We construct a solution to the ¯∂-equation on a strongly pseudo- convex domain of a complex manifold

Electronic Journal of Differential Equations, Vol. 2004(2004), No. 73, pp. 1–9.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

SOLUTIONS TO ∂-EQUATIONS ON STRONGLY¯ PSEUDO-CONVEX DOMAINS WITH L^p-ESTIMATES

OSAMA ABDELKADER & SHABAN KHIDR

Abstract. We construct a solution to the ¯∂-equation on a strongly pseudo- convex domain of a complex manifold. This is done for forms of type (0, s), s≥1, with values in a holomorphic vector bundle which is Nakano positive and for complex valued forms of type (r, s), 1≤r≤n, when the complex manifold is a Stein manifold. Using Kerzman’s techniques, we find theL^p-estimates, 1≤p≤ ∞, for the solution.

1. Introduction

The existence of solutions to the equation ¯∂g = f, on strongly pseudo-convex domains inCⁿ, withL^p-estimates whenfis a form of type (0, s); ¯∂f = 0,s≥1, and satisfies L^p-estimates, 1≤ p≤ ∞, has been a central theme in complex analysis for many years. Øvrelid [5] has obtained a solution with L^p-estimates for this equation. Abdelkader [1] has extended Øvrelid’s results to forms of type (n, s) on strongly pseudo-convex domains in ann−dimensional Stein manifold. In this paper we extend Abdelkader’s results to forms of type (r, s); 0≤r≤n. For this purpose, we first study the equation ¯∂g = f, on strongly pseudo-convex domains in ann- dimensional complex manifold M when f is a form of type (0, s); ¯∂f = 0, s ≥1, with values in a holomorphic vector bundle. Then, we apply this results to the vector bundle Vr

T^?(M) (the r^th-exterior product of the holomorphic cotangent vector bundle T^?(M)) and using the fact that anyC−valued differential form of type (r, s) onM is a differential form of type (0, s) onM with values in the vector bundle Vr

T^?(M). When r = n, the vector bundle K(M) = Vn

T^?(M) is the canonical line bundle of M. Therefore it is sufficient in this case to study the equation ¯∂g=f forf with values in a holomorphic line bundle which is the case in [1]. In fact, the main aim of this paper is to establish the following existence theorem withL^p-estimates:

Theorem 1.1(Global theorem). Let M be a complex manifold of complex dimen- sion n and let E →M be a holomorphic vector bundle, of rankN, over M. Let DbM be a strongly pseudo-convex domain with smooth C⁴-boundary. Then

2000Mathematics Subject Classification. 32F27, 32C35, 35N15.

Key words and phrases. L^p-estimates, ¯∂-equation, strongly pseudo-convex, smooth boundary, complex manifolds.

c

2004 Texas State University - San Marcos.

Submitted March 01, 2004. Published May 20, 2004.

1

(1) If the holomorphic vector bundle E is Nakano positive, then, there exists an integer k0 = k0(D) > 0 such that for any f ∈ L¹_0,s(D, E^k); ∂f¯ = 0, s ≥ 1 and k ≥ k0 there is a form g = T_N^skf ∈ L¹_0,s−1(D, E^k) satisfies ∂g¯ = f, where T_N^sk

is a bounded linear operator and E^k =E⊗E⊗ · · · ⊗E (k-times). Moreover, if f ∈ L^p_0,s(D, E^k); 1 ≤p≤ ∞, there is a constant C_s^k such that kgk_L^p

0,s−1(D,E^k) ≤ C_s^kkfk_L^p

0,s(D,E^k). The constantC_s^k is independent of f and p. If f isC^∞, then g is also C^∞.

(2) If M is a Stein manifold, then, for any f ∈ L¹_r,s(D); ∂f¯ = 0, 0 ≤ r ≤ n, and s ≥ 1, there is a form g = T^sf ∈ L¹_r,s−1(D) such that ∂g¯ = f, where T^s is a bounded linear operator. Moreover, if f ∈ L^p_r,s(D); 1 ≤ p ≤ ∞, we have kgk_L^p

r,s−1(D)≤Cskfk_L^p_r,s(D). The constant Cs is independent of f and p. If f is C^∞, then g is alsoC^∞.

