$L^p$-theory for the tangential Cauchy-Riemann equation

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Christine Laurent-Thiébaut

To cite this version:

Christine Laurent-Thiébaut. L^p-theory for the tangential Cauchy-Riemann equation. 2013. �hal- 00802093�

Christine LAURENT-THI ´EBAUT

Our purpose is to extend to the tangential Cauchy-Riemann operator the L^p theory, for p ≥ 1, developed for the Cauchy-Riemann operator in [6]. We will replace complex manifolds bys-concave locally embeddable generic CR manifolds and the Cauchy-Riemann complex by the tangential Cauchy-Riemann complex. We have to restrict ourself to locally embeddable CR manifolds since, up to now, very few is known in abstract CR manifolds outside the case of compact CR manifolds, where some results were obtained by C.D. Hill and M. Nacinovich [4] and by M.-C. Shaw and L. Wang [16].

Compare to the complex case some new difficulties arise in the CR setting since the Poincar´e lemma for the tangential Cauchy-Riemann operator fails to hold in all degrees.

In particular the Dolbeault isomorphism does not hold in all degrees and its classical proof works no more for large degrees.

We are particularly interested in solving the tangential Cauchy-Riemann equation with exact support in L^p spaces. We consider the following Cauchy problem.

Let M be a C^∞-smooth, generic CR submanifold of real codimension k in a complex manifold X of complex dimension nand Da relatively compact domain in M with C^∞- smooth boundary. Forp≥1, given any (r, q)-formf with coefficients inL^p(M), 0≤r≤n and 1≤q ≤n−k, such that

supp f ⊂D and ∂bf = 0 in the weak sense in M, does there exists a (r, q−1)-forme g with coefficients inL^p(M) such that

suppg⊂D and ∂_bg=f in the weak sense inM ?

We prove (see Corollary 3.10) that the answer is yes as soon as M is s-concave, s≥ 2, D is the transversal intersection of M with a relatively compact completely strictly pseudoconvex domain in X withC^∞-smooth boundary,p > 1 and 1≤q ≤s−1.

The paper is organized as follows. In section one we recall the main definitions of the theory of CR manifolds.

In the second section we develop theL^p_loc theory, we prove a Poincar´e lemma with L^p estimates for the tangential Cauchy-Riemann operator and study the Dolbeault isomor- phism. We prove

UJF-Grenoble 1, Institut Fourier, Grenoble, F-38041, France

CNRS UMR 5582, Institut Fourier, Saint-Martin d’H`eres, F-38402, France 2010 A.M.S. Classification: 32V20, 32W10 .

Key words: tangential Cauchy-Riemann equation, Serre duality,L^pestimates.

1

Th´eor`eme 0.1. Let M be a C^∞-smooth, generic, s-concave, s ≥ 1, CR submanifold of real codimension k of a complex manifold X of complex dimension n and p ≥ 1 a real number. If 0≤r ≤n, the natural maps

Φ^p_q : H_∞^r,q(M)→H_L^r,qp loc

(M) and Ψ^p_q : H_L^r,qp loc

(M)→H_cur^r,q(M) satisfy:

(i) Φ^pq andΨ^pq are surjective, if 0≤q≤s−1 or n−k−s+ 1≤q≤n−k.

(ii) Φ^pq andΨ^pq are injective, if 0≤q≤sor n−k−s+ 2≤q≤n−k.

If 0≤r≤n, the natural maps

Φ^p_c,q : H_c,∞^r,q (M)→H_c,L^r,qp(M) and Ψ^p_c,q : H_c,L^r,qp(M)→H_c,cur^r,q (M) satisfy:

(i) Φ^pc,q and Ψ^pc,q are surjective, if0≤q ≤s−1 or n−k−s+ 1≤q≤n−k.

(ii) Φ^pc,q and Ψ^pc,q are injective, if0≤q ≤s or n−k−s+ 2≤q ≤n−k.

Then we extend the Serre duality introduced in [7] to the L^p_loc tangential Cauchy- Riemann complexes,p >1, and get some density results and a theorem on the solvability inL^p spaces of the tangential Cauchy-Riemann equation with compact support in bidegree (0,1).

The last section is devoted to the L^p theory. First using local L^p results proved in [12], we develop anL^p Andreotti-Grauert theory for large degrees. Then using once again Serre duality, we solve the weak Cauchy problem inL^p for the tangential Cauchy-Riemann operator.

I would like to thank E. Russ who pointed me out the reference [5] to study the compacity of operators between L^p spaces.

1 CR manifolds

LetM be aC^∞-smooth, paracompact differential manifold, we denote byTMthe tangent bundle of Mand by TCM=C⊗TM the complexified tangent bundle.

D´efinition 1.1. An almost CR structure on M is a subbundle H0,1M of TCM such that H_0,1M∩H_0,1M={0}.

If the almost CR structure is integrable, i.e. for allZ, W ∈Γ(M, H_0,1M) then [Z, W]∈ Γ(M, H_0,1M) , it is called aCR structure.

IfH_0,1Mis a CR structure, the pair (M, H_0,1M) is called anabstract CR manifold.

TheCR-dimension of M is defined by CR-dimM= rkC H_0,1M.

We set H_1,0M = H_0,1M and HM =H_1,0M⊕H_0,1M. For X ∈ HM, let X^0,1 denote the projection of X on H0,1M and X^1,0 its projection on H1,0M.

From now on letnandkbe integers such that the CR manifoldMis of real dimension 2n−k and CR-dimensionn−k.

D´efinition 1.2. Let (M, H_0,1M) and (M^′, H_0,1M^′) be two CR manifolds andF : M→M^′ a C¹-map. The map F is called a CR map if and only if for each x∈M,dF((H_0,1M)_x)⊂ (H0,1M^′)_F(x).

In particular, if (M, H_0,1M) is a CR manifold and f a complex valued function, then f is a CR function if and only if for anyL∈H_0,1M we have Lf = 0.

We denote byH^0,1M the dual bundle (H_0,1M)^∗ of H_0,1M. Let Λ^0,qM =Vq

(H^0,1M), thenC_0,q^s (M) = Γ^s(M,Λ^0,qM) is called the space of (0, q)-forms of classC^s,s≥0 onM.

If the almost CR structure is a CR structure, i.e. if it is integrable, and ifs≥1, then we can define an operator

∂_b : C_0,q^s (M)→ C_0,q+1^s−1 (M) (1.1) called the tangential Cauchy-Riemann operator by setting ∂bf = df_|H

0,1M×···×H0,1M. It satisfies ∂b ◦∂b = 0 and a complex valued function f is a CR function if and only if

∂bf = 0

Let us consider the quotient bundleTeM=TCM/H0,1Mand the canonical bundle map π : TCM→TeM.

