)𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓²_−1(𝑈). (37) Summing over a partition of unity subordinate to an open covering of 𝑀 by patches {𝑈_𝑖}^𝑚_𝑖=1, we obtain estimate like (37) on each of these patches and using the interior regularity properties, we get 𝐺_𝑏𝑓²_𝑘(𝑀)≤𝜖𝑓_𝑘(𝑀)𝐺_𝑏𝑓_𝑘(𝑀)+𝐶(𝜖)𝐺_𝑏𝑓²_{𝑘−1(𝑀)}. (38) The first term in the right-hand side of (38) is estimated by 𝜖(s.c.)‖𝐺_𝑏𝑓‖²_𝑘(𝑀)+𝜖(l.c.)‖𝑓‖²_𝑘(𝑀), where s.c. and l.c. denote a small and a large constants, respectively, in the inequality |𝑎𝑏|≤(s.c.)𝑎²+(l.c.)𝑏². The second term is estimated by interpolation of Sobolev norms (‖𝐺_𝑏𝑓‖²_{𝑘−1(𝑀)}≤𝜀‖𝐺_𝑏𝑓‖²_𝑘(𝑀)+𝐶(𝜀)‖𝐺_𝑏𝑓‖²) and then by using the continuity of 𝐺_𝑏 in 𝐿²_0,𝑠(𝑀) with 𝐿²-bounded norm.
Adding up the analogues terms and absorbing, by choosing 𝜖 and 𝜀 to be small enough, ‖𝐺_𝑏𝑓‖²_𝑘(𝑀) into the left, this gives 𝐺_𝑏𝑓²_𝑘(𝑀)≤𝐶𝑓²_𝑘(𝑀)+𝐾𝑓², (39) where 𝐶=𝐶(𝜖,𝑘)>0 and 𝐾=𝐾(𝜖,𝑘)>0. The embedding Sobolev space implies (31) for 𝑘=0,1,2,3,…. The general case is obtained from interpolation of linear operators. As mentioned above, the density of 𝒞^∞_0,𝑠(𝑀) in 𝑊^𝑘_0,𝑠(𝑀) passes (31) to forms 𝑓 in 𝑊^𝑘_0,𝑠(𝑀). This proves the continuity of 𝐺_𝑏 in 𝑊^𝑘_0,𝑠(𝑀).

Corollary 9. Let 𝑀 be given as in Theorem 8, then the canonical solution operators 𝜕_𝑏𝐺_𝑏 and 𝜕^∗_𝑏𝐺_𝑏 are continuous on 𝑊^𝑘_0,𝑠(𝑀) for all 𝑘≥0.

Proof. We argue by induction on 𝑘. The case when 𝑘=0 follows from (25). Suppose that the assertions hold for positive integers less than 𝑘 and assume that 𝜁, 𝑈, 𝜂, and 𝜂₁ are given as in the proof of Theorem 8. By the interior elliptic regularity properties, we prove first a priori estimate for 𝜕_𝑏𝐺_𝑏𝑓 and 𝜕^∗_𝑏𝐺_𝑏𝑓 with 𝑓∈𝒟_0,𝑠(𝑈) as follows: 𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓²+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝑓²=𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓,𝜕_𝑏𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝑓,𝜕^∗_𝑏𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓+𝒪𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝑓𝐺_𝑏𝑓_𝑘(𝑈)=𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓+𝜂Λ^𝑘𝜂₁𝜕_𝑏𝜕^∗_𝑏𝐺_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓+𝒪𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝑓𝐺_𝑏𝑓_𝑘(𝑈)+𝐺_𝑏𝑓²_𝑘(𝑈)=𝜂Λ^𝑘𝜂₁□_𝑏𝐺_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝐺_𝑏𝑓+𝒪𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝑓𝐺_𝑏𝑓_𝑘(𝑈)+𝐺_𝑏𝑓²_𝑘(𝑈)≤𝐶₁𝑓_𝑘(𝑈)𝐺_𝑏𝑓_𝑘(𝑈)+𝐶₂𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝑓𝐺_𝑏𝑓_𝑘(𝑈)+𝐺_𝑏𝑓²_𝑘(𝑈). (40) Summing over a partition of unity, using the small and large constants for the resulting terms ‖𝑓‖_𝑘‖𝐺_𝑏𝑓‖_𝑘, ‖𝜕_𝑏𝐺_𝑏𝑓‖_𝑘‖𝐺_𝑏𝑓‖_𝑘, and ‖𝜕^∗_𝑏𝐺_𝑏𝑓‖_𝑘‖𝐺_𝑏𝑓‖_𝑘, using (31) and adding up the analogues terms, we see that the terms on the right-hand side containing ‖𝜕_𝑏𝐺_𝑏𝑓‖²_𝑘 and ‖𝜕^∗_𝑏𝐺_𝑏𝑓‖²_𝑘 can be absorbed into the left hand side. We therefore obtain 𝜕_𝑏𝐺_𝑏𝑓²_𝑘(𝑀)+𝜕^∗_𝑏𝐺_𝑏𝑓²_𝑘(𝑀)≤𝐶𝑓²_𝑘(𝑀),𝑓∈𝒟_0,𝑠(𝑀). (41) This completes the induction on 𝑘 for the norms of 𝜕_𝑏𝐺_𝑏 and 𝜕^∗_𝑏𝐺_𝑏. By the density of 𝒞^∞_0,𝑠(𝑀) in 𝑊^𝑘_0,𝑠(𝑀), the estimates extend to forms in 𝑊^𝑘_0,𝑠(𝑀). As before, the general case is obtained from interpolation of linear operators. Then 𝜕_𝑏𝐺_𝑏 and 𝜕^∗_𝑏𝐺_𝑏 are continuous on 𝑊^𝑘_0,𝑠(𝑀).

