aa r X i v : . [ m a t h . C V ] O c t A Dolbeault Lemma for Temperate Currents.
Henri Skoda ∗ Dedicated to the memory of P ierre Dolbeault.
Abstract
We consider a bounded open Stein subset Ω of a complex Steinmanifold X of dimension n . We prove that if f is a current on X of bidegree ( p, q + 1), ∂ -closed on Ω, we can find a current u on X ofbidegree ( p, q ) which is a solution of the equation ∂u = f in Ω. In otherwords, we prove that the Dolbeault complex of temperate currents onΩ (i.e. currents on Ω which extend to currents on X ) is concentratedin degree 0. Moreover if f is a current on X = IC n of order k , then wecan find a solution u which is a current on IC n of order k + 2 n + 1. Keywords : Stein open subset of IC n or of a Stein manifold, L esti-mates, ¯ ∂ -operator, Dolbeault ¯ ∂ -complex, temperate distributions and cur-rents, temperate cohomology, Sobolev spaces. We will prove the following result in the same way as the famous “ Dolbeault-Grothendieck ” lemma for ¯ ∂ . Theorem 1.
Let Ω be a bounded Stein open subset of IC n and let f be agiven current of bidegree ( p, q + 1) on IC n (with compact support) which is ¯ ∂ -closed on Ω . Then there exists a current u of bidegree ( p, q ) (with compactsupport) in IC n such that: (1) ¯ ∂u = f,in Ω .Moreover if f is of order k (resp. if f ∈ H − s ( p,q +1) ( IC n ) for some s > ), wecan find a solution u is of order at most k +2 n +1 (resp. u ∈ H − s − n − p,q ) ( IC n ) ,more precisely if k is the integer such that: s ≤ k < s + 1 , for every r > k ,we can find u ∈ H − r − n ( p,q ) ( IC n ) ). ∗ Corresponding author: Henri Skoda, Sorbonne University, IMJ-PRG Campus Pierreet Marie Curie, 4 Place Jussieu, 75005 Paris France, E-mail: [email protected]
1e say that a current T on Ω ⊂ IC n is temperate if and only if it can beextended to IC n . In other words, we have: Corollary 2.
For a given relatively compact open Stein subset of IC n , theDolbeault ¯ ∂ -cohomology of temperate currents on Ω vanishes. As usual, we denote by H s ( p,q ) (IC n ) the space of current on IC n of bidegree( p, q ) the coefficients of which are distributions in the Sobolev space H s (IC n ).A distribution T ∈ D ′ (IR n ) is of order k ∈ IN if it is locally a finite linearcombination of derivatives of order at most k of Radon measures on IR n or equivalenty if T can be extended as a continuous linear form defined onall functions of class C k with compact support in IR n or equivalently if forevery relatively compact open subset Ω ⊂ IR n , all functions φ ∈ D (Ω) verifyan inequality : | < T, φ > | ≤ C (Ω , T ) sup x ∈ Ω Σ | α |≤ k | D αx φ ( x ) | , in which theconstant C (Ω , T ) only depends on Ω and T . Of course a current is of order k , if its coefficients are distributions of order k .The preceding results are still valid replacing IC n by a Stein manifold(section 4, theorem 12) and for currents taking their values in a given holo-morphic vector bundle. But for the sake of simplicity, we begin with thecase of IC n as in the Dolbeault-Grothendieck lemma : the general case of aStein manifold does only need more difficult technical tools but no truly newideas or methods. In the case of a Stein manifold the loss of regularity islarger than 2 n + 1 because we have to iterate several times the constructionmade in the case of IC n .This result answers a question raised by Pierre Schapira in a personaldiscussion. He hopes it can be useful to make significant progress in theMicrolocal Analysis theories highlighted for instance in the papers of M.Kashiwara and P. Schapira, [KS1996] and [Scha2017] in which such a tem-perate cohomology naturally appears.Even though the result is essentially a consequence of L. H¨ormander’s L estimates for ¯ ∂ (corollary 4), it seems that it can not be explicitly foundin the literature on the subject (with complete proof). Let us observe thefollowing features of the result. No assumption of smoothness is required forΩ. The given current current f and the solution u have coefficients in spacesof distribution H s (IC n ) with s <
0. Hence they are never supposed to besmooth but with temperate singularities as for instance derivatives of Diracmeasures and the result is quite different from the most usual regularityresults for ¯ ∂ involving C k regularity up to the boundary of Ω ( k ≥
0) bothfor Ω and for the given differential forms on Ω. If f ∈ H s ( p,q +1) (IC n ) for some s ≥
