SOBOLEV SPACE ESTIMATES FOR SOLUTIONS OF THE EQUATION ∂u = f ON POLYCYLINDERS

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SOBOLEV SPACE ESTIMATES FOR SOLUTIONS OF THE EQUATION ∂u = f ON POLYCYLINDERS

PATRICK W. DARKO and CLEMENT H. LUTTERODT Received 10 August 2003

In trying to improve Weinstock’s results on approximation by holomorphic functions on certain product domains, we are led to estimates in Sobolev spaces for the∂-operator on polycylinders for(γ, q)-forms. This generalizes our results for the same operator on poly- cylinders previously obtained, and can be applied to a number of other problems such as the Corona problem.

2000 Mathematics Subject Classification: 32A99.

1. Introduction. Had we the uniform estimates of Grauert and Lieb for the Cauchy- Riemann equation [5] on polycylinders, we would have improved Weinstock’s result [6, Theorem 1.1] on polycylinders a long time ago. On the other hand, we have had several estimates for the∂-operator in Sobolev spaces [2,3,4] on polycylinders, which we have been trying to improve. At the same time, we noticed that because some Sobolev norms dominate the uniform norm, if we did the approximation in those Sobolev norms, we would get the desired improvement of the above-mentioned theorem of Weinstock on polycylinders. We therefore went ahead here and jazzed up our previous estimates to get new estimates. We also applied our results to solve the Sobolev-Corona problem.

2. Delta-bar inW_(γ,q+1)^k,p (Ω). LetL^p_(γ,q)(U )denote the space of forms of type(γ, q) with coefficients inL^p(U ). Then,

f=

|J|=γ

|J|=q

fI,Jdz^I∧dz^J, (2.1)

where

means that the summation is performed only over strictly increasing multi- indices,I=(i1, . . . , iγ),J=(j1, . . . , jq),dz^I=dzi₁∧ ··· ∧dzi_γ,dz^J=dzj₁∧ ··· ∧dzj_q, Uis open inCⁿ, and 1≤p≤ ∞.

The norm of the(γ, q)-form in (2.1) is defined by

f_L^p

(γ,q)(U )=

I

J

fI,J_{Lp(U )}1/p

, for 1≤p <∞, fL^∞_(γ,q)(U )=max

I,J

fI,J_{L∞(U )}.

(2.2)

LetW^k,p(U ), 1≤p≤ ∞,k=1,2, . . . ,be the space of functions which together with their distributional derivatives of order throughkare inL^p(U ), with the usual norm, and W_(γ,q)^k,p (U )the space of (γ, q)-forms with coefficients inW^k,p(U ), with the norm defined by

f_W^k,p

(γ,q)(U )=

I

J

fI,J_{Wk,p(U )}1/p

, 1≤p <∞, f_W^k,p

(γ,q)(U )=max

I,J

fI,J_{Wk,p(U )}.

(2.3)

A bounded open setΩinCⁿis called a polycylinder ifΩcan be expressed as a product ofnbounded open sets inC, that is,Ω=U1×U2×···×Un, where eachUjis open and bounded inC. AndΩis called admissible if eachUjhas boundary with plane measure zero.

Our main result is the following theorem.

Theorem2.1. LetΩbe an admissible polycylinder inCⁿand letf∈W_(γ,q^k,p₊₁₎(Ω)be

∂-closed,1≤p≤ ∞,k=1,2, . . . ,then there is au∈W_(γ,q)^k,p (Ω)such that∂u=f and

u_W^k,p

(γ,q)(Ω)≤δf_W^k,p

(γ,q+1)(Ω), (2.4)

whereδdepends onΩ.

3. The Corona problem. The Corona problem is stated in [1]. LetX be a relatively compact domain in a topological spaceY. Letf0, . . . , fN be complex-valued continuous functions onX;f0, . . . , fN verify the Corona assumption if there isδ >0 such that

S

fS≥δ >0 onX. (3.1)

LetAbe a function algebra onX. The Corona problem is solvable inA(onX) when each setf0, . . . , fN∈A, which verifies the Corona assumption, represents 1 inA, that is, there areg0, . . . , gN inAsuch that

f0g0+···+fNgn=1 onX. (3.2)

FromTheorem 2.1through cohomology with bounds (see [4, Theorem 2.3]), we have the following theorem.

