Abstract and Applied Analysis

Volume 2014 (2014), Article ID 326434, 11 pages

http://dx.doi.org/10.1155/2014/326434

Research Article

Global Regularity for the -Equation on Manifolds of Arbitrary Codimension

Shaban Khidr1,2 and Osama Abdelkader³

¹Mathematics Department, Faculty of Science, King Abdulaziz University, North Jeddah, Jeddah 21589, Saudi Arabia

²Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt

³Mathematics Department, Faculty of Science, Minia University, El-Minia 61915, Egypt

Received 9 April 2014; Accepted 12 May 2014; Published 12 June 2014

Academic Editor: Dumitru Baleanu

Copyright © 2014 Shaban Khidr and Osama Abdelkader. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a compact manifold of -codimension and -dimension in a complex manifold of complex dimension . In this paper, assuming that satisfies condition for some with , we prove an -existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for -forms on .

1. 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 -sense on boundaries of smooth bounded pseudoconvex domains in due to Shaw [2] for all and Boas and Shaw [3] for . 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 , at least five, embedded in , , 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 -Sobolev spaces , , and they further obtained a global solution with -regularity for solutions of the -equation for -forms.

This paper is concerned with proving an -existence theorem for the -Neumann problem on a compact manifold of real dimension () that satisfies condition for some with 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 -setting with no weights using our arguments in [8, 9]. Namely, via a partition of unity, we globalize first the local maximal -Sobolev estimates obtained by [10] for and patching them together to obtain global ones on . Further, we explore an -existence theorem for the -equation on . These results allow us to prove that the complex Green operator and the Szegö projection operators are continuous in the Sobolev spaces for some such that and . Furthermore, we obtain a global smooth solution for the -equation given smooth data on . Before we proceed, we recall first some basic definitions and notations on manifolds.

Definition 1. Let be a -manifold of real dimension . Then a structure on is given by a complex subbundle of the complexified tangent bundle such that the following conditions are satisfied.

(1), where is the fiber at each .(2)If we define , then .(3) is involutive (or formally integrable); that is, if and are two smooth sections of , defined on an open subset of , then so is their Lie bracket , for every open subset of .

A manifold endowed with this structure is called a manifold of -dimension and codimension .

Let be a generic manifold of real dimension embedded in an -dimensional complex manifold . Such a manifold can be represented locally in the following form: for each there exists an open neighborhood of in such that

where are real-valued functions on such that

The complex subbundle which defines the induced structure on is given by . Denote by the space of -forms with -coefficients on . The involution condition (3) of Definition 1 implies that there is a restriction of the de Rham exterior derivative to , which is defined by .

Let us equip with a Hermitian metric such that and consider on the induced metric, then . Let be the space of -forms whose coefficients are with compact support in . We then can define a Hermitian inner product on by

where is the volume element associated with the induced metric on and is the pointwise inner product induced on by the metric on at each . Let be the corresponding norm and the -completion of with respect to this norm. Let be the maximal closed extension of the original on . A form is in the domain of if , defined in the sense of distributions, belongs to . In this way, defines a linear, closed, densely defined operator. Let be the -Hilbert space adjoint of such that for all in and in . The Kohn-Laplacian is defined by

where

We recall that the Kohn-Laplacian is not elliptic, so it has a characteristic set of dimension . Let be the -dimensional bundle such that

Let be the dual bundle of . Let , then annihillates . Thus is called the characteristic bundle. The Levi form of at a point is defined as the Hermitian form on with values in such that

where is the projection of onto .

The Levi form of at a point in the direction is the scalar Hermitian form denoted and is given by

Definition 2 (see [10, Definition 1.2]). A manifold of real dimension and codimension in a complex manifold of complex dimension is said to satisfy condition , , at a point in the direction if the Levi form has at least positive eigenvalues or at least negative eigenvalues. is said to satisfy condition at if it satisfies condition for all directions .

Note that in the hypersurface case, that is, , 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 .

2. -Existence Theory for

Let be a generic manifold of real dimension and codimension in a complex manifold of complex dimension . For each point , there is then a neighborhood of in and a local orthonormal basis consisting of smooth vector fields for (see, e.g., [12, Section 7.2; Theorem 3]). The collection of vector fields forms a local orthonormal basis for . Let be real vector fields on such that the set forms a local orthonormal basis for . Denote by , , the basis for dual to . In terms of this basis, an element in can be uniquely expressed as a sum:

where is an -tuple of integers with and .