0, then f ∈ L p,q +1) (Ω) and the result is an immediate consequence ofH¨ormander’s theorem which provides a solution u in L p,q ) (Ω). Then u hasa trivial extension in L p,q ) (IC n ) (by 0 outside Ω).The gap 2 n + 1 of regularity for the solution u does not depend on Ω. In the2asic example Ω = B (0 , R ) \ H in which H is a complex analytic hypersur-face of the ball B (0 , R ) of center 0 and radius R , the result does not dependat all on the complexity of the singularities of H (and on the degree of H when H is algebraic). The gap 2 n + 1 is an automatic consequence of themethod of proof. To improve the gap 2 n + 1 does not seem to immediatelyhave a major interest for the purpose in [Scha2017].We need four steps to prove theorem 1. At first, as P. Dolbeault in [Dol1956],by solving an appropriate Laplacian equation ∆ v = ¯ ∂ ⋆ f on IC n (∆ is theusual Laplacian on IC n defined on differential forms and currents and ¯ ∂ ⋆ isthe operator adjoint of ¯ ∂ for the usual Hermitian structure on IC n ) and re-placing f by f − ¯ ∂v , we reduce the problem to the case of a current f whichhas harmonic coefficients on Ω. As f is temperate, the mean value proper-ties of harmonic functions imply that f grows at the boundary of Ω like anegative power of the distance d ( z, ∂ Ω) to the boundary of Ω (for z ∈ Ω).Then H¨ormander’s L estimates for ¯ ∂ give a solution u of ¯ ∂u = f such that R Ω | u | [ d ( z, ∂ Ω)] l dλ ( z ) < + ∞ (for some l > u can be extended asa current on IC n .Similar methods, were already used by P. Lelong [Le1964] for the Lelong-Poincar´e ∂ ¯ ∂ -equation and by H. Skoda [Sk1971] for the ¯ ∂ -equation to obtainsolutions explicitly given on IC n by integral representations and with precisepolynomial estimates. Y.T. Siu has already studied holomorhic functions ofpolynomial growth on bounded open domain of IC n using H¨ormander’s L estimates for ¯ ∂ in [Siu1970].We establish the preliminary results we need in Section 2 and we prove the-orem 1 in Section 3. We extend the results to a Stein manifold in section 4theorem 12 (using J-P. Demailly’s theorem 7 extending H¨ormander’s resultsto manifolds).In the case of a subanalytic bounded open Stein subset Ω in a Stein manifold X , Pierre Schapira in [Scha2020] gives independently a proof of Theorem 1(i.e of corollary 2) and of theorem 12. His proof is basically founded on co-homological methods which are particularly well adapted to the subanalyticcase. It also heavily depends on H¨ormander’s L estimates for ¯ ∂ that heuses in the case of a bounded Stein open subset of IC n after embedding thegiven Stein manifold in some space IC n . He also uses Lojasiewcz inequalitiesand another H¨ormander’s inequality for subanalytic subsets.A first version of this article only treating the case of IC n was submitted toArXiv [Sk2020] on march 2020.I thank Pierre Schapira very much for raising his insightful question whichhas strongly motivated this research.3 Preliminary definitions and results
Before proving theorem 1, we need to remind several classical results. Wehave sometimes given direct proof to establish the results in the appropriateform we wish.An open subset of IC n is called Stein if it is holomorphically convex: forall compact K in Ω the holomorphic hull ˆ K Ω of K is compact ( x ∈ ˆ K Ω if and only if x ∈ Ω and for all holomorphic function f on Ω, | f ( x ) | ≤ max ξ ∈ K | f ( ξ ) | ).Let us recall the following fundamental H¨ormander’s L existence theo-rem for ¯ ∂ [H¨or1966] or [H¨or1965] . We can also use J.P. Demailly’s book[Dem2012], Chapter VIII, paragraph 6, Theorem 6.9, p. 379. We denoteby L p,q ) (Ω , loc ) the vector space of current of bidegree ( p, q ) in Ω the co-efficients of which are in L (Ω , loc ) for the usual Lebesgue measure dλ onIC n . Theorem 3.
Let Ω be an open pseudoconvex subset of IC n and φ a plurisub-harmonic function defined on Ω . For every g ∈ L p,q +1) (Ω , loc ) with ¯ ∂g = 0 such that: R Ω | g | e − φ dλ < + ∞ , there exists u ∈ L p,q ) (Ω , loc ) such that: (2) ¯ ∂u = g in Ω and: (3) Z Ω | u | e − φ (1 + | z | ) − dλ ≤ Z Ω | g | e − φ dλ. If Ω is bounded, u verifies the L estimate:(4) Z Ω | u | e − φ dλ ≤ C (Ω) Z Ω | g | e − φ dλ with C (Ω) := (1 + max z ∈ Ω | z | ) . The classical Oka-Norguet-Bremerman theorem ([H¨or1966] paragraph 2.6and theorem 4.2.8) claims that the following assertions are equivalent:1) Ω is Stein,2) Ω is pseudoconvex: i.e there exists a plurisubharmonic function φ on Ωwhich is exhaustive (for all c ∈ IR the subset { z ∈ Ω | φ ( z ) < c } is relativelycompact in Ω),3) the function − log d ( z, ∂ Ω) is plurisubharmonic in Ω.Therefore for a given k ≥
0, we can choose φ ( z ) = − k log d ( z, ∂ Ω) in theinequality (4) and we will only need to use the following special case oftheorem 3 (see also [H¨or1965] theorem 2.2.1’).