The plan of this paper is as follows: In section 1, we state the main theorem. In section 2, we set the notation and recall some useful facts. In section 3, we prove an existence theorem with L²−estimates. In section 4, we give local solution for the

∂-equation with¯ L^p-estimates for 1≤p≤ ∞. In section 5, we prove the existence theorem withL^p-estimates.

2. Notation and Preliminaries

Let M be an n-dimensional complex manifold and let π : E → M be a holo- morphic vector bundle, of rankN, overM. Let {uj};j ∈I, be an open covering of M consisting of coordinates neighborhoods uj with holomorphic coordinates z_j = (z_j¹, z_j², . . . , zⁿ_j) over which E is trivial, namely π⁻¹(u_j) = u_j ×C^N. The N-dimensional complex vector space Ez = π⁻¹(z); z ∈ M, is called the fiber of E over z. Let h = {hj}; hj = (hjµ¯η) be a Hermitian metric along the fibers of E and let (h^µ¯_j^η) be the inverse matrix of (hjµ¯η). Let θ = {θj}; θj = (θ_jµ^ν );

θ^ν_jµ = ∂logh_j = Pn α=1

PN

η=1h^ν_j^η¯∂h_∂z^jµ¯α^η j

dz_j^α = Pn

α=1Ω^ν_jµαdz^α_j and Θ = {Θ_j};

Θ_j = (Θ^ν_jµ); Θ^ν_jµ = √

−1 ¯∂∂logh_j = √

−1Pn

α,β=1Θ^ν_jµαβ¯dz^α_j ∧d¯z_j^β be the con- nection and the curvature forms associated to the metric h respectively, where Θ^ν_jµαβ¯ = −^{∂Ωνjµα}

∂d¯z^β_j , 1 ≤ µ ≤ N; 1 ≤ ν ≤ N. The associated curvature matrix is given by

(H_jη¯β,να¯ ) =

N

X

µ=1

hjµ¯ηΘ^µ_jναβ¯ .

LetT(M) (resp. T^?(M)) be the holomorphic tangent (resp. cotangent) bundle of M.

Definition 2.1. E is said to be Nakano positive, atz∈uj, if the Hermitian form XH_j¯ηβ,να¯ (z)ζ_α^νζ¯_β^η

is positive definite for anyζ= (ζ_α^ν)∈Ez⊗Tz(M);ζ6= 0.

The notationX bM means thatX is an open subset ofM such that its closure is a compact subset ofM.

Definition 2.2. A domain D b M is said to be strongly pseudo-convex with smoothC⁴-boundary if there exist an open neighborhood U of the boundary ∂D ofD and aC⁴ functionλ:U →Rhaving the following properties:

(i) D∩U ={z∈U;λ(z)<0}.

(ii) Pn α,β=1

∂²λ(z)

∂z_j^α∂z¯^β_jµ_αµ¯_β≥L(z)|µ|²;z∈U∩u_j,µ= (µ₁, µ₂, . . . , µ_n)∈Cⁿ and L(z)>0.

(iii) The gradient ∇λ(z) = (^∂λ(z)_∂x1 j

,^∂λ(z)_∂y1 j

,^∂λ(Z)_∂x2 j

,^∂λ(z)_∂y2 j

, . . . ,^∂λ(z)_∂xn j ,^∂λ(z)_∂yn

j ) 6= 0 for z= (z_j¹, z_j², . . . , zⁿ_j)∈uj∩U;z_j^α=x^α_j +iy^α_j.

Letγ= (µ1, ν1, . . . , µn, νn) be any multi-index and|γ|=Pn

i=1(µi+νi), where µ_i andν_i are non-negative integers. LetD^γ =∂^|γ|/∂x^µ₁¹∂y^ν₁¹. . . ∂x^µ_nⁿ∂y_n^νn. Remark 2.3. By shrinkingU we can assume thatU bU, where ˜˜ U is an open,λ isC⁴ on ˜U and the properties (i), (ii) and (iii) of Definition 2.2 hold on ˜U. Thus, we can choose a neighborhood V of ∂D such that V b U and for any z ∈ V there exist positive constants L, F and F⁰ satisfy L(z) > L, |∇˜λ(z)| ≥ F and

|D^γ˜λ(z)| ≤ F⁰ < ∞ for any multi-index γ with |γ| ≤ 4, where ˜λ is the a slight perturbation ofλ.