We then have

where is zero if as sets and is the sign of the permutation that reorders as if , and the stands for terms of order zero. Using integration by parts, we obtain

For in , the subspace of smooth -forms on that can be extended smoothly up to and including the boundary, we set

If we further assume that satisfies condition for some with , for each , we can find a constant such that

uniformly for all (see, e.g., [10]).

Set ; . The condition implies that the real vector and their commutators of length at most two span the tangent space at each point in . Thus 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 satisfying the following -subelliptic estimate:

Here and always denotes the Sobolev space -norm, is the norm of its dual space, and is the usual -norm. We may omit the subscript from the norm notation when there is no danger of confusion.

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

Theorem 3. Let be a manifold of real dimension and codimension in a complex manifold of complex dimension . Suppose that satisfies condition for some with . For each point , there is then an open neighborhood on which the Kohn Laplacian satisfies the -subelliptic estimate

uniformly for all in .

In addition, if is compact, the estimate (15) holds uniformly on for all in .

Theorem 4 (see [10]). Let be given as in Theorem 3 and the unique solution of the equation for , where is the identity operator. Let be a relatively compact subset of . If the restriction of to is in , the restriction of to is then in . In addition, suppose that and are two cut-off functions supported in such that on the support of ; then if the restriction of to is in the -Sobolev space for some nonnegative integer , the restriction of to is in and there is a constant (independent of ) such that

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

Theorem 5. Let be a compact manifold of real dimension and codimension in an -dimensional complex manifold . Suppose that satisfies condition for some with . Let such that for in , , then is in and there exists a constant (independent of ) such that

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 , , then is in and there is a constant (independent of ) such that

Let

be the closed subspace of consisting of harmonic forms and

The main -result is the following theorem.

Theorem 7. Let be a compact manifold of real dimension and codimension in an -dimensional complex manifold . Suppose that satisfies condition for some such that . Then the following holds.

(1)The space of harmonic -forms is of finite dimensional.(2)The operators , , and have closed ranges.(3)The complex Green operator exists and is a compact operator in .(4)For any in , we have

 where is the orthogonal projection of onto .(5). on .(6)If is defined on (resp., , on (resp., on .(7)If is in such that and , then and is the unique solution to the equation which is orthogonal to and satisfies .(8), and for each there is a positive constant such that the estimate holds uniformly for all in .

Proof. Since is compact, via a partition of unity, the estimate (15) holds globally on . Suppose that is a sequence in such that is bounded, in the -norm and in the -norm as . Thus, we have for some constant . By Rellich’s Lemma, the inclusion map is compact; we can then extract a subsequence of which converges in . Then the hypotheses of Theorem in Hörmander [15] are satisfied which implies that is finite dimensional and the estimate

holds for every in with .

By Theorem   in [15], we then conclude that the operators and have closed ranges. We obtain also from (22) that

This estimate implies that is one-to-one and in view of Theorem   in [15] that the range of is closed. It forces, since is self-adjoint, the strong Hodge decomposition:

Thus is one-to-one and onto. This implies the existence of the complex Green operator as a unique operator that inverts on . The operator is defined as follows: if is in , we define , where is the unique solution of with . is extended to the whole space by setting on . The boundedness of in follows from (23).

To show that is compact in , it suffices to show compactness on (since on ). When (and hence ), the integration by parts, Cauchy-Schwarz inequality (), and (23) imply

By applying (15) to and using (23), we get

where is a positive constant. Thus the compactness of in follows from Rellich’s Lemma.

The assertions in (5) follow immediately from the definition of . For assertion , if and is also defined on , by (21) and the first assertion of (5), we have

A similar equation holds for . Assertions (1)–(6) have been established.

To show assertion , if and , then as well (from (21)). Consequently, , since , and hence . Thus and is orthogonal to . Following assertion and the fact that is bounded, satisfies the following -estimate:

Finally, we show assertion ; if , then and, since is compact, . On other hand, from assertion (5), . Since is hypoelliptic, by Proposition 6, .

Again Proposition 6 implies

Here we have used the fact that is of finite dimension to conclude the estimate

for some constant . The theorem is proved.

3. 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 -Sobolev spaces , , for some with . 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 -Sobolev spaces () to themselves and globally regular if it maps the space continuously to itself.

3.1. Continuity of the Complex Green Operator

We prove first the continuity of the complex Green operator on , .

Theorem 8. Let be a compact manifold of real dimension and codimension in an -dimensional complex manifold . Suppose that satisfies condition for some with . Then the complex Green operator is continuous on the Sobolev space , ; that is, there is a constant such that

Proof. We consider the special case when . Indeed the general case is then derived by means of interpolation of linear operators. Since is compact, it is easy to show that is a dense subspace in . Further, by Theorem 7 (8), we have for . Thus it suffices to establish (31) for . For , (31) follows from (23).