3.2. Exact and Global Regularity Theorems

We now show the expression of the complex Green operator by Szegö projections.

Theorem 10. The Szegö projections 𝑆_𝑠∶𝐿²_0,𝑠(𝑀)→Ker(𝜕_𝑏) are given by the following relations: 𝑆_𝑠=𝐼𝑑−𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏=𝐼𝑑−𝐺_𝑏𝜕^∗_𝑏𝜕_𝑏,𝑠≥0,𝑆_𝑠−1=𝐼𝑑−𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏,𝑠≥1. (42)(43)

Proof. We first show that 𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏=𝐺_𝑏𝜕^∗_𝑏𝜕_𝑏. For 𝛼,𝛽∈^⟂ℋ^𝑏_0,𝑠(𝑀), we observe that 𝜕_𝑏𝛼=0⟹𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝛼=0⟹𝛼=𝜕_𝑏𝜕^∗_𝑏𝐺_𝑏𝛼=𝐺_𝑏𝜕_𝑏𝜕^∗_𝑏𝛼,𝜕^⋆_𝑏𝛽=0⟹𝜕_𝑏𝜕^∗_𝑏𝐺_𝑏𝛽=0⟹𝛽=𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝛽=𝐺_𝑏𝜕^∗_𝑏𝜕_𝑏𝛽. (44)(45) As Range(𝜕_𝑏)⟂Ker(𝜕^∗_𝑏) and Range(𝜕^∗_𝑏)⟂Ker(𝜕_𝑏), one has 𝜕_𝑏𝛼=0⟹𝜕_𝑏𝐺_𝑏𝛼=0,𝜕^∗_𝑏𝛽=0⟹𝜕^∗_𝑏𝐺_𝑏𝛽=0. (46)(47)
Any 𝑓⟂ℋ^𝑏_0,𝑠(𝑀) can then be written as 𝑓=𝛼+𝛽 so that 𝜕_𝑏𝛼=0 and 𝜕^∗_𝑏𝛽=0. By (45) and (46), we then have 𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓=𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝛼+𝛽=𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝛽=𝐺_𝑏𝜕^⋆_𝑏𝜕_𝑏𝛽=𝐺_𝑏𝜕^⋆_𝑏𝜕_𝑏𝑓. (48) This implies the second equality in (42). Now, If 𝑓∈Ker(𝜕_𝑏), then (𝐼𝑑−𝐺_𝑏𝜕^∗_𝑏𝜕_𝑏)𝑓=𝑓, so the expression for 𝑆_𝑠 holds. Next, if 𝑓⟂Ker(𝜕_𝑏) and hence 𝑓⟂ℋ^𝑏_0,𝑠(𝑀), so 𝑓=𝜕_𝑏𝜕^⋆_𝑏𝐺_𝑏𝑓+𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓 and 𝑢=𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓 is the canonical solution to the equation 𝜕_𝑏𝑢=𝜕_𝑏𝑓. Thus 𝜕_𝑏(𝑓−𝑢)=0, that is, 𝑓−𝑢∈Ker(𝜕_𝑏). We claim that 𝑢⟂Ker(𝜕_𝑏). Indeed, for all 𝑔∈Ker(𝜕_𝑏) one has (𝑢,𝑔)=(𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝑔)=(𝜕_𝑏𝐺_𝑏𝑓,𝜕_𝑏𝑔)=0. Since 𝑓⟂Ker(𝜕_𝑏), it turns out that 𝑓−𝑢⟂Ker(𝜕_𝑏) so 𝑓−𝑢=0 and then 0=𝑓−𝑢=(𝐼𝑑−𝜕^⋆_𝑏𝜕_𝑏𝐺_𝑏𝑓). This proves (42). Similarly, we get (43).