Theorem3.1. LetΩbe an admissible polycylinder and suppose thatΩhas a Lipschitz boundary, andpk > n(in which case Γ(Ω,ᏻ)∩W^k,p(Ω)is an algebra with members extending continuously to the boundary ofΩ). Then the Corona problem is solvable in Γ(Ω,ᏻ⁾∩W^k,p(Ω)(hereᏻis the structure sheaf ofCⁿ, andΓ(Ω,ᏻ⁾is the set of sections ofᏻ^overΩ).

SOBOLEV SPACE ESTIMATES FOR SOLUTIONS OF THE EQUATION∂u=f . . . 971 4. Approximation. LetK=Ωbe the closure inCⁿof an admissible polycylinder, let C(K)denote the Banach space of continuous complex-valued functions onKwith the uniform norm, and letH(K)denote the closure inC(K)of the space of functions which are holomorphic in some neighborhood ofK.

Our last result is then the following theorem.

Theorem4.1. IfUis a neighborhood ofK,f∈C²(U ), and∂f /∂zj=0onK,1≤j≤ n, thenf∈H(K).

In this paper, we prove the(0,1)version of Theorems2.1and4.1only.Theorem 3.1 follows when considered as the weak Corona theorem as in [1]. The general version of Theorem 2.1can be proved using the induction process in [3].

5. Solution of∂u=f ((γ, q)=(0,1)). For allfsatisfying the hypothesis ofTheorem 2.1in this case, extendfto all ofCⁿby zero outsideΩand call it againf. Then∂f=0 in the distribution sense inCⁿ. Then the following is true.

Lemma 5.1. If u(z) = (2π i)⁻¹

C(ξ−z1)⁻¹f1(ξ, z2, . . . , zn)dξ∧dξ, where f = n

j=1fjdzj, withf1 ≡0, then∂u=f and uW^k,p(Ω)≤δf_W^k,p

(0,1)(Ω), 1≤p≤ ∞, k=1,2, . . . , (5.1)

whereδdepends only onΩ.

Proof. We regularizefcoefficientwise:

fm= n

j=1

fj mdzj,

fj m(z)=

fj

z− ξ

m

φ(ξ)dλ(ξ)=m²ⁿ

fj(ξ)φ

m(z−ξ) dλ(ξ),

(5.2)

where φ∈C₀^∞(Cⁿ),

φdλ=1, φ ≥0, suppφ = {z:|z| ≤1}, and λ is a Lebesgue measure. ThenfmL^p_(0,1)≤ fL^p_(0,1)for 1≤p≤ ∞,fm→f in

L¹_(0,1)(Ω) (5.3)

asm→ ∞for 1≤p≤ ∞, andfmis∂-closed inCⁿ. Now, let

um(z)=(2π i)⁻¹

C

ξ−z1 −1 f1 m

ξ, z2, . . . , zn dξ∧dξ. (5.4)

Then

um(z)= −(2π i)⁻¹

Cⁿξ⁻¹

f1 z1−ξ, z2, . . . , zn dξ∧ξ, (5.5)

and from (5.4) and (5.5),

∂um(z)

∂zl =(2π i)⁻¹

C

ξ−z1 −1∂ fl m

ξ, z2, . . . , zn

∂ξ dξ∧dξ=

fl m(z). (5.6)

Therefore,∂um=fm, and sincefm→f inL¹_(0,1)(Ω)for 1≤p≤ ∞,um→uinL¹(Ω)if 1≤p≤ ∞, and we have∂u=f.

For 1≤p≤ ∞, it is clear from [2] that

uL^p(Ω)≤δf_L^p

(0,1)(Ω), (5.7)

withδdependingΩ.