Definition 2.4. Let X be an n-dimensional complex manifold and let Φ be an exhaustive function onX, that is, the setsX_c={z∈X; Φ(z)< c}bX;c∈Rand X =∪X_c. We say thatXis weakly 1-complete (resp. Stein) manifold if Φ is aC^∞ plurisubharmonic (resp. strictly plurisubharmonic), that is, ifPn

α,β=1

∂²Φ(z)

∂z_j^α∂z¯_j^βµ^αµ¯^β is positive semi-definite (resp. positive definite) onX forµ = (µ¹, . . . , µⁿ) ∈Cⁿ; µ6= 0.

We will use the standard notation of H¨ormander [5] for differential forms. Thus aC-valued differential formϕ={ϕ_j} of type (r, s) onM can be expressed, onu_j, asϕj(z) =P

A_r,B_sϕjA_rB_s(z)dz_j^Ar∧d¯z^B_j^s, whereAr andBsare strictly increasing multi-indices with lengths r and s, respectively. An E-valued differential form ϕ of type (r, s), on M, is given locally by a column vector ^tϕ_j = (ϕ¹_j, ϕ²_j, . . . , ϕ^N_j ) whereϕ^a_j, 1≤a≤N, areC-valued differential forms of type (r, s) onuj. Λ^r,s(M) denotes the space ofC-valued differential forms of type (r, s) and of classC^∞ on M. Let Λ^r,s(M, E) (resp. D^r,s(M, E)) be the space ofE-valued differential forms (resp. with compact support) of type (r, s) and of classC^∞ onM.

Leth⁰={h⁰_j},h⁰_j = (h⁰_jµ¯η), be the initial Hermitian metric along the fibers ofE and let Θ⁰={Θ⁰_j}be the associated curvature form. The induced Hermitian metric along the fibers of the line bundleB =VN

Eis given by the system of positiveC^∞ functions {a⁰_j}, wherea⁰_j = det(h⁰_jµ¯η)⁻¹. Hence, the system{1/a⁰_j} also defines a Hermitian metric along the fibers of B whose curvature matrix (H_jη¯β,να¯ ) is given by (1/a⁰_j)(∂²loga⁰_j/∂z^α_j∂z¯^β_j). IfEis Nakano positive, with respect toh⁰, thenB is positive, with respect to {1/a⁰_j}, that is, the Hermitian matrix (∂²loga⁰_j/∂z^α_j∂¯z_j^β) is positive definite. Hence,

ds²₀=

n

X

α,β=1

g_jα⁰ β¯dz_j^αd¯z_j^β ; g⁰_jαβ¯=∂²loga⁰_j/∂z^α_j∂z¯^β_j

defines a K¨ahler metric on M. For ϕ, ψ ∈ Λ^r,s(M, E), we define a local inner product, atz∈uj, by

(2.1)

N

X

ν,µ=1

h⁰_jν¯µϕ^ν_j(z)∧?ψ^µ_j(z) =a(ϕ(z), ψ(z))dv₀,

where the Hodge star operator ? and the volume element dv0 are defined by ds²₀ anda(ϕ, ψ) is a function, onM, independent ofj.

LetL^p_r,s(M, E) (resp. L^∞_r,s(M, E)) be the Banach space ofE-valued differential forms f on M, of type (r, s), such that kfk_L^p_r,s(M,E) = (R

M|f(z)|^pdv0)^1/p < ∞ for 1 ≤ p < ∞ (resp. kfk_L^∞

r,s(M,E) = ess sup_z∈M|f(z)| < ∞), where |f(z)| = pa(f(z), f(z)).