Foru∈Γ^∞(M, H_1,0M), we define the tangential Cauchy-Riemann operator by setting

∂bu(L) =π[L, W],

if L∈Γ(M, H0,1M) andW ∈Γ^∞(M, TCM) satisfiesπ(W) =u. By the integrability of the CR structure, this does not depend on the choice of W with π(W) =u.

Let us consider the vector bundle Λ^p,qM= Λ^p(TeM)^∗⊗Λ^0,qM of (p, q)-forms onM and set E^p,q(M) = Γ^∞(M,Λ^p,qM). We can define the tangential Cauchy-Riemann operator on E^p,q(M) by setting foru∈ E^p,0(M)

(∂_bu(L))(L₁, . . . , L_p) =L(u(L₁, . . . , L_p))− Xp

i=1

u(L₁, . . . , L_i−1, L(L_i), L_i+1, . . . , L_p),

for any L ∈ Γ(M, H_0,1M) and Li ∈ Γ(M,TeM), 1 ≤ i ≤ p, where L(Li) = ∂bLi(L) is defined as above, and

(∂_bθ(L₁, . . . , L_q+1))(L₁, . . . , L_p) =

q+1X

j=1

(−1)^j+1(∂_b(θ(L₁, . . . ,Lc_j, . . . , L_q+1))(L_j))(L₁, . . . , L_p)

+X

j<k

(−1)^j+kθ([L_j, L_k], L₁, . . . ,Lc_j, . . . ,Lc_k, . . . , L_q+1)(L₁, . . . , L_p),

if θ∈ E^p,q(M),L₁, . . . , L_q+1 ∈Γ(M, H_0,1M) andL_i ∈Γ(TeM), 1≤i≤p.

So we get a differential complex (E^p,•(M), ∂_b) called the tangential Cauchy-Riemann complex for smooth forms. We denote byH_∞^p,q(M) the associated cohomology groups.

Let D^p,q(M) be the space of compactly supported elements of E^p,q(M). We put on E^p,q(M) the topology of uniform convergence on compact sets of the sections and all their derivatives. Endowed with this topology E^p,q(M) is a Fr´echet-Schwartz space. Let K be a compact subset ofM andD^p,q_K (M) the closed subspace ofE^p,q(M) of forms with support in K endowed with the induced topology. Choose (K_n)_n∈N an exhausting sequence of compact subsets of M. Then D^p,q(M) = ∪n≥0D^p,q_Kn(M). We put on D^p,q(M) the strict

inductive limit topology defined by the FS-spaces D^p,q_Kn(M). By restricting the tangen- tial Cauchy-Riemann operator to D^p,q(M) we get a new complex (D^p,•(M), ∂b), whose cohomology groups are denoted Hc,∞^p,q(M).

The space of currents onM of bidimension (p, q) or bidegree (n−p, n−k−q) is the dual of the space D^p,q(M) and is denoted either by D^′_p,q(M), if we use the graduation given by the bidimension, or by D^{′n−p,n−k−q}(M), if we use the graduation given by the bidegree. An element of D^{′n−p,n−k−q}(M) can be identified with a distribution section of Λ^{n−p,n−k−q}M. We define the tangential Cauchy-Riemann operator on D^{′n−p,n−k−q}(M) as the transpose map of the ∂b operator from D^p,q−1(M) intoD^p,q(M), we still denote it by

∂b since its restriction to E^{n−p,n−k−q}(M) coincides with the∂b operator on smooth forms and we get a new complex (D^′n−p,•(M), ∂_b), the tangential Cauchy-Riemann complex for currents, which is the dual complex of the complex (D^p,•(M), ∂b) (see section 2 in [7]).

The cohomology groups are denoted by Hcur^{n−p,n−k−q}(M). Finally the dual of E^p,q(M) denoted either byE_p,q^′ (M) orE^{′n−p,n−k−q}(M) is the space of currents onMof bidimension (p, q) or bidegree (n−p, n−k−q) with compact support. Restricting the ∂_b operator from D^{′n−p,n−k−q}(M) toE^{′n−p,n−k−q}(M), we get a fourth complex (E^′n−p,•(M), ∂b), whose cohomology groups are denoted by Hc,cur^{n−p,n−k−q}(M). This complex is the dual complex of the complex (E^p,•(M), ∂b) (see section 2 in [7]).

The annihilator H⁰M of HM = H_1,0M⊕H_0,1M in TC^∗M is called the characteristic bundle of M. Given p∈M,ω ∈H_p⁰M and X, Y ∈HpM, we choose eω ∈Γ(M, H⁰M) and X,e Ye ∈ Γ(M, HM) with ωep = ω, Xep = X and Yep = Y. Then dω(X, Ye ) = −ω([X,e Ye]).

Therefore we can associate to each ω ∈H_p⁰M an hermitian form

Lω(X) =−iω([X,e X])e (1.2)

on (H1,0)pM. This is called theLevi form of Mat ω ∈H_p⁰M.

In the study of the ∂b-complex two important geometric conditions were introduced for CR manifolds of real dimension 2n−k and CR-dimension n−k. The first one by Kohn in the hypersurface case, k = 1, the condition Y(q), the second one by Henkin in codimensionk, k≥1, the q-concavity.

A CR manifoldM satisfies Kohn’s condition Y(q) at a point p∈M for some 0≤q≤ n−1, if the Levi form of M at p has at least max(n−q, q+ 1) eigenvalues of the same sign or at least min(n−q, q+ 1) eigenvalues of opposite signs.

A CR manifold M is said to be q-concave at p ∈ M for some 0 ≤ q ≤ n−k, if the Levi form Lω at ω ∈H_p⁰M has at least q negative eigenvalues on HpM for every nonzero ω ∈H_p⁰M.

In [16] the condition Y(q) is extended to arbitrary codimension.

D´efinition 1.3. An abstract CR manifold is said to satisfy condition Y(q) for some 1≤q ≤n−katp∈M if the Levi formLω atω ∈H_p⁰Mhas at leastn−k−q+ 1 positive eigenvalues or at least q+ 1 negative eigenvalues on H_pM for every nonzero ω∈H_p⁰M.

Note that in the hypersurface type case, i.e. k= 1, this condition is equivalent to the classical condition Y(q) of Kohn for hypersurfaces and in particular if the CR structure is strictly peudoconvex, i.e. the Levi form is positive definite or negative definite, condition

Y(q) holds for all 1≤q < n−1. Moreover, ifM isq-concave atp∈M, thenq≤(n−k)/2 and condition Y(r) is satisfied atp∈Mfor any 0≤r≤q−1 andn−k−q+ 1≤r≤n−k.