For each , let be a pseudodifferential operator of order with symbol . Let be an open neighborhood of in and let and be two cutoff functions with supports in such that on supp ; then whenever .

Recall that the compactness of in is equivalent to the compactness estimate: for every there is a constant such that for every

where . For this estimate and further results on the compactness of the complex Green operator see, e.g., [16–19].

Applying (32) for , we obtain

We sometimes use for 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 [20, Corollary 3.1] (or [11, Lemma ]) that

Here we have used the fact that the tangential derivative of order satisfies the tangential Sobolev estimate .

Taking in the form , we get

The Cauchy-Schwarz inequality implies

Inequality (33) becomes

Summing over a partition of unity subordinate to an open covering of by patches , we obtain estimate like (37) on each of these patches and using the interior regularity properties, we get

The first term in the right-hand side of (38) is estimated by , where . and . denote a small and a large constants, respectively, in the inequality . The second term is estimated by interpolation of Sobolev norms () and then by using the continuity of in with -bounded norm.

Adding up the analogues terms and absorbing, by choosing and to be small enough, into the left, this gives

where and . The embedding Sobolev space implies (31) for . The general case is obtained from interpolation of linear operators. As mentioned above, the density of in passes (31) to forms in . This proves the continuity of in .

Corollary 9. Let be given as in Theorem 8, then the canonical solution operators and are continuous on for all .

Proof. We argue by induction on . The case when follows from (25). Suppose that the assertions hold for positive integers less than and assume that , , , and are given as in the proof of Theorem 8. By the interior elliptic regularity properties, we prove first a priori estimate for and with as follows:

Summing over a partition of unity, using the small and large constants for the resulting terms , , and , using (31) and adding up the analogues terms, we see that the terms on the right-hand side containing and can be absorbed into the left hand side. We therefore obtain

This completes the induction on for the norms of and . By the density of in , the estimates extend to forms in . As before, the general case is obtained from interpolation of linear operators. Then and are continuous on .

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 are given by the following relations:

Proof. We first show that . For , we observe that

As and , one has

Any can then be written as so that and . By (45) and (46), we then have

This implies the second equality in (42). Now, If , then , so the expression for holds. Next, if and hence , so and is the canonical solution to the equation . Thus , that is, . We claim that . Indeed, for all one has . Since , it turns out that so and then . This proves (42). Similarly, we get (43).

Theorem 11. Let be given as in Theorem 8. Then the Szegö projections operators and are continuous in the Sobolev spaces and for all , respectively.

Proof. We investigate first the continuity of . For the case , when , we have

Here we have used the fact that , because . The relation (43) thus implies that . This proves the continuity in .

The case . Applying (32) for on , we obtain

The first term on the right-hand side of (50) is estimated as

The sum of the last two terms on the right-hand side of the preceding equality is estimated by

We then have

The first term on the right-hand side of (53) equals zero due to the fact that .

We now analyze the second term as follows:

Thus

where

As above, the three terms on the right-hand side of (56) are estimated, respectively, by

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 . By choosing the . small enough we can absorb the first term in the right-hand side of the last inequality into . This completes the estimation of the first term on the right-hand side of (50). Therefore (50) becomes

By summing over a partition of unity subordinate to an open covering of by patches so that on each of these patches an estimate like (58) is satisfied, using the interior regularity properties, we get

By using the small and large constants, the first term on the right-hand side in (59) is estimated as

Then adding and choosing and the . small enough we can absorb the third term on the right-hand side of (59) into the left-hand side; we obtain

Applying this inequality with replaced by to the last term on the right-hand side and repeating, we obtain

We have

Again summing over a partition of unity, using the interior regularity properties and the small and large constants technique, we obtain

Substituting (62) into (64), we obtain

Choosing small enough allows us to absorb the first term on the right-hand side into the left, we then get

As the operator has -closed range, it follows from Theorem in Hörmander [15] that there is a positive constant such that

Then, by (49), we obtain

Substituting (68) into (66), we get

By (43), the Szegö projection is therefore continuous on for each . The general case is obtained from interpolation of linear operators.

For the continuity of the Szegö projection , in view of (42), it suffices to show that

For , we have

For , as before, an elliptic regularity argument implies

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 , we conclude (70) which proves the continuity of on .

Corollary 12. Let be a compact manifold of real dimension and codimension in an -dimensional complex manifold . Suppose that satisfies condition for some with . Then for any in () such that and , there exists in which solves the equation .