The Hermitian metric along the fibers of E^k =E⊗E⊗ · · · ⊗E, associated to h⁰, is defined byh^0k ={h^0k_j }, whereh^0k_j =h⁰_jh⁰_j. . . h⁰_j (k-factors). The transition functions ofK(M) are the Jacobian determinant

kij= ∂(z_j¹, z_j², . . . , zⁿ_j)

∂(z_i¹, z_i², . . . , zⁿ_i)

onui∩uj. We see that|kij|²=gig⁻¹_j onui∩uj, where gi = det(∂²loga⁰_i/∂z_i^α∂¯z_i^β) . Therefore, the system of positiveC^∞ functions{g_j⁻¹}(resp. g={gj}) determines a Hermitian metric along the fibers ofK(M) (resp. the dual bundleK⁻¹(M)).

3. Existence Theorems with L²-Estimates

LetY bM be weakly 1-complete domain of M with respect to a plurisubhar- monic function Φ andλ(t) be a realC^∞function onRsuch thatλ(t)>0,λ⁰(t)>0 andλ⁰⁰(t)>0 fort >0 andλ(t) = 0 fort≤0. Leth_j =e^−λ(Φ)h⁰_j, onu_j∩Y, and a_j= det(h_j)⁻¹. Thus, the Hermitian matrix (∂²loga_j/∂z^α_j∂¯z_j^β) is positive definite onu_j∩Y. Hence,

ds²=

n

X

α,β=1

g_jαβ¯dz_j^αd¯z_j^β ; g_jαβ¯=∂²loga_j/∂z^α_j∂¯z_j^β

defines a K¨ahler metric on Y. The Hermitian metrics h^k ={h^k_j} and g induce a Hermitian metric b^k ={h^k_jgj}; k≥1, along the fibers ofK⁻¹(M)⊗E^k|Y, where h^k_j =hjhj. . . hj (k-factors).

LetL²_r,s(Y, K⁻¹(M)⊗E^k,loc, gh^0k, ds²₀) be the space of allK⁻¹(M)⊗E^k−val- ued differential forms of type (r, s) which has measurable coefficients and square integrable on compact subsets of Y with respect to ds²₀ and gh^0k. For ϕ, ψ ∈ Λ^r,s(Y, K⁻¹(M)⊗E^k) we define a local inner product a(ϕ(z), ψ(z))kdv by re- placing g_jh^k_j and ds² instead of h⁰_j and ds²₀, respectively, in (2.1). For ϕ or ψ∈ D^r,s(Y, K⁻¹(M)⊗E^k), we define a global inner product by

(3.1) hϕ, ψik=

Z

Y

a(ϕ, ψ)k dv.

Let ω = √

−1Pn

α,β=1g_jαβ¯dz_j^α ∧d¯z^β_j be the fundamental form of ds² and let L = e(ω) be the wedge multiplication by ω. Let Γ : Λ^r,s(Y, K⁻¹(M)⊗E^k) → Λ^r−1,s−1(Y, K⁻¹(M)⊗E^k) be the operator locally defined by Γ = (−1)^r+s? L?, where the ? operator is defined by ds². Let ϑk be the formal adjoint of ¯∂ : Λ^r,s(Y, K⁻¹(M)⊗E^k) → Λ^r,s+1(Y, K⁻¹ ⊗E^k) with respect to the inner prod- uct (3.1) and ^k = ¯∂ϑk +ϑk∂¯ be the Laplace-Beltrami operator. The curvature form associated tob^k is given by

Θ^k={Θ^k_j}; Θ^k_j =√

−1 ¯∂∂logb^k_j =kΘ⁰_j+√

−1(k∂∂λ(Φ)¯ −∂∂¯logg_j).

Since the Levi form√

−1∂∂λ(Φ) is positive semi-definite,¯ Eis Nakano positive with respect toh⁰and ¯Y is compact subset ofM, there exists an integerk0=k0(Y)>0 such thatK⁻¹(M)⊗E^k|Y is Nakano positive, with respect tob^k, fork≥k0. Hence as in Nakano [4] we can prove the following lemma:

Lemma 3.1. Letf ∈L²_n,s(Y, K⁻¹(M)⊗E^k,loc, gh^0k, ds²₀);k≥k0,s≥1be given, then we can choose the function λ(t)such that ds² is complete, hf, fik <∞, and there is a constant c >0 such that

(3.2) h∂ϕ,¯ ∂ϕi¯ k+hϑkϕ, ϑkϕik ≥chϕ, ϕik, for any ϕ∈ D^n,s(Y, K⁻¹(M)⊗E^k).