Theorem 11. Let 𝑀 be given as in Theorem 8. Then the Szegö projections operators 𝑆_𝑠−1 and 𝑆_𝑠 are continuous in the Sobolev spaces 𝑊^𝑘_0,𝑠−1(𝑀) and 𝑊^𝑘_0,𝑠(𝑀) for all 𝑘≥0, respectively.

Proof. We investigate first the continuity of 𝑆_𝑠−1. For the case 𝑘=0, when 𝑓∈𝐿²_0,𝑠(𝑀), we have 𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²=𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝐺_𝑏𝜕_𝑏𝑓,𝜕_𝑏𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝐺_𝑏𝜕_𝑏𝑓,𝜕_𝑏𝑓=𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏≤𝑓,𝑓𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓𝑓. (49) Here we have used the fact that 𝜕_𝑏𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝜕_𝑏𝑓, because 𝜕²_𝑏=0. The relation (43) thus implies that ‖𝑆_𝑠−1𝑓‖≤𝐶‖𝑓‖. This proves the continuity in 𝐿²_0,𝑠−1(𝑀).
The case 𝑘≥1. Applying (32) for 𝜑=𝜂Λ^𝑘𝜂₁𝐺_𝑠𝜕_𝑏𝑓 on 𝑈, we obtain 𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓²≤𝜖𝑄_𝑏𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓+𝐶(𝜖)𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓²_−1(𝑈). (50) The first term on the right-hand side of (50) is estimated as 𝑄_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓=𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓²+𝜕^∗𝐴𝐺_𝑏𝜕_𝑏𝑓²=𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓,𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓,𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓=𝐴𝜕_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕_𝑏,𝐴𝐺_𝑏𝜕_𝑏𝑓,𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕^∗_𝑏,𝐴𝐺_𝑏𝜕_𝑏𝑓,𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓. (51) The sum of the last two terms on the right-hand side of the preceding equality is estimated by 𝜕_𝑏,𝐴𝐺_𝑏𝜕_𝑏𝑓𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕^∗_𝑏,𝐴𝐺_𝑏𝜕_𝑏𝑓𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓≤𝐷𝐺_𝑏𝜕_𝑏𝑓_{𝑘−1(𝑈)}𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝐷𝐺_𝑏𝜕_𝑏𝑓_{𝑘−1(𝑈)}𝜕^∗_𝑏𝐴𝐺_𝑠𝜕_𝑏𝑓≤𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓=𝒪(l.c.)𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈)+(s.c.)𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓²=𝒪𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈). (52)
We then have 𝑄_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓≤𝜕_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝐴^∗𝜕_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓+𝒪𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈). (53) The first term on the right-hand side of (53) equals zero due to the fact that 𝜕_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝜕²_𝑏𝐺_𝑏𝑓=0.
We now analyze the second term as follows: 𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓=𝜕_𝑏𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓=𝐴𝜕_𝑏𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕_𝑏,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓=𝐴𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕_𝑏,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓=𝜕_𝑏𝐴𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓+𝐴,𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕_𝑏,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓=𝐴𝑓,𝜕^∗_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓=+⋯𝐴𝑓,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓+𝐴𝑓,𝜕^∗_𝑏,𝐴𝐺_𝑏𝜕_𝑏𝑓+𝐴,𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕_𝑏,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓. (54) Thus 𝑄_𝑏𝐴𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓≤𝐴𝑓,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓+𝐸+𝒪𝐺_𝑏𝜕𝑓²_𝑘(𝑈)≤𝐴𝑓,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓+|𝐸|+𝒪𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈), (55) where 𝐸=𝐴𝑓,𝜕^∗_𝑏,𝐴𝐺_𝑏𝜕_𝑏𝑓+𝐴,𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓+𝜕_𝑏,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝐴𝐺_𝑏𝜕_𝑏𝑓. (56) As above, the three terms on the right-hand side of (56) are estimated, respectively, by 𝐴𝑓𝜕^∗_𝑏,𝐴𝐺_𝑏𝜕_𝑏𝑓≤𝑓_𝑘(𝑈)𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)≤(s.c.)𝑓²_𝑘(𝑈)+(l.c.)𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈),𝑓_𝑘(𝑈)𝐴𝐺_𝑏𝜕_𝑏𝑓≤(s.c.)𝑓²_𝑘(𝑈)+(l.c.)𝐴𝐺_𝑏𝜕_𝑏𝑓²=𝒪𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈),𝜕_𝑏,𝐴𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓𝐴𝐺_𝑏𝜕_𝑏𝑓≤(s.c.)𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈)+(l.c.)𝐴𝐺_𝑏𝜕_𝑏𝑓². (57) Now we are left with the first term in the right-hand side of (55) which, by applying the Cauchy-Schwarz inequality, is estimated by ‖𝑓‖_𝑘(𝑈)‖𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓‖_𝑘(𝑈). By choosing the s.c. small enough we can absorb the first term in the right-hand side of the last inequality into ‖𝑓‖_𝑘(𝑈)‖𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓‖_𝑘(𝑈). This completes the estimation of the first term on the right-hand side of (50). Therefore (50) becomes 𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓²≤𝜖𝑓_𝑘(𝑈)𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)+𝜖(s.c.)𝑓²_𝑘(𝑈)+𝜖𝐶𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈)+𝐶(𝜖)𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓²_−1(𝑈)≤𝜖𝑓_𝑘(𝑈)𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)+𝜖(s.c.)