Global Regularity for the ∂b-Equation on CRManifolds of Arbitrary Codimension

and Applied Analysis 3 of complex dimension n. For each point p 0 ∈ M, there is then a neighborhood U of p 0 in X and a local orthonormal basis consisting of smooth vector fields L 1 , . . . , L n−l for T 1,0 (U) (see, e.g., [12, Section 7.2;Theorem 3]).The collection of vector fields {L 1 , . . . , L n−l } forms a local orthonormal basis for T(U). Let T 1 , . . . , T l be real vector fields on U such that the set {L 1 , . . . , L n−l , L 1 , . . . , L n−l , T 1 , . . . , T l } forms a local orthonormal basis for CT(U). Denote by {ω 1 , . . . , ω , ω1, . . . , ωn−l, γ 1 , . . . , γ l } the basis for CT⋆(U) dual to {L 1 , . . . , L n−l , T 1 , . . . , T l }. In terms of this basis, an element φ inC 0,s (U) can be uniquely expressed as a sum:


Introduction and Basic Notations
The tangential Cauchy-Riemann complex (or   -complex) was first introduced by Kohn and Rossi [1] for studying the holomorphic extension of  functions from the boundary of a complex manifold.The closed range property is related to existence and regularity theorems for   and for  manifolds to a reason of embedding.It is worth then to mention that the   -operator has closed range in the  2 -sense on boundaries of smooth bounded pseudoconvex domains in C  due to Shaw [2] for all 1 ≤  <  − 2 and Boas and Shaw [3] for  =  − 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  manifold of real dimension 2 − 1, at least five, embedded in C  ,  ≥ , leading to global regularity for the   -equation on such  manifolds.The main tool in his proof is that of microlocalizations using a new type of weight functions called strongly  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   -operator on  manifold of hypersurface type satisfying weak () condition.More precisely, by using the weighted -theory, they showed that the complex Green's operator is continuous in the  2 -Sobolev spaces   ,  ∈ N, and they further obtained a global solution with C ∞ -regularity for solutions of the   -equation for (0, )forms.
This paper is concerned with proving an  2 -existence theorem for the   -Neumann problem on a C ∞  compact manifold  of real dimension 2 − ℓ (ℓ ≥ 1) that satisfies condition () for some  with 1 ≤  ≤  − ℓ − 1 in an dimensional complex manifold  and with establishing the global regularity properties of the   -equation.In particular, our   -problem is set up in the usual  2 -setting with no weights using our arguments in [8,9].Namely, via a partition of unity, we globalize first the local maximal  2 -Sobolev estimates obtained by [10] for ◻  and patching them together to obtain global ones on .Further, we explore an  2existence theorem for the   -equation on .These  2 results allow us to prove that the complex Green operator   and the Szegö projection operators   are continuous in the Sobolev spaces   0, () for some  such that 1 ≤  ≤  − ℓ − 1 and  ≥ 0. Furthermore, we obtain a global smooth solution for (3)  1,0 () is involutive (or formally integrable); that is, if  1 and  2 are two smooth sections of  A C ∞ manifold  endowed with this  structure is called a  manifold of -dimension  − ℓ and  codimension ℓ.
We then can define a Hermitian inner product on D 0, () by where V is the volume element associated with the induced metric on  and ⟨, ⟩  is the pointwise inner product induced on C ∞ 0, () by the metric on C() at each  ∈ .Let ‖‖ 2 = (, ) be the corresponding norm and  2 0, () the  2 -completion of D 0, () with respect to this norm.Let   :  2 0, () →  2 0,+1 () be the maximal closed extension of the original   on C ∞ 0, ().A form  ∈ where We recall that the Kohn-Laplacian ◻  is not elliptic, so it has a characteristic set of dimension ℓ.Let () be the ℓdimensional bundle such that Let  * () be the dual bundle of ().Let  ∈  * (), then  annihillates  1,0 () ⊕  0,1 ().Thus  * () is called the characteristic bundle.The Levi form of  at a point  ∈  is defined as the Hermitian form on  1,0 () with values in () such that where   is the projection of C  () onto   ().
The Levi form of  at a point  ∈  in the direction  ∈  * () is the scalar Hermitian form denoted L  () and is given by Note that in the hypersurface case, that is, ℓ = 1, the condition () defined above is equivalent to the classical () condition of Kohn for hypersurfaces (see, e.g., [11] for more details).In particular, if the  structure is strictly pseudoconvex; that is, the Levi form of  is positive or negative definite, condition () holds for all 1 ≤  ≤  − 2.
uniformly for all  in D 0, ().
In addition, if  is compact, the estimate (15) holds uniformly on  for all  in C ∞ 0, ().
Theorem 4 (see [10]).Let  be given as in Theorem 3 and  the unique solution of the equation (◻  + ) =  for  ∈  2 0, (), where  is the identity operator.Let  ⊂⊂  be a relatively compact subset of .If the restriction of  to  is in C ∞ 0, (), the restriction of  to  is then in C ∞ 0, ().In addition, suppose that  and  1 are two cut-off functions supported in  such that  = 1 on the support of  1 ; then if the restriction of  to  is in the  2 -Sobolev space   0, () for some nonnegative integer , the restriction of  1  to  is in  +1 0, () and there is a constant   > 0 (independent of ) such that Patching the above local estimates, we obtain the following global one.
Using Theorem 5 and following an induction argument on , we get the following result.Proposition 6.Let  be given as in Theorem 5. Then the Kohn Laplacian ◻  is hypoelliptic.Moreover, if ◻   =  for  in   0, (),  ≥ 0, then  is in  +1 0, () and there is a constant Let be the closed subspace of  2 0, () consisting of harmonic forms and The main  2 -result is the following theorem.