Remark 3.2. We note that whenE is a line bundle Lemma 3.1 is valid for forms inD^r,s(Y, K⁻¹(M)⊗E^k) withr+s≥n+ 1.

From Lemma 3.1 and the Hilbert space technique of H¨ormander [5], as in the proof of [1, Theorem 2.1], we can prove the following theorem:

Theorem 3.3. Let Y b M be weakly 1-complete domain and let E → M be a holomorphic vector bundle over M. IfE is Nakano positive, overM, then for any f ∈L²_n,s(Y, K⁻¹(M)⊗E^k, b^k, ds²)with ∂f¯ = 0, s≥1 andk≥k₀ there exists a formg=T f ∈L²_n,s−1(Y, K⁻¹(M)⊗E^k, b^k, ds²)satisfies∂g¯ =f and two constants C=C(Y)andck=ck(G, Y)such that

kgk_L2

n,s−1(Y,K⁻¹(M)⊗E^k,b^k,ds²)≤Ckfk_L2

n,s(Y,K⁻¹(M)⊗E^k,b^k,ds²), kgk_L2

n,s−1(G,K⁻¹(M)⊗E^k)≤c_kkfk_L2

n,s(G,K⁻¹(M)⊗E^k), whereT is a bounded linear operator andGbY.

4. Local solution for the ∂¯-equation with L^p-estimates

LetDbM be a strongly pseudo-convex domain withλandU of Definition 2.2.

Letx∈∂Dbe an arbitrary fixed point and let Wa be an open neighborhood ofx such that Wa buj ⊂U, for a certain j∈I, andzj(Wa) is the ballB(0, a)bCⁿ, where (uj, zj) is a holomorphic chart. Then, Wa can be considered as strongly pseudo-convex domain inCⁿ and the volume elementdv0can be considered as the Lebesgue measure onB(0, a).

Theorem 4.1 ([5]). Let G b Cⁿ be a strongly pseudo-convex domain and u ∈ L¹_0,s(G); s≥1. Then, there exist kernels Ks(ξ, z)such that the integral R

Gu(ξ)∧ K_s−1(ξ, z)dµ(ξ) is absolutely convergent for almost all z ∈ G¯ and the operator T^s : L^p_0,s(G) → L^p_0,s−1(G), defined by T^su(z) = R

Gu(ξ)∧K_s−1(ξ, z)dµ(ξ), with norm≤c;1≤p≤ ∞. Moreover, if∂u¯ = 0, then, there is a formg=T^susatisfies

∂g¯ =u, wheredµ(ξ)is the Lebesgue measure on Cⁿ.

Now, we extend the operatorT^stoL^p_0,s(D∩Wa, E). For this purpose, we define an operatorT_N^s :f ∈L¹_0,s(D∩Wa, E)→T_N^sf ∈L¹_0,s−1(D∩Wa, E);s≥1, by

(4.1) T_N^sf(z) =

N

X

λ=1

T^sf^λ(z)bλ(z), where f(z) = PN

λ=1f^λ(z)b_λ(z), that is, f^λ(z) are the components of f|uj with respect to an orthonormal basisb_λ(z) onE_z;z∈u_j.

We consider the following situation: In the notation of Definition 2.2, from Remark 2.3, let y ∈ ∂V⁻, where V⁻ = {z ∈ V; ˜λ(z) < 0} and let Wa be a neighborhood of y such that Wa b uj ⊂ V, for a certain j ∈ I, and zj(Wa) is the ball B(0, a)⊂Cⁿ, a≤˜a, where ˜adepends continuously onL, F, F⁰ and the distance d(y,CV) fromy to the complement of V. In the above notation, as the local theorem in [3], we can prove the following theorem:

Theorem 4.2 (Local theorem). Let T_N^s_k be the linear operator defined by (4.1) and let f ∈ L¹_0,s(V⁻, E^k); ∂f¯ = 0, where N^k is the rank ofE^k. Then, there is a formg=T_N^sf ∈L¹_0,s−1(V⁻∩W_a, E^k)such that ∂g¯ =f. If f isC^∞, then so is g.