D´efinition 1.4. Let (M, H_0,1M) be an abstract CR manifold, X be a complex manifold and F : M → X be an embedding of class C^∞, then F is called a CR embedding if dF(H_0,1M) is a subbundle of the bundleT_0,1X of the holomorphic vector fields on X and dF(H_0,1M) =T_0,1X∩TCF(M).

LetF be a CR embedding of an abstract CR manifold into a complex manifoldX and set M =F(M), thenM is a CR manifold with the CR structure H_0,1M =T_0,1X∩TCM.

LetU be a coordinate domain in X, then F_|

F−1(U) = (f₁, . . . , f_N), with N = dimCX, and F is a CR embedding if and only if, for all 1≤j ≤N,∂_bfj = 0.

A CR embedding is calledgeneric if dimCX−rkCH_0,1M = codimRM.

Most of the known results on CR manifolds concern CR manifolds which are locally embeddable at each point in Cⁿ and s-concave, s ≥ 1. Moreover by a theorem of C.D.

Hill and M. Nacinovich ([4], Proposition 3.1), each s-concave, s≥1, CR manifold, which is locally embeddable at each point, can be generically embedded in a complex manifold.

Consequently in most of the cases assuming that the abstract CR manifold M is in fact a generic CR submanifoldM of a complex manifold X is not a restriction.

2 L

-cohomology for the tangential Cauchy-Riemann op- erator

Let M be a C^∞-smooth CR manifold of real dimension 2n−k and CR-dimension n−k.

We equip M with an hermitian metric such that H_1,0M ⊥ H_0,1M. Ifp is a real number such that p ≥ 1, let us consider the subspace (L^p_loc)^r,q(M) of D^′r,q(M) consisting in the L^p_loc-sections of the vector bundle Λ^r,qM, 0≤r ≤nand 0≤q ≤n−k, endowed with the topology ofL^p convergence on compact subsets ofM. Taking the restriction to (L^p_loc)^r,q(M) of the tangential Cauchy-Riemann operator for currents we get an unbounded operator whose domain of definition is the set of forms f with L^p_loc coefficients such that ∂_bf has also L^p_loc coefficients and since ∂b ◦ ∂b = 0 we get a complex of unbounded operators

(L^p_loc)^r,•(M), ∂b

. We denote byH_L^r,qp loc

(M) the associated cohomology groups.

Let us consider the dual space of (L^p_loc)^r,q(M), when 1 < p <∞. Since the injection of E^r,q(M) into (L^p_loc)^r,q(M) has a dense image, the dual space of (L^p_loc)^r,q(M) is included in E^{′n−r,n−k−q}(M) the space of (n−r, n−k−q)-currents with compact support on M, moreover by the H¨older inequality it coincides with the space (L^pc^′)^{n−r,n−k−q}(M) of L^p′ forms with compact support in M with p^′ such that ¹_p + _p¹′ = 1 and the duality pairing is defined by < f, g >=R

Mf ∧g. Taking again the restriction to (L^pc^′)^r,q(M) of the tan- gential Cauchy-Riemann operator for currents, we get a complex of unbounded operators

(L^pc^′)^n−r,•(M), ∂_b

. We denote by H^p,q

c,L^p′(M) the associated cohomology groups. The complexes (L^p_loc)^r,•(M), ∂_b

and (L^pc^′)^n−r,•(M), ∂_b

, with p^′ such that ¹_p + _p¹′ = 1, are clearly dual to each other.

2.1 Dolbeault isomorphism

Let M be a C^∞-smooth abstract CR manifold of real dimension 2n−k, k > 0, and CR-dimension n−k.

As in the complex case some relations between the smooth ∂_b-cohomology groups of M, the ∂b-cohomology groups in the sense of currents of M and the L^p_loc ∂b-cohomology groups ofM will result from a CR analog of the Dolbeault isomorphism. Let Ω^rM be the sheaf of CR (r,0)-forms, 0≤r≤nonM. If we can prove that the complexes (E^r,•(M), ∂_b), (D^′r,•(M), ∂b) and (L^p_loc)^r,•(M), ∂b

are resolutions of the sheaf Ω^rM, these groups will be isomorphic to the sheaf cohomology groups H^•(M,Ω^rM). This will be true if M is CR-hypoelliptic, which means that all CR (r,0)-forms on an open subset of M are C^∞- smooth and if the Poincar´e lemma for the ∂b operator holds in the C^∞-smooth category, the current category and the L^p_loc category in degree q, 1≤ q ≤ t(M), for some integer t(M) depending on the geometry ofM. In that case for 1≤q ≤t(M), one gets

H^q(M,Ω^rM)∼H_∞^r,q(M)∼H_cur^r,q(M)∼H_L^r,qp loc

(M).

Unfortunately results on CR-hypoellipticity or on the validity of the Poincar´e lemma for the ∂_b operator are only proven under the hypothesis that M is locally embeddable and satisfies some concavity conditions. Therefore it comes from the remark at the end of section 1 that it is not a real restriction to assume that M is generically embedded in a complex manifold.

From now on we assume thatM=M is aC^∞-smooth, generic CR submanifold of real codimensionk in a complex manifold X of complex dimensionn.

In that case the ∂_b operator is CR-hypoelliptic as soon as M is 1-concave (cf. [1], Corollaire 0.2) and the Poincar´e lemma holds for the ∂_b operator in the smooth and in the current categories in bidegree (r, q), 0≤r ≤nand 1≤q ≤s−1 orn−k−s+1≤q≤n−k, if M is s-concave, s≥2 (cf. [3], [13], [14], [1]).

Moreover in this setting, the Poincar´e lemma holds also in theL^p_loc category,

Lemme 2.1. Let M be a C^∞-smooth, generic, s-concave, s≥1, CR submanifold of real codimension kof a complex manifold X of complex dimensionnand p≥1a real number.