𝑓²_𝑘(𝑈)+𝜖𝐶𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑈)+𝐶^′(𝜖)𝐺_𝑏𝜕_𝑏𝑓²_{𝑘−1(𝑈)}. (58) By summing over a partition of unity subordinate to an open covering of 𝑀 by patches {𝑈_𝑖}^𝑚_𝑖=1 so that on each of these patches an estimate like (58) is satisfied, using the interior regularity properties, we get 𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)≤𝜖𝑓_𝑘(𝑀)𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑀)+𝜖s.c.𝑓²_𝑘(𝑀)+𝜖𝐶𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑀)+𝐶^′(𝜖)𝐺_𝑏𝜕_𝑏𝑓²_{𝑘−1(𝑀)}. (59) By using the small and large constants, the first term on the right-hand side in (59) is estimated as 𝜖(s.c.)𝑓²_𝑘(𝑀)+(l.c.)𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀). (60) Then adding and choosing 𝜖 and the s.c. small enough we can absorb the third term on the right-hand side of (59) into the left-hand side; we obtain 𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)≤𝜖𝐶𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)+𝐶^′(𝜖)𝑓²_𝑘(𝑀)+𝐺_𝑏𝜕_𝑏𝑓²_{(𝑀)𝑘−1}. (61) Applying this inequality with 𝑘 replaced by 𝑘−1 to the last term on the right-hand side and repeating, we obtain 𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)≤𝜖𝐶𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)+𝐶^′(𝜖)𝑓²_𝑘(𝑀)+𝐺_𝑏𝜕_𝑏𝑓². (62) We have 𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²=𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝜕^∗_𝑏𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓+𝒪𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝜕_𝑏𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓+𝒪𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝜕_𝑏𝑓+𝒪𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓,𝜕_𝑏𝜂Λ^𝑘𝜂₁𝑓+𝒪𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)𝑓_𝑘(𝑈)+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝜕^∗_𝑏𝜂Λ^𝑘𝜂₁𝐺_𝑏𝜕_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝑓+𝒪𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)𝑓_𝑘(𝑈)+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓=𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝑓+𝒪𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)𝑓_𝑘(𝑈)+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓≤𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓𝜂Λ^𝑘𝜂₁𝑓+𝒪𝐺_𝑏𝜕_𝑏𝑓_𝑘(𝑈)𝑓_𝑘(𝑈)+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓. (63) Again summing over a partition of unity, using the interior regularity properties and the small and large constants technique, we obtain 𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)≤𝐶𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)+𝑓²_𝑘(𝑀). (64)
Substituting (62) into (64), we obtain 𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)≤𝐾𝜖𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)+𝐶^′(𝜖)𝑓²_𝑘(𝑀)+𝐺_𝑏𝜕_𝑏𝑓². (65) Choosing 𝜖>0 small enough allows us to absorb the first term on the right-hand side into the left, we then get 𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)≤𝐶^′(𝜖)𝑓²_𝑘(𝑀)+𝐺_𝑏𝜕_𝑏𝑓². (66) As the operator 𝜕^∗_𝑏 has 𝐿²(𝑀)-closed range, it follows from Theorem 1.1.1 in Hörmander [15] that there is a positive constant 𝐶 such that 𝐺_𝑏𝜕_𝑏𝑓≤𝐶𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓. (67) Then, by (49), we obtain 𝐺_𝑏𝜕_𝑏𝑓≤𝐶𝑓. (68) Substituting (68) into (66), we get 𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝑓²_𝑘(𝑀)≤𝐶𝑓²_𝑘(𝑀). (69) By (43), the Szegö projection 𝑆_𝑠−1 is therefore continuous on 𝑊^𝑘_0,𝑠−1(𝑀) for each 𝑘=0,1,2…. The general case is obtained from interpolation of linear operators.
For the continuity of the Szegö projection 𝑆_𝑠, in view of (42), it suffices to show that 𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓²_𝑘(𝑀)≤𝐶𝑓²_𝑘(𝑀),𝑘≥0. (70) For 𝑘=0, we have 𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓²=𝜕_𝑏𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝜕_𝑏𝐺_𝑏𝑓=𝜕_𝑏𝑓,𝜕_𝑏𝐺_𝑏𝑓=𝑓,𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓≤𝐶𝑓𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓. (71) For 𝑘≥1, as before, an elliptic regularity argument implies 𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓²=𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓=𝜂Λ^𝑘𝜂₁𝜕_𝑏𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓+𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝜂Λ^𝑘𝜂₁,𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓+𝜕_𝑏,𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓=𝜂Λ^𝑘𝜂₁𝜕_𝑏𝑓,𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓+𝒪𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓=𝜂Λ^𝑘𝜂₁𝑓,𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓+𝒪𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓+𝒪𝑓_𝑘(𝑈)𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓≤𝜂Λ^𝑘𝜂₁𝑓𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓+𝒪𝑓_𝑘(𝑈)+𝜂Λ^𝑘𝜂₁𝜕_𝑏𝐺_𝑏𝑓𝜂Λ^𝑘𝜂₁𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓. (72) Summing over a partition of unity, using the small and large constants argument, absorbing the terms containing ‖𝜕^∗_𝑏𝜕_𝑏𝐺_𝑏𝑓‖_𝑘(𝑀), and finally using the fact that 𝜕_𝑏𝐺_𝑏 is continuously bounded on 𝑊^𝑘_0,𝑠(𝑀), we conclude (70) which proves the continuity of 𝑆_𝑠 on 𝑊^𝑘_0,𝑠(𝑀).