If f ∈L^p_0,s(V⁻, E^k), theng∈L^p_0,s−1(V⁻∩W_a, E^k)and satisfies kgk_L^p

0,s−1(V⁻∩Wa,E^k)≤Ckfk_L^p

0,s(V⁻,E^k); 1≤p≤ ∞,

whereC=C(s, k, N)is a constant which depends continuously onL,F,F⁰ anda.

5. Global solution for the ∂¯-equation with L^p-estimates The local result yields Lemma 5.1 (An extension lemma) which in turn enables one to solve ¯∂η= ˆf (with bounds) in a strongly pseudoconvex domain ˆD which is larger thanD, ¯D⊆D. Here we make use of theˆ L²−estimates for solutions of the

∂−equation as presented in Theorem 3.3.¯

Lemma 5.1 (An extension lemma). Let DbM be a strongly pseudo-convex do- main with smooth C⁴-boundary. Then, there exists another slightly larger strongly pseudo-convex domain Dˆ b M with the following properties: D¯ b D, for anyˆ f ∈ L¹_0,s(D, E^k) with s ≥1 and ∂f¯ = 0, there exist two bounded linear operators L₁,L₂, a form fˆ=L₁f ∈L¹_0,s( ˆD, E^k) and a formu=L₂f ∈L¹_0,s−1(D, E^k)such that:

(i) ¯∂fˆ= 0in D.ˆ (ii) ˆf =f−∂u¯ inD.

(iii) If f ∈L^p_0,s(D, E^k), then fˆ∈L^p_0,s( ˆD, E^k) andu∈L^p_0,s−1(D, E^k) with the estimates

kfˆk_Lp

0,s( ˆD,E^k)≤C1kfk_L^p

0,s(D,E^k), (5.1)

kuk_L^p

0,s−1(D,E^k)≤C₂kfk_L^p

0,s(D,E^k) 1≤p≤ ∞, (5.2)

where the constants C₁ andC₂ are independent of f andp.

If f isC^∞ inD, thenfˆisC^∞ inDˆ anduisC^∞ in D.

Since ∂D is compact, we can Cover ∂D by finitely many neighborhoods Wi,a_i

of xi ∈ ∂D, i = 1,2, . . . , m, such that for each xi we have Wi,a_i buj bV bU for a certaini∈I. Puta= min_1≤i≤mai. Then as Lemma 2.3.3 and the Claim on page 321 in Kerzman [3] (see also [1, Proposition 3.2]), we can prove the following proposition:

Proposition 5.2. Let Dˆ be as in the extension lemma and let Wi,a be an open set of Dˆ such that Wi,a b uj ⊂ D, for a certainˆ j ∈ I and zj(Wi,a) is the ball B(0, a) b Cⁿ. Then, for any f ∈ L¹_0,s(W_i,a, E^k); ∂f¯ = 0 there is α = T f ∈

L¹_0,s−1(Wi,a/2, E^k) such that ∂α¯ = f, where T is a bounded linear operator. If f ∈L^p_0,s(W_i,a, E^k);1≤p≤2, then, we haveα∈L^p+1/4n_0,s−1 (W_i,a/2, E^k)and

kαk_Lp+1/4n

0,s−1 (W_i,a/2,E^k)≤ckfk_L^p

0,s(W_i,a,E^k), and for any p,1≤p≤ ∞, we have

kαk_L^p

0,s−1(W_i,a/2,E^k)≤ckfk_L^p

0,s(W_i,a,E^k), wherec=c(n, a, k, N)is a constant independent off andp.