For each x ∈ M, there exists an open neighborhood U of x and an open neighborhood V ⊂⊂U ofxsuch that for any∂b-closed formf ∈(L^p_loc)^r,q(U),0≤r ≤nand1≤q ≤s−1 or n−k−s+ 1≤q≤n−k, there existsg∈(L^p_loc)^r,q−1(V) such that f =∂bg on V. Proof. LetU be an open neighborhood ofxsuch that the Bochner-Martinelli type formula from [1] holds. We get from Theorem 0.1 in [1] that there exists some continuous operators Rb^r,_M^eq from D^r,qe(U) intoE^r,q−1e (U), 0≤r ≤nand 1≤qe≤sor n−k−s+ 1≤qe≤n−k, such that, for any C^∞-smooth (r, q)-form u with compact support in U, 0 ≤ r ≤n and 1≤q ≤s−1 orn−k−s+ 1≤q≤n−k,

u=∂bRb^r,q_Mu+Rb^r,q+1_M ∂bu. (2.1)

The operatorsRb^r,q_M are integral operators whose kernelR^r,q_M are of weak type _2n−1²ⁿ (cf. [12], Lemma 4.2.2). Hence the operators Rb^r,q_M extend as continuous operators from (L^pc)^r,q(U) into (L^p_loc)^r,q−1(U) and (2.1) still holds ifu∈(L^pc)^r,q(U)∩Dom(∂b).

Let f ∈(L^p_loc)^r,q(U) a ∂b-closed form on U and χ a C^∞-smooth, positive function on M with compact support inU and constant equal to 1 on some neighborhoodVe ⊂⊂U of X, then

χf =∂bRb^r,q_M(χf) +Rb^r,q+1_M ∂b(χf)

=∂bRb^r,q_M(χf) +Rb^r,q+1_M ((∂bχ)∧f).

Since the singularities of the kernels R^r,q_M are concentrated on the diagonal of U ×U and the support of∂bχis contained inU\Ve, the differential formRb^r,q+1_M ((∂bχ)∧f) is a smooth

∂_b-closed form on Ve. By the Poincar´e lemma for the ∂_b operator for C^∞-smooth forms there exists an open neighborhood V ⊂⊂ Ve of x and a differential form h ∈ E^r,q−1(V) such that Rb_M^r,q+1((∂_bχ)∧f) =∂_bh. If we setg=Rb^r,q_M(χf) +h theng∈(L^p_loc)^r,q−1(V) and f =∂_bgon V.

Th´eor`eme 2.2. Let M be a C^∞-smooth, generic, s-concave, s ≥ 1, CR submanifold of real codimension k of a complex manifold X of complex dimension n and p ≥ 1 a real number.

If 0≤r≤n and 0≤q≤s−1, the natural maps Φ^p_q : H_∞^r,q(M)→H_L^r,qp

loc

(M) and Ψ^p_q : H_L^r,qp loc

(M)→H_cur^r,q(M) are isomorphisms, still called Dolbeault isomorphisms.

More precisely for all0≤r≤n we have:

(i) If 1 ≤q ≤s−1, for any f ∈(L^p_loc)^r,q(M) with ∂bf = 0 and any neighborhood U of the support of f, there exists g∈(L^p_loc)^r,q−1(M) with support in U such that f−∂bg∈ E^r,q(M).

(ii) If 1 ≤ q ≤ s and if f ∈ (L^p_loc)^r,q(M) satisfies f = ∂bS for some current S ∈ D^′r,q−1(M), then, for any neighborhoodU of the support ofS, there existsg∈(L^p_loc)^r,q−1(M) with support in U such thatf =∂_bg on M.

Proof. For 0≤q ≤s−1, the proof works as in the complex case, see Propositions 1.1, 1.2 and 1.4 in [6].

Let us consider the caseq=s. Following the proof of Theorem 5.1 in [7], it is sufficient to prove that for eachx∈M, there exists an open neighborhoodV ofxsuch that for any (r, s−1)-currentT onM with∂bT ∈(L^p_loc)^r,s(M), there existsg∈(L^p_loc)^r,s−1(V) such that

∂_bT =∂_bg on V. Let U be an open neighborhood ofx such that the Bochner-Martinelli type formula from [1] holds. For 0≤r ≤n and n−k−s+ 1 ≤qe≤ n−k, there exists some continuous operatorsRb^r_M^qefromD^r,qe(U) intoE^r,eq−1(U), such that, for anyC^∞-smooth (r,q)-forme u with compact support inU,, 0≤r≤nand n−k−s+ 1≤qe≤n−k,

u=∂bRb^r,e_M^qu+Rb^r,e_M^q+1∂bu. (2.2) Let us denote by Re^r,q_M from E^′r,q(U) into D^′r,q−1(U) the transpose of Rbn−r,n−k−q+1

M , 0 ≤

r ≤n and 1≤q ≤s, then by duality, since the ∂_b operator for currents is the transpose of the ∂b operator for C^∞-smooth forms, we have for any (r, q)-current T with compact support in U, 0≤r≤nand 0≤q≤s−1

T =∂bRe^r,q_MT +Re^r,q+1_M ∂bT. (2.3)

where we have setRe^r,0_M = 0. Moreover since, by the proof of Lemma 2.1, the operatorsRb^r,_M^qe extend as continuous operators from (L^pc)^r,qe(U) into (L^p_loc)^r,q−1e (U) for any p ≥ 1, their transpose operators Re_M^r,q define continuous operators from (L^pc)^r,q(U) into (L^p_loc)^r,q−1(U) for anyp >1. Forp= 1 note thatRe_M^r,qis still an integral operator whose kernel is of weak type _2n−1²ⁿ hence continuous from (L¹_c)^r,q(U) into (L¹_loc)^r,q−1(U).

Assume T is a (r, s−1)-current on M with ∂_bT ∈ (L^p_loc)^r,s(M). Let V ⊂⊂ U be an open neighborhood of x and χ a C^∞-smooth, positive function on M with compact support inU and constant equal to 1 on some neighborhood ofV, then we can apply (2.3) to χT and take the∂_b of it, so we get

∂b(χT) =∂b(Re^r,q+1_M ∂b(χT)). (2.4)

Note that ∂_b(χT) = χ∂_bT +∂_b(χ)∧T with χ∂_bT ∈ (L^p_loc)^r,s(M), if ∂_bT ∈(L^p_loc)^r,s(M), and support of ∂_b(χ)∧T is contained inU \V. Since the operators are continuous from (L^pc)^r,q(U) into (L^p_loc)^r,q−1(U) and the singularity of their kernel is contained in the diagonal ofU×U, the restiction toV of the currentRe^r,q+1_M ∂_b(χT) is a formg∈(L^p_loc)^r,q−1(V) such that ∂bT =∂bg on V.