Theorem 13. Let be a compact manifold of real dimension and codimension in an -dimensional complex manifold . Suppose that satisfies condition for some with . Then for any in , with and , there exists a global solution in to the equation .

Proof. By Corollary 12, for each , there exists some such that . We modify each by an element of in order to construct a telescoping series that belongs to for each . To conclude the proof, we first claim that is dense in for any . Since is dense in , , in the -norm, then for a given there is a sequence converging to in the -norm; that is, as . implies that , so . Let . since the Szegö projection is a bounded operator on . By the same reason we have as . This implies that in the -norm. Thus, indeed, is dense in for any .

Next, using this result and following the inductive argument due to [21, page 230], we can construct a sequence , , and as follows:

where is such that

and in general

where is such that

Clearly , so set

It follows that for each , and hence and . 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.

References

J. J. Kohn and H. Rossi, “On the extension of holomorphic functions from the boundary of a complex manifold,” Annals of Mathematics: Second Series, vol. 81, pp. 451–472, 1965.

M.-C. Shaw, “-estimates and existence theorems for the tangential Cauchy-Riemann complex,” Inventiones Mathematicae, vol. 82, no. 1, pp. 133–150, 1985.

H. P. Boas and M.-C. Shaw, “Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries,” Mathematische Annalen, vol. 274, no. 2, pp. 221–231, 1986.

J. J. Kohn, “The range of the tangential Cauchy-Riemann operator,” Duke Mathematical Journal, vol. 53, no. 2, pp. 525–545, 1986.

A. C. Nicoara, “Global regularity for on weakly pseudoconvex CR manifolds,” Advances in Mathematics, vol. 199, no. 2, pp. 356–447, 2006.

J. J. Kohn and A. C. Nicoara, “The equation on weakly pseudo-convex CR manifolds of dimension 3,” Journal of Functional Analysis, vol. 230, no. 2, pp. 251–272, 2006.

P. S. Harrington and A. Raich, “Regularity results for on CR-manifolds of hypersurface type,” Communications in Partial Differential Equations, vol. 36, no. 1, pp. 134–161, 2011.

S. Khidr and O. Abdelkader, “Global regularity and -estimates for on an annulus between two strictly pseudoconvex domains in a Stein manifold,” Comptes Rendus Mathématique. Académie des Sciences: Paris, vol. 351, no. 23-24, pp. 883–888, 2013.

S. Khidr and O. Abdelkader, “The -equation on an annulus between two strictly -convex domains with smooth boundaries,” Complex Analysis and Operator Theory, 2013.

M.-C. Shaw and L. Wang, “Hölder and estimates for on CR manifolds of arbitrary codimension,” Mathematische Annalen, vol. 331, no. 2, pp. 297–343, 2005.

G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, vol. 75 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1972.

A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1991.

J. J. Kohn, “Hypoellipticity and loss of derivatives,” Annals of Mathematics: Second Series, vol. 162, no. 2, pp. 943–986, 2005.

M. Derridj, “Subelliptic estimates for some systems of complex vector fields,” in Hyperbolic Problems and Regularity Questions, Trends in Mathematics, pp. 101–108, Birkhäuser, Basel, Switzerland, 2007.

L. Hörmander, “ estimates and existence theorems for the operator,” Acta Mathematica, vol. 113, pp. 89–152, 1965.

E. J. Straube, “The complex Green operator on CR-submanifolds of of hypersurface type: compactness,” Transactions of the American Mathematical Society, vol. 364, no. 8, pp. 4107–4125, 2012.

A. Raich, “Compactness of the complex Green operator on CR-manifolds of hypersurface type,” Mathematische Annalen, vol. 348, no. 1, pp. 81–117, 2010.

A. S. Raich and E. J. Straube, “Compactness of the complex Green operator,” Mathematical Research Letters, vol. 15, no. 4, pp. 761–778, 2008.

S. Munasinghe and E. J. Straube, “Geometric sufficient conditions for compactness of the complex Green operator,” Journal of Geometric Analysis, vol. 22, no. 4, pp. 1007–1026, 2012.

J. J. Kohn and L. Nirenberg, “Non-coercive boundary value problems,” Communications on Pure and Applied Mathematics, vol. 18, pp. 443–492, 1965.

J. J. Kohn, “Methods of partial differential equations in complex analysis, complex variables (Williamstown, Mass., 1975),” in Proceedings of Symposia in Pure Mathematics, vol. 30, pp. 215–237, American Mathematical Society, 1977.