The proof of Proposition 5.2 is purely local. Using Proposition 5.2, as [1, Propo- sition 3.2], we prove the following proposition:

Proposition 5.3. Let Dˆ be as in the extension lemma. Then, there is a strongly pseudo-convex domainD₁bDˆ such that for everyfˆ∈L¹_0,s( ˆD, E^k);∂¯fˆ= 0, there are two bounded linear operatorsL1andL2and two formsf1=L1fˆ∈L¹_0,s(D1, E^k) andη₁=L₂fˆ∈L¹_0,s−1(D₁, E^k)such that:

(i) ¯∂f₁= 0 onD₁, (ii) ˆf =f1+ ¯∂η1 on D1, (iii) kf1k_Lp+1/4n

0,s (D1,E^k)≤ckfˆk_L^p

0,s( ˆD,E^k) forfˆ∈L^p_0,s( ˆD, E^k);1≤p≤2, (iv) For every open set W bD1 and for every p,1≤p≤ ∞, we have

kf1k_L^p

0,s(W,E^k)≤ckfkˆ _Lp

0,s( ˆD,E^k), kη1k_L^p

0,s−1(W,E^k)≤ckfˆk_Lp

0,s( ˆD,E^k),

wherec=c( ˆD, W, n, k, N)is a constant independent offˆandp.

Since every strongly pseudo-convex domain is weakly 1-complete and noting that Λ^n,s(D, K⁻¹(M)⊗E^k) ≡ Λ^0,s(D, E^k); k ≥ 1. Then, using Theorem 3.3, Proposition 5.3, and the interior regularity properties of the ¯∂-operator, as [1, Theorem 3.1], we prove the following theorem:

Theorem 5.4. LetDˆ be the strongly pseudo-convex domain of the extension lemma andW bD. Then, for any formˆ fˆ∈L¹_0,s( ˆD, E^k)with ∂¯fˆ= 0, there exists a form η∈L¹_0,s−1(W, E^k),η=Tfˆsuch that∂η¯ = ˆf, whereT is a bounded linear operator.

If fˆ∈L^p_0,s( ˆD, E^k) with1≤p≤ ∞andk≥k0, thenη∈L^p_0,s−1(W, E^k)and kηk_L^p

0,s−1(W,E^k)≤Ckfˆk_L^p

0,s( ˆD,E^k)

whereC=C( ˆD, W, k)is a constant independent of fˆandp. IffˆisC^∞, thenη is C^∞.

Proof. Proposition 5.3 yieldsD₁. A new application of Proposition 5.3 toD₁yields D₂. We iterate 4ntimes and obtain

Dˆ ⊇D1⊇D2⊇ · · · ⊇D4ncW

Hence, for any f ∈ L¹_0,s( ˆD, E^k); ¯∂f = 0, there exist fj ∈ L¹_0,s(Dj, E^k) and υj ∈ L¹_0,s−1(D_j, E^k);j= 1,2, . . . ,4n. Clearly, we have:

fˆ=f₁+ ¯∂υ₁=f₂+ ¯∂υ₁+ ¯∂υ₂=f₃+ ¯∂υ₁+ ¯∂υ₂+ ¯∂υ₃=· · ·=f_4n+ ¯∂(

4n

X

j=1

υ_j)

inD4n,f4n ∈L²_0,s(D4n, E^k) andkf4nk_L2

0,s−1(D_4n,E^k)≤Kkfˆk_L1

0,s( ˆD,E^k).

Now we apply Theorem 3.3 with ˆD =D4n and W ⊂Y bD4n. Let υ be the solution of ¯∂υ=f4n obtained from Theorem 3.3, with

kυk_L2

0,s−1(Y,E^k)≤Kkf_4nk_L2

0,s(D4n,E^k)≤Kkfkˆ _L1

0,s( ˆD,E^k). Setη=υ+P4n

j=1υ_j, then we obtain ¯∂η= ¯∂υ+ ¯∂(P4n

j=1υ_j) =f_4n+ ¯∂(P4n

j=1υ_j) = ˆf in Y (hence in W). Using (iv) of Proposition 5.3, collecting estimates and the estimatesk.k_L1

0,s( ˆD,E^k)≤Kk.k_Lp

0,s( ˆD,E^k) (since ˆD is bounded), we obtain:

(5.3) kηk_L10,s−1(Y,E^k)≤Kkfˆk_Lp

0,s( ˆD,E^k), 1≤p≤ ∞.