Corollaire 2.3. Let M be aC^∞-smooth, connected, non compact, generic, s-concave,s≥ 1, CR submanifold of real codimension k of a complex manifold X of complex dimension n and p≥1 a real number. If0≤r ≤n and 0≤q ≤s−1, the natural maps

Φ^p_c,q : H_c,∞^r,q (M)→H_c,L^r,qp(M) and Ψ^p_c,q : H_c,L^r,qp loc

(M)→H_c,cur^r,q (M) are isomorphisms. Moreover Φ^pc,s and Ψ^pc,s are only injective.

In particular Hc,∞^r,q (M) = 0 implies H_c,L^r,qp(M) = 0, when 0≤r≤n and 1≤q ≤s−1 and Hc,cur^r,q (M) = 0 implies H_c,L^r,qp(M) = 0, when 0≤r ≤n and1≤q ≤s.

Proof. First assume 0 ≤q ≤s−1. As in the complex case, sinceH∞^r,q(M) ∼ Hcur^r,q(M), the natural map Θ^pc,q = Ψ^pc,q◦Φ^pc,q from Hc,∞^r,q (M) → Hc,cur^r,q (M) is an isomorphism, con- sequently the map Φ^pc,q is injective and the map Ψ^pc,q is surjective. Moreover by Theorem 5.1 in [7] the map Θ^pc,s is injective and so is Φ^pc,s. On the other hand the assertion (i) of Theorem 2.2 implies that Φ^pc,q is surjective and the assertion (ii) of Theorem 2.2 implies that Ψ^pc,q is injective. Moreover the last statement remains true ifq =s.

Unfortunately the Poincar´e lemma for the∂_b operator fails to hold in any category in bidegree (r, q) for 0≤r ≤nands≤q≤n−k−s, whenMiss-concave, hence the classical cohomological methods can no more be applied to get the Dolbeault isomorphisms when q ≥n−k−s+ 1, even if the Poincar´e lemma holds for suchq. The method developed in [10] cannot either be used to get the Dolbeault isomorphisms with the L^p_loc cohomology groups in high degrees since the notion of trace is not well defined in L^p. We will use the regularization method introduced by S. Sambou in [15], which depends on the fundamental solution for the∂b operator constructed in [1].

Th´eor`eme 2.4. Let M be a C^∞-smooth, generic, s-concave, s ≥ 1, CR submanifold of real codimension k of a complex manifold X of complex dimension n and p ≥ 1 a real

number. If 0≤r ≤n, the natural maps Φ^p_q : H_∞^r,q(M)→H_L^r,qp

loc

(M) and Ψ^p_q : H_L^r,qp loc

(M)→H_cur^r,q(M) satisfy:

(i) Φ^pq andΨ^pq are surjective, if n−k−s+ 1≤q≤n−k.

(ii) Φ^pq andΨ^pq are injective, if n−k−s+ 2≤q ≤n−k.

In particular H∞^r,q(M) = 0 implies H_L^r,qp loc

(M) = 0, when 0 ≤ r ≤ n and n−k− s+ 1 ≤ q ≤ n−k and Hcur^r,q(M) = 0 implies H_L^r,qp

loc

(M) = 0, when 0 ≤ r ≤ n and n−k−s+ 2≤q≤n−k.

Proof. In [15], S. Sambou proved that for any 0≤r≤nand 1≤q ≤s−1 orn−k−s+1≤ q ≤n−kthere exists linear operators

Rr,q : D^′r,q(M)→ E^r,q(M) and Ar,q : D^′r,q(M)→ D^′r,q−1(M)

such that Ar,q has the same regularity properties than the operators Re^r,q_M defined in the proof of Theorem 2.2, in particularA_r,qis continuous from (L^p_loc)^r,q(M) into (L^p_loc)^r,q−1(M) forp≥1 and from E^r,q(M) intoE^r,q−1(M), and, for any T ∈ D^′r,q(M),

T =R_r,qT+A_r,q+1∂_bT+∂_bA_r,qT, if 1≤q ≤s−2 orn−k−s+ 1≤q≤n−k and

T =Rr,qT +∂bAr,qT, if q =s−1 and∂bT = 0.

It follows that for f ∈ (L^p_loc)^r,q(M) with ∂bf = 0 we get f = Rr,qf +∂bAr,qf where R_r,qf is C^∞-smooth on M and A_r,qf is L^p_loc. Hence Φ^pq is surjective for 1 ≤q ≤s−1 or n−k−s+ 1≤q ≤n−k. In the same way we get that Ψ^pq is surjective if 1≤q ≤s−1 or n−k−s+ 1≤q≤n−k, sinceE^r,q(M)⊂(L^p_loc)^r,q(M).

If f ∈ (L^p_loc)^r,q(M) satisfies f = ∂_bT, where T is a (r, q −1)-current on M, then f =∂b(R_r,q−1T+Ar,qf) for 1≤q≤sor n−k−s+ 2≤q ≤n−kandR_r,q−1T+Ar,qf ∈ (L^p_loc)^r,q(M). Hence Ψ^pq is injective, if 1≤q ≤sorn−k−s+ 2≤q≤n−k. The proof of the injectivity of Φ^pq uses the same arguments since the operatorsA_r,q is continuous from E^r,q(M) intoE^r,q−1(M).

Note that the proof of Theorem 2.4 gives also an alternative proof for the Dolbeault isomorphism in small degrees.

As in the complex case, it follows from Theorem 2.4 that the assertion (ii) from The- orem 2.2 holds also for n−k−s+ 2≤q≤n−k.

Corollaire 2.5. Let M be a C^∞-smooth, generic, s-concave, s ≥1, CR submanifold of real codimension k of a complex manifold X of complex dimension n and p ≥ 1 a real number. Let (r, q) such that 0≤r ≤n and 1≤q ≤sor n−k−s+ 2≤q ≤n−k, and f ∈(L^pc)^r,q(M)withf =∂_bSfor some currentS ∈ D^′r,q−1(M), then, for any neighborhood U of the support of S, there existsg∈(L^pc)^r,q−1(M) with∂bg=f and suppg⊂U.

Moreover the Dolbeault isomorphism holds also in large degrees for the cohomology groups with compact support.

Corollaire 2.6. Let M be a C^∞-smooth, generic, s-concave, s ≥1, CR submanifold of real codimension k of a complex manifold X of complex dimension n and p > 1 a real number. If 0≤r ≤n and n−k−s+ 2≤q≤n−k, the natural maps

Φ^p_c,q : H_c,∞^r,q (M)→H_c,L^r,qp(M) and Ψ^p_c,q : H_c,L^r,qp(M)→H_c,cur^r,q (M) are isomorphisms. Moreover Φ^p_{c,n−k−s+1} and Ψ^p_{c,n−k−s+1} are only surjective.