Now, let∂^α=∂^|α|/∂x^α₁¹∂y₁^α2···∂xn^∂2n−1∂yn^α2n,α=(α1, α2, . . . , α2n),z=(x1+iy1, . . . , xn+iyn), then [2,3] we see, whereγ(α)is a power of−1, that

γ^(α)∂^αum(z)=(2π i)⁻¹

C

ξ−z1 −1

∂^α f1 m

ξ, z2, . . . , zn dξ∧dξ. (5.8)

Since it is clear that for 1≤p≤ ∞, whenk≥ |α|andf∈W_(0,1)^k,p (Ω),∂^α(f1)m→∂^αfin L¹(Ω)whenm→ ∞, it follows that, fork=1,2, . . . ,

γ^(α)∂^αu(z)=(2π i)⁻¹

U₁

ξ−z1 −1∂^αf1

ξ, z2, . . . , zn dξ∧ξ. (5.9)

Therefore, fork=1,2, . . . ,1≤p≤ ∞,

uW^k,p(Ω)≤δf_W^k,p

(0,1)(Ω). (5.10)

Iff1≡0 andf ≡0, there is anfj₀ which we can use in place off1.

6. Proof ofTheorem 4.1. Letfsatisfy the hypothesis ofTheorem 4.1. We may sup- pose thatfhas compact support inU. Regularizef as we did inSection 5:

fm=

f

z− ξ m

φ(ξ)dλ(ξ)=m²ⁿ

f (ξ)φ

m(z−ξ) dλ(ξ). (6.1)

Then,fm∈C^∞(Cⁿ),fm→finW^1,∞(Cⁿ)asm→ ∞, and for eachj,

∂fm

∂zj

(z)= ∂f

∂zj

z− ξ m

φ(ξ)dλ(ξ), (6.2)

so ifGis open inCⁿ,

fm_W1,∞(G)≤ fW^1,∞(G^ν), ∂fm

∂zj

_W1,∞(G)≤ ∂f

∂zj

_W1,∞(Gν), (6.3) whereG^ν= {z−νξ:z∈G, |ξ| ≤1}.

SOBOLEV SPACE ESTIMATES FOR SOLUTIONS OF THE EQUATION∂u=f . . . 973 Now, we can find a sequence{Ωλ}of decreasing admissible polycylinders such that K=Ω= ∩Ωλand such thatδinTheorem 2.1is the same for eachΩλ. Note also that

∂f_W1,∞(Ω

λ) →0 asλ → ∞. (6.4)

Let >0 be given. Choosem0such that f−fm_W1,∞(K)<

2 form > m0. (6.5)

Chooseλ0such that

∂f

W_(0,1)^1,∞(Ωλ0)<δ⁻¹

4 . (6.6)

Then, for smallν,

∂f

W_(0,1)^1,∞(Ω_λ^ν

0)<δ⁻¹

2 . (6.7)

ByTheorem 2.1, we can chooseu∈W^1,∞(Ωλ₀)such that∂u=∂fm, m > m0 (fixed), and

uW^1,∞(Ωλ0)≤δ∂fm

W_(0,1)^1,∞(Ωλ0)

≤δ∂f

W_(0,1)^1,∞(Ω^ν_λ

0) (by (6.3))

<

2 (by (6.7)).

(6.8)

Thenh=fm−uis holomorphic in a neighborhood ofKand with · K the uniform norm, we have

f−hK≤f−fm_K+fm−h_K

≤f−fm_W1,∞(K)+uW^1,∞(K)

<

2+uW^1,∞(Ωλ0)

< .

(6.9)

References

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[2] P. W. Darko,OnL^p-Sobolev space estimates for the inhomogeneous Cauchy-Riemann equa- tion on polycylinders, Complex Variables Theory Appl.38(1999), no. 4, 367–373.

[3] ,L^pSobolev-space estimates for the∂-operator on polycylinders, Complex Variables Theory Appl.40(2000), no. 4, 387–393.

[4] ,Sobolev cohomology for locally polycylindrical domains, Int. J. Math. Math. Sci.29 (2002), no. 4, 187–194.

[5] H. Grauert and I. Lieb,Das Ramirezsche Integral und die Lösung der Gleichung∂f¯ =αim Bereich der beschränkten Formen, Rice Univ. Studies56(1970), no. 2, 29–50 (Ger- man).

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Patrick W. Darko: Department of Mathematics and Computer Science, Lincoln University, PA 19352, USA

E-mail address:[email protected]

Clement H. Lutterodt: Department of Mathematics, Howard University, Washington, DC 20059, USA

E-mail address:[email protected]