Finally, an application of the interior regularity properties for solutions of the el- liptic ¯∂−operator yields

kηk_L^p

0,s−1(W,E^k)≤K(kηk_L1

0,s−1(Y,E^k) +kfˆk_L^p

0,s(Y,E^k)), 1≤p≤ ∞, which together with (5.3) give the estimates in Theorem 5.4.

Proof of Theorem 1.1. Let ˆD⊇D¯ be the strongly pseudo-convex domain furnished by Lemma 5.1 (An extension lemma). If f ∈L¹_0,s(D, E^k) withs≥1 and ¯∂f = 0, then Lemma 5.1 yields a form ˆf = L1f ∈ L¹_0,s( ˆD, E^k) and a form u = L2f ∈ L¹_0,s−1(D, E^k) such that: ¯∂fˆ= 0; ˆf =f −∂u¯ in D, and (i), (ii), (iii), (5.1), (5.2) in that lemma are valid.

We solve ¯∂η = ˆf using Theorem 5.4 (withW =D). Hence, η ∈L¹_0,s−1(D, E^k) and

∂η¯ = ˆf =f−∂u¯ in D.

the desired solution is g =η+u. The estimates in the first part of Theorem 1.1 follows from those in Lemma 5.1 and Theorem 5.4. η and u are linear in f and they areC^∞ iff isC^∞. The first part of Theorem 1.1 is proved.

Now, we prove the second part of Theorem 1.1. In fact, Theorem 4.2, Lemma 5.1, Proposition 5.2, and Proposition 5.3 are valid if we replace the vector bundle E^k by the vector bundleVr

T^?(M). IfM is a Stein manifold, then every strongly pseudo-convex domain ofM is also a Stein manifold. Hence, as [2, Theorem 5.2.4], we can prove the following auxiliary theorem:

Theorem 5.5. LetM be a Stein manifold of complex dimensionnand letDbM be strongly pseudo-convex domain. Then, for everyf ∈L²_r,s(D, E^k,loc) with∂f¯ = 0, 0≤r≤nand s≥1 there exists a form g=T f ∈L²_r,s−1(D, E^k,loc); ∂g¯ =f, and a constant c=c(D)such that

kgk_L2

r.s−1(D,E^k,loc)≤ckfk_L2

r,s(D,E^k,loc),

where T is a bounded linear operator. Moreover, for any G b D there exists a constant c1=c1(G, D)such that

kgk_L²

r,s−1(G,E^k,loc)≤c₁kfk_L2

r,s(D,E^k).

Then, we can apply the result of Theorem 5.5 instead of that of Theorem 3.3, we conclude that Theorem 5.4 is valid if we replace E^k byVr

T^?(M); 0≤r≤n.

Using this result and the identity

Λ^r,s(M)≡Λ^0,s(M,∧^rT^?(M)), 1≤r≤n

we obtain the second part of our results.

References

[1] O. Abdelkader,L^p-estimates for solution of∂-equation in strongly pseudo-convex domains,¯ Tensor. N. S,58, (1997), 128-136.

[2] L. H¨ormander, An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton. N. J., (1990).

[3] N. Kerzman, H¨older andL^p-estimates for solutions of ∂u¯ = f in strongly pseudo-convex domains, Comm. Pure and Applied Math.24, (1971), 301-380.

[4] S. Nakano, Vanshing theorems for weakly 1-complete manifolds, Number theory, Algebraic Geometry and commutative Algebra. In honor of Akizuki, Y., Kinokuniyia, Tokyo, (1973), 169-179.

[5] N. Øvrelid,Integral representation formulas and L^p-estimates for∂-equation, Mat. scand.¯ 29, (1971), 137-160.

Osama Abdelkader

Mathematics Department, Faculty of Science, Minia University, El-Minia, Egypt E-mail address:[email protected]

Shaban Khidr

Mathematics Department, Faculty of Science, Cairo University, Beni- Suef, Egypt E-mail address:[email protected]