In particular H_c,∞^r,q (M) = 0 implies H_c,L^r,qp(M) = 0, when 0 ≤ r ≤ n and n−k− s+ 1 ≤ q ≤ n−k and Hc,cur^r,q (M) = 0 implies H_c,L^r,qp(M) = 0, when 0 ≤ r ≤ n and n−k−s+ 2≤q≤n−k .

To end this section we will precise the support of the solution of the tangential Cauchy- Riemann equation ∂_bg=f, whenf is a (r,1)-form, 0≤r ≤n, with compact support in M and L^p coefficients.

Proposition 2.7. LetM be aC^∞-smooth, connected, generic, 1-concave CR submanifold of real codimension k of a complex manifold X of complex dimension n, D a relatively compact subset of M and p ≥ 1 a real number. Assume Hc,cur^r,1 (M) = 0 for 0 ≤ r ≤ n, and M \D is connected, then for any ∂b-closed (r,1)-form f with support in D and L^p coefficients there exists a unique (r,0)-form g with L^p coefficients and such that ∂bg=f and supp g⊂D.

Proof. Since M is 1-concave, it follows from Corollary 2.3 that for any 0 ≤ r ≤ n, Hc,cur^r,1 (M) = 0 implies H_c,L^r,1p(M) = 0. Hence there exists an (r,0)-form g with com- pact support in M and L^p coefficients such that ∂_bg =f. The form g vanishes on some open subset of M \D and is CR on M \supp f, which contains M \D. By analytic continuation, which holds in 1-concave manifolds, g vanishes on M \D, since M \D is connected, therefore suppg⊂D. Leth be another solution with compact support in M, theng−his CR with compact support in M, hence vanishes onM sinceM is connected and g=h.

2.2 Serre duality

In this sectionM will always be aC^∞-smooth, generic CR submanifold of real codimension k in an-dimensional complex manifold X.

As we already mentioned in the previous section, ifpandp^′satisfy ¹_p+_p¹′ = 1, the spaces (L^p_loc)^r,q(M) and (L^pc^′)^{n−r,n−k−q}(M) are dual from each other, with the duality pairing<

f, g >=R

Mf∧gand we get two dual complexes (L^p_loc)^r,•(M), ∂b

and (L^pc^′)^n−r,•(M), ∂b

for each 0≤r≤n.

Fix some integer r, 0 ≤ r ≤ n. We can apply Theorem 1.5 in [7] to the complexes (D^r,•, ∂b), (E^′r,•, ∂b) and (L^pc^′)^n−r,•(M), ∂b

. We deduce that if ¹_p + _p¹′ = 1, for any

0≤q ≤n−k, the natural maps

σH_cur^r,q(M)→ H_c,∞^{n−r,n−k−q}(M)′ σH_∞^r,q(M)→ H_c,cur^{n−r,n−k−q}(M)′ σH_c,cur^r,q (M)→ H_∞^{n−r,n−k−q}(M)′ σH_c,∞^r,q (M)→ H_cur^{n−r,n−k−q}(M)′ σH_L^r,qp

loc

(M)→ H^{n−r,n−k−q}

c,L^p′ (M)′ σH_c,L^r,qp(M)→ H^{n−r,n−k−q}

L^p_loc^′ (M)′

. are isomorphisms.

Moreover, from Theorem 3.2 in [7], we get that Hc,cur^{n−r,n−k−q}(M) is Hausdorff if and only if H∞^r,q+1(M) is Hausdorff and if these assertions hold, then

H_c,cur^{n−r,n−k−q}(M)∼ H_∞^r,q(M)′

H_∞^r,q+1(M)∼ Hn−r,n−k−q−1

c,cur (M)′

.

We deduce easily that, if there exists an integer s(M) ≥ 1 such that H∞^r,q(M) is Hausdorff for 0≤q≤s(M)−1 andn−k−s(M) + 1≤q ≤n−k, then

H_c,cur^r,q (M) is Hausdorff andH_c,cur^r,q (M)∼ H_∞^{n−r,n−k−q}(M)′

, if 1≤q ≤s(M) andn−k−s(M) + 2≤q ≤n−k. In particular

H_∞^r,q(M) = 0 for 0≤q≤s(M)−1 and n−k−s(M) + 1≤q≤n−k implies

H_c,cur^r,q (M) = 0 for 0≤q ≤s(M)−1 andn−k−s(M) + 2≤q ≤n−k.

Associated with the Corollaries 2.3 and 2.6, this gives

Proposition 2.8. Let M be a C^∞-smooth, generic, s-concave, s ≥ 1, CR submanifold of real codimension k in a n-dimensional complex manifold X and p > 1 a real number.

Assume that H∞^r,q(M) = 0 for 0≤r≤nand 0≤q ≤s−1 or n−k−s+ 1≤q ≤n−k, then H_c,L^r,qp(M) = 0 for 0≤r≤n and 0≤q≤s−1 or n−k−s+ 2≤q≤n−k.

We will use the following notations:

Z_∞^r,q(M) =E^r,q(M)∩ker∂b, E_∞^r,q(M) =E^r,q(M)∩Im∂b

Z_c,∞^r,q (M) =D^r,q(M)∩ker∂b, E_c,∞^r,q (M) =D^r,q(M)∩Im∂b

Z_cur^r,q(M) =D^′r,q(M)∩ker∂b, E_cur^r,q(M) =D^′r,q(M)∩Im∂b

Z_c,cur^r,q (M) =E^′r,q(M)∩ker∂_b, E_c,cur^r,q (M) =E^′r,q(M)∩Im∂_b Z_L^r,qp

loc

(M) = (L^p_loc)^r,q(M)∩ker∂_b, E_L^r,qp loc

(M) = (L^p_loc)^r,q(M)∩Im∂_b Z_c,L^r,qp(M) = (L^p_c)^r,q(M)∩ker∂b, E_c,L^r,qp(M) = (L^p_c)^r,q(M)∩Im∂b

In theL^p setting the Serre duality can be expressed in the following way:

Th´eor`eme 2.9. Let M be a C^∞-smooth, generic CR submanifold of real codimension k in a n-dimensional complex manifold X. Let r, q ∈N with 0≤r ≤n and 0≤q≤n−k, then for p, p^′ ∈Rwith ¹_p +_p¹′ = 1 the following assertions are equivalent:

(i) E_c,L^{n−r,n−k−q}p (M) ={g∈(L^pc)^{n−r,n−k−q}(M) | < g, f >= 0,∀f ∈Z^r,q

L^p_loc^′ (M)}

(ii) H_c,L^{n−r,n−k−q}p (M) is Hausdorff (iii) E^r,q+1

L^p_loc^′ (M) ={g∈(L^p_loc^′ )^r,q+1(M) | < g, f >= 0,∀f ∈Zn−r,n−k−q−1

c,L^p (M)}

(iv)H^r,q+1

L^p_loc^′ (M) is Hausdorff.