Theorem 7.
Let  be a C ∞ compact  manifold of real dimension 2 − ℓ and codimension ℓ ≥ 1 in an -dimensional complex manifold .Suppose that  satisfies condition () for some  such that 1 ≤  ≤  − ℓ − 1.Then the following holds.
To show assertion (7) for some constant   .The theorem is proved.

Sobolev Space Estimates
In this section, we prove that the complex Green operator   , the canonical solution operators     and  *    , and the Szegö projection   operators enjoy some regularity properties in the  2 -Sobolev spaces   0, (),  ≥ 0, for some  with 1 ≤  ≤  − ℓ − 1.Furthermore, we obtain a global regularity for the solutions of the   -equation.
By the same way for bounded pseudoconvex domains, a differential operator is said to be exactly regular if it maps all  2 -Sobolev spaces   0, () ( ≥ 0) to themselves and globally regular if it maps the space C ∞ 0, () continuously to itself.
Applying (32) for Λ   1   , we obtain We sometimes use  for Λ   1 and  * for its formal adjoint, which is also a tangential operator of order .We estimate the first term on the right hand side in (33), it is a standard consequence of [ Any  ⊥ H  0, () can then be written as  =  +  so that    = 0 and  *   = 0.By ( 45) and ( 46), we then have Proof.We investigate first the continuity of  −1 .For the case  = 0, when  ∈  (49) Here we have used the fact that    *       =   , because  2  = 0.The relation (43) thus implies that ‖ −1 ‖ ≤ ‖‖.This proves the continuity in  2 0,−1 ().The case  ≥ 1. Applying (32) for  = Λ   1      on , we obtain The first term on the right-hand side of (50) is estimated as The first term on the right-hand side of (53) equals zero due to the fact that        =  2     = 0. We now analyze the second term as follows: where As above, the three terms on the right-hand side of ( 56 ) .
Then adding and choosing  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 Summing over a partition of unity, using the small and large constants argument, absorbing the terms containing ‖ *      ‖ () , and finally using the fact that     is continuously bounded on   0, (), we conclude (70) which proves the continuity of   on   0, ().