Corollary 12. Let 𝑀 be a 𝒞^∞ compact 𝐶𝑅 manifold of real dimension 2𝑛−ℓ and codimension ℓ≥1 in an 𝑛-dimensional complex manifold 𝑋. Suppose that 𝑀 satisfies condition 𝑌(𝑠) for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1. Then for any 𝑓 in 𝑊^𝑘_0,𝑠(𝑀) (𝑘≥0) such that 𝜕_𝑏𝑓=0 and 𝑓⟂ℋ^𝑏_0,𝑠(𝑀), there exists 𝑢 in 𝑊^𝑘_0,𝑠−1(𝑀) which solves the equation 𝜕_𝑏𝑢=𝑓.

Theorem 13. Let 𝑀 be a 𝒞^∞ compact 𝐶𝑅 manifold of real dimension 2𝑛−ℓ and codimension ℓ≥1 in an 𝑛-dimensional complex manifold 𝑋. Suppose that 𝑀 satisfies condition 𝑌(𝑠) for some 𝑠 with 1≤𝑠≤𝑛−ℓ−1. Then for any 𝑓 in 𝒞^∞_0,𝑠(𝑀), with 𝜕_𝑏𝑓=0 and 𝑓⟂ℋ^𝑏_0,𝑠(𝑀), there exists a global solution 𝑢 in 𝒞^∞_0,𝑠−1(𝑀) to the equation 𝜕_𝑏𝑢=𝑓.