If these assertions hold, then

H_c,L^{n−r,n−k−q}p (M)∼ H^r,q

L^p_loc^′ (M)′

H_L^r,q+1p loc

(M)∼ Hn−r,n−k−q−1 c,L^p′ (M)′

. Proof. Since for each 0≤r ≤n the complexes (L^p_loc)^r,•(M), ∂b

and (L^pc^′)^n−r,•(M), ∂b

are dual complexes and (L^p_loc)^r,•(M), ∂b

is a complex of Fr´echet-Schwartz spaces, we can apply Theorem 1.6 of [9].

As in [9], we will use Serre duality to get some density theorems for the spaces of L^p forms.

Th´eor`eme 2.10. LetM be aC^∞-smooth, connected, non compact, generic,1-concave CR submanifold of real codimensionkin an-dimensional complex manifoldX andp >1a real number. For any integer r, 0 ≤r ≤n, the space Z_∞^r,n−k−1(M) of C^∞-smooth, ∂_b-closed, (r, n−k−1)-forms is dense in the space Z_L^r,n−k−1p

loc

(M) of ∂b-closed, (r, n−k−1)-forms with L^p_loc coefficients for the L^p_loc topology.

Proof. By the Hahn-Banach theorem, it is sufficient to prove that for anyg∈(L^pc^′)^n−r,1(M) such that < g, ϕ >= 0 for all ϕ ∈ Z∞^r,n−k−1(M), we have < g, f >= 0 for all f ∈ Z_L^r,n−k−1p

loc

(M). Let

g∈(L^p_c^′)^n−r,1(M)∩ {T ∈ E^′n−r,1(M) | < T, ϕ >= 0, ∀ϕ∈Z_∞^r,n−k−1(M)}.

Malgrange’s theorem [8] claims that, under the assumptions of the theorem,H_∞^r,n−k(M) = 0, then by Serre duality (see Theorem 3.2 in [9]) we get thatg∈(L^pc^′)^n−r,1(M)∩Ec,cur^n−r,1(M).

From assertion (ii) in Theorem 2.2, we deduce that there exists h ∈ (L^pc^′)^n−r,0(M) such that g=∂_bh. Letf ∈Z_L^r,n−k−1p

loc

(M), then< g, f >=< ∂_bh, f >=< h, ∂_bf >= 0.

Adding the assumption that some cohomology group is Hausdorff to replace Mal- grange’s theorem the previous theorem can be extended tos-concave CR manifold,s≥1.

Th´eor`eme 2.11. Let M be a C^∞-smooth, generic, s-concave, 1 ≤ s ≤ n−k, CR sub- manifold of real codimension k in an n-dimensional complex manifold X and p >1 a real number. For any integers r, q, 0≤r ≤n and 1 ≤q ≤s, assume that H∞^{r,n−k−q+1}(M) is Hausdorff, then the space Z∞^r,n−k−q(M) of C^∞-smooth, ∂b-closed, (r, n−k−q)-forms is

dense in the space Z_L^r,n−k−qp loc

(M) of ∂b-closed,(r, n−k−q)-forms with L^p_loc coefficients for the L^p_loc topology.

Let us give some geometrical conditions on M which ensure that, for all 0 ≤r ≤ n, H_∞^{r,n−k−q+1}(M) is Hausdorff for some 1≤q≤n−k.

1) IfM is compact ands-concave, by Theorem 4 in [3] or Theorem 4.1 in [4],H∞^{r,n−k−q+1}(M) is finite dimensional hence Hausdorff for 1 ≤q ≤ s and n−k−s+ 2 ≤ q ≤n−k and Hausdorff for q=n−k−s+ 1.

2) IfM is connected, non compact and 1-concave, then by Theorem 0.1 in [8],H∞^r,n−k(M) = 0.

3) The cohomology groupsH_∞^{r,n−k−q+1}(M) are also finite dimensional hence Hausdorff for 1 ≤ q ≤ s, as soon as M is s-concave and admits an exhausting function with good pseudoconvexity properties (see for example Theorem 6.1 in [4]). This holds in particular whenM of real dimension 2n−kis generically embedded in an n-dimensional pseudoconvex complex manifold X.

Let us end by a separation theorem for the L^p cohomology with compact support in bidegree (r,1), 0≤r≤n.

Th´eor`eme 2.12. LetM be aC^∞-smooth, connected, non compact, generic,1-concave CR submanifold of real codimensionkin an-dimensional complex manifoldX andp >1a real number. For any integer r, 0 ≤ r ≤n, the cohomology groups H_c,L^r,1p(M) are Hausdorff.

More precisely, if f ∈ (L^pc)^r,1(M) is a ∂b-closed form such that < f, ϕ >= 0 for all ∂b- closed, C^∞-smooth (n−r, n−k−1)-form ϕin a neighborhood of the support of f, there exists a unique g ∈(L^pc)^r,0(M) such that ∂_bg =f and if the support of f is contained in the closure of some relatively compact domain D of M then suppg⊂D.

Proof. Under the hypotheses of the theorem, by Theorem 0.1 in [8], H_∞^n−r,n−k(M) = 0, for all 0≤r ≤n. Using Theorem 2.4, we get H_L^n−r,n−kp

loc

(M) = 0 for any p >1 and Serre duality (Theorem 2.9) implies that the cohomology groups H_c,L^r,1p(M) are Hausdorff for all 0 ≤ r ≤ n and any p > 1. In the last assertion the existence of g follows from (i) in Theorem 2.9 and from the density of Z_∞^{n−r,n−k−1}(M) in Z_L^{n−r,n−k−1}p

loc

(M) for the L^p_loc topology proved in Theorem 2.10. The uniqueness of g results from analytic continuation in connected 1-concave CR manifolds and of the non compactness of M. To get the support condition let us replace the CR manifold M by Mf=M\ {z₁, . . . , zl}, where the points z₁, . . . , z_l are chosen such that zi ∈Di,i= 1, . . . , l, and the Di’s, i= 1, . . . , l, are the relatively compact connected components of M \D. The new manifold Mf has the same properties than M and we get that in fact the support of g is a compact subset of Mf, moreover g is CR in M \D and vanishes in a neighborhood of each zi, i= 1, . . . , l, therefore by analytic continuation it vanishes on M\D.

Note that, if the orthogonality condition is only satisfied for the formsϕ∈Z∞^{n−r,n−k−1}(M) we have to assume as in Proposition 2.7 thatM\Dis connected to get that the support of g is contained inD.

3 L

-theory for the tangential Cauchy-Riemann operator

Throughout this section M will always be a C^∞-smooth, generic CR submanifold of real codimensionkin a complex manifoldXof complex dimensionnandDa relatively compact domain in M with rectifiable boundary. We equipX with an hermitian metric such that T_1,0X ⊥ T_0,1X and we consider on M the induced metric then H_1,0M ⊥ H_0,1M. We denote by dV the volume form on M associated to that metric.

For any real number p≥1, we define the spaceL^p(D) as the set of all functions f on D with complex values such that|f|^p is integrable onD. SinceD⊂⊂X, we have

D(D)⊂ E(X)_|D ⊂L^p(D),

where E(X)_|D is the space of restrictions to D of C^∞-smooth functions on X and D(D) the subspace of C^∞-smooth functions with compact support in D. The spaceL^p(D) is a Banach space, it is the completed space ofD(D) for thek.kp-norm defined by

kfkp = ( Z

D

|f|^pdV)^1p.

For any fixed r, 0≤r ≤n, we consider the spaces L^pr,q(D) of (r, q)-forms, 0≤q ≤n−k, with L^p coefficients and we are going to define unbounded operators from L^pr,q(D) into L^p_r,q+1(D) whose restriction to the spaceD^r,q(D) will coincide with the tangential Cauchy- Riemann operator assocated to the CR structure onM.

Let us define∂b,c : L^pr,q(D)→L^p_r,q+1(D) as the closed minimal extension of∂b|_D^r,q_(D), it is a closed operator by definition andf ∈Dom(∂_b,c) if and only if there exists a sequence (fν)_ν∈N of forms inD^r,q(D) and a form g ∈ D^r,q+1(D) such that (fν)_ν∈N converges to f inL^pr,q(D) and (∂bfν)ν∈N converges tog inL^p_r,q+1(D), and we have g=∂b,cf.

We can also consider ∂b,s : L^pr,q(D) → L^p_r,q+1(D), the closed minimal extension of

∂b|_E^r,q_(M)|

D

, it is also a closed operator and f ∈ Dom(∂b,s) if and only if there exists a sequence (fν)_ν∈Nof forms inE^r,q(M) and a formg∈ E^r,q+1(M) such that (fν)_ν∈Nconverges to f inL^pr,q(D) and (∂_bf_ν)_ν∈N converges tog inL^p_r,q+1(D), and we have g=∂_b,sf.

Since L^pr,q(D) is a subspace of the space of currents on D, the ∂b operator is defined on L^pr,q(D) in the weak sense, so we can define two operators∂_b,ec : L^pr,q(D)→L^p_r,q+1(D) and ∂b : L^pr,q(D) → L^p_r,q+1(D) which coincides with the ∂b in the sense of currents and whose domains of definition are respectively

Dom(∂b,ec) ={f ∈L^p_r,q(M) |suppf ⊂D, ∂bf ∈L^p_r,q(M)}

and

Dom(∂b) ={f ∈L^p_r,q(D) |∂bf ∈L^p_r,q(D)}.

When the boundary ofDis sufficiently smooth (at least Lipschitz), we get the following proposition, which is a direct consequence of Friedrichs’ Lemma, which holds inL^p,p≥1.

Its proof is the same as in the complex case (see lemma 4.3.2 in [2] and lemma 2.4 in [11]).

Proposition 3.1. Let D be a relatively compact domain with Lipschitz boundary in a CR submanifold M of a complex manifold X and r, q be integers with 0 ≤ r ≤ n and 0≤q ≤n−k.

(i) A differential formf ∈L^pr,q(D)belongs to the domain of definition of the∂boperator if and only if there exists a sequence (fν)_ν∈N of forms in E^r,q(D) such that the sequences (f_ν)_ν∈Nand(∂f_ν)_ν∈Nconverge respectively tof and∂_bf ink.k_p norm. Therefore∂_b,s=∂_b. (ii) A differential form f ∈ L^pr,q(D) belongs to the domain of definition of the ∂_b,c operator if and only if the forms f₀ and ∂bf₀ are both in L^p_∗(M), where f₀ denotes the extension by 0 of f on M \D. Therefore ∂_b,c=∂_b,ec.

We will denote respectively byH_L^r,qp(D) andH_ec,L^r,qp(D) the cohomology groups associated to the complexes (L^p_r,•(D), ∂_b) and (L^p_r,•(D), ∂_b,ec).

3.1 Andreotti-Grauert theory in L^p for large degrees

In this section we will extend to the CR case a part of the results obtained in section 2.2 in [6].

We will consider the case whenDis the intersection ofM with a strictly pseudoconvex domain of X, more precisely

D´efinition 3.2. LetM be aC^∞-smooth,s-concave, s≥1, generic CR submanifold of real codimension k in a complex manifold X of complex dimensionn. A relatively compact domainDinM is said to becompletely strictlys-convex if there exists a real,C^∞-smooth, strictly plurisubharmonic function ρdefined on a neighborhood U of Din X such that

D=M∩ {x ∈U |ρ(x)<0}.

We will say thatDisstrictly s-convex if the functionρ is only defined on a neighborhood of the boundary ofD.

When X = Cⁿ, the solvability with L^p-estimates of the ∂_b-equation in completely strictly s-convex domains of a s-concave, s ≥ 1, generic CR submanifold M has been studied in [12]. In particular the following result is proven

Th´eor`eme 3.3. LetM be a C^∞-smooth,s-concave,s≥1, generic CR submanifold of real codimension k inCⁿ and D a relatively compact strictly s-convex domain inM withC^∞- smooth boundary. If p≥1, there exists some bounded linear operators T_q−1^r from L^pr,q(D) into L^p_r,q−1(D), 0≤r≤n, n−k−s+ 1≤q≤n−k, such that for any f ∈L^pr,q(D) with

∂bf = 0 in D, we have ∂bT_q−1^r f =f.

Let us go back to the caseX is a complex manifold.

Th´eor`eme 3.4. Let M be a C^∞-smooth, s-concave, s ≥ 1, generic CR submanifold of real codimension k in a complex manifold X of complex dimension n and D a relatively compact strictly s-convex domain in M with C^∞-smooth boundary. If p≥1, there exists some bounded linear operators Te_q−1^r from L^pr,q(D) into L^p_r,q−1(D) and compact operators