Proof. By Corollary 12, for each 𝑘≥0, there exists some 𝑢_𝑘∈𝑊^𝑘_0,𝑠−1(𝑀) such that 𝜕_𝑏𝑢_𝑘=𝑓. We modify each 𝑢_𝑘 by an element of Ker(𝜕_𝑏) in order to construct a telescoping series that belongs to 𝑊^𝑘_0,𝑠(𝑀) for each 𝑘≥1. To conclude the proof, we first claim that 𝑊^𝑘_0,𝑠(𝑀)∩Ker(𝜕_𝑏) is dense in 𝑊^𝑚_0,𝑠(𝑀)∩Ker(𝜕_𝑏) for any 𝑘>𝑚≥0. Since 𝒞^∞_0,𝑠(𝑀) is dense in 𝑊^𝑚_0,𝑠(𝑀), 𝑚≥0, in the 𝑊^𝑚-norm, then for a given 𝜂∈𝑊^𝑚_0,𝑠(𝑀)∩Ker(𝜕_𝑏) there is a sequence 𝜂_𝑗∈𝒞^∞_0,𝑠(𝑀) converging to 𝜂 in the 𝑊^𝑚_0,𝑠(𝑀)-norm; that is, ‖𝜂_𝑗−𝜂‖_𝑚(𝑀)→0 as 𝑗→∞. 𝜕_𝑏𝜂=0 implies that 𝜂−𝑆_𝑠𝜂=𝜕^∗_𝑏𝐺_𝑏𝜕_𝑏𝜂=0, so 𝜂=𝑆_𝑠𝑢. Let 𝜂_𝑗=𝑆_𝑠𝜂_𝑗. 𝜂_𝑗∈𝑊^𝑘_0,𝑠(𝑀)∩Ker(𝜕_𝑏) since the Szegö projection 𝑆_𝑠 is a bounded operator on 𝑊^𝑘_0,𝑠(𝑀). By the same reason we have ‖𝜂_𝑗−𝜂‖_𝑚(𝑀)=‖𝑆_𝑠(𝜂_𝑗−𝜂)‖_𝑚(𝑀)≤𝐶‖𝜂_𝑗−𝜂‖_𝑚(𝑀)→0 as 𝑗→∞. This implies that 𝜂_𝑗→𝜂 in the 𝑊^𝑚-norm. Thus, indeed, 𝑊^𝑘_0,𝑠(𝑀)∩Ker(𝜕_𝑏) is dense in 𝑊^𝑚_0,𝑠(𝑀)∩Ker(𝜕_𝑏) for any 𝑘>𝑚≥0.
Next, using this result and following the inductive argument due to [21, page 230], we can construct a sequence 𝑢_𝑘∈𝑊^𝑘_0,𝑠−1(𝑀), 𝜕_𝑏𝑢_𝑘=𝑓, and ‖𝑢_𝑘+1−𝑢_𝑘‖_𝑘(𝑀)≤2^−𝑘 as follows: 𝑢₁=𝑢₁,𝑢₂=𝑢₂+𝑣₂, (73) where 𝑣₂∈𝑊²_0,𝑠−1(𝑀)∩Ker(𝜕_𝑏) is such that 𝑢₂−𝑢_11(𝑀)≤2⁻¹ (74) and in general 𝑢_𝑘+1=𝑢_𝑘+1+𝑣_𝑘+1, (75) where 𝑣_𝑘+1∈𝑊^𝑘+1_0,𝑠(𝑀)∩Ker(𝜕_𝑏) is such that 𝑢_𝑘+1−𝑢_{𝑘𝑘(𝑀)}≤2^−𝑘. (76)
Clearly 𝜕_𝑏𝑢_𝑘=𝑓, so set 𝑢=𝑢_𝑗+^∞_𝑘=𝑗𝑢_𝑘+1−𝑢_𝑘,𝑗∈ℕ. (77) It follows that 𝑢∈𝑊^𝑘_0,𝑠−1(𝑀) for each 𝑘∈ℕ, and hence 𝑢∈𝒞^∞_0,𝑠−1(𝑀) and 𝜕_𝑏𝑢=𝑓. The general case is obtained from interpolation of linear operators.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 130-248-D1435. The authors, therefore, acknowledge with thanks DSR technical and financial support.