Corollary 4.
Let Ω be a bounded Stein open subset of IC n and k ≥ be agiven real number. Then for every g ∈ L p,q +1) (Ω , loc ) with ¯ ∂g = 0 such hat: R Ω | g | [ d ( z, ∂ Ω)] k dλ < + ∞ , there exists u ∈ L p,q ) (Ω , loc ) such that: (5) ¯ ∂u = g in Ω and: (6) Z Ω | u | [ d ( z, ∂ Ω)] k dλ ≤ C (Ω) Z Ω | g | [ d ( z, ∂ Ω)] k dλ If we denote by L ,k ( p,q ) (Ω) the space of u ∈ L p,q ) (Ω , loc ) such that R Ω | u | [ d ( z, ∂ Ω)] k dλ < + ∞ , by:(7) L ,k , ( p,q ) (Ω) := { u ∈ L ,k ( p,q ) (Ω) | ¯ ∂u ∈ L ,k ( p,q +1) (Ω) } and by O ,k ( p, (Ω) := { u ∈ L ,k ( p, (Ω) | ¯ ∂u = 0 } , corollary 4 means that thefollowing Dolbeault-complex is exact:(8)0 → O ,k ( p, (Ω) → L ,k , ( p, (Ω) ¯ ∂ −→ L ,k , ( p, (Ω) ¯ ∂ −→ . . . ¯ ∂ −→ L ,k , ( p,q ) (Ω) ¯ ∂ −→ L ,k , ( p,q +1) (Ω) ¯ ∂ −→ . . . ¯ ∂ −→ L ,k , ( p,n ) (Ω) → . We also need two results of real analysis.
Lemma 5.
Let w be a distribution on IR n of order k which is harmonic (forthe usual Laplacian on IR n ) on the bounded open subset Ω of IR n . Then w is of polynomial growth on Ω : | w ( z ) | ≤ C (Ω , w ) [ d ( z, IR n \ Ω)] − k − n wherethe constant C (Ω , w ) only depends on Ω and w .If w ∈ H − s ( IR n ) for s ≥ we have: | w ( z ) | ≤ C (Ω , w ) [ d ( z, IR n \ Ω)] − k − n where k is the integer such that s ≤ k < s + 1 .Proof. Let ρ be a non negative regularizing function in D (IR n ) which onlydepends on | ζ | , has its support in the Euclidean ball of radius 1 and verifies: R IR n ρ ( ζ ) dλ ( ζ ) = 1 where dλ is the Lebesgue measure on IR n .Let ρ ǫ ( ζ ) := ǫ n ρ ( ζǫ ) be the associatef family of regularizing functions in D (IR n ) so that ρ ǫ has its support in the ball of radius ǫ and verifies too R IR n ρ ǫ ( ζ ) dλ ( ζ ) = 1.As w is harmonic in Ω, for every z ∈ Ω, w ( z ) coincide with its mean-valueon every Euclidean sphere of center z and radius r < d ( z, ∂ Ω). Thereforeusing Fubini’s theorem we get for every ǫ < d ( z, ∂ Ω):(9) w ( z ) = Z IR n w ( z + ζ ) ρ ǫ ( ζ ) dλ ( ζ ) = Z IR n w ( ζ ) ρ ǫ ( z − ζ ) dλ ( ζ ) . i.e. w = w ⋆ ρ ǫ on Ω ǫ := { z | d ( z, ∂ Ω) < ǫ } (in which ⋆ represents aconvolution product) . 5esting w as a distribution on the test function (in the variable ζ ): ρ ǫ ( z − ζ )with ǫ < d ( z, ∂ Ω) ≤
1, equation (9) becomes:(10) w ( z ) = < w ( ζ ) , ρ ǫ ( z − ζ ) > ζ . As w is a distribution of order k , we have for every function φ ∈ D (IR n ) aninequality:(11) | < w, φ > | ≤ C ( w ) sup ζ ∈ IC n Σ | α |≤ k | D αζ φ ( ζ ) | , in which C ( w ) > w .Taking φ ( ζ ) = ρ ǫ ( z − ζ ), we get:(12) | w ( z ) | ≤ C ( w ) sup | ζ |≤ ǫ Σ | α |≤ k | D αζ ρ ǫ ( z − ζ ) | , and:(13) | w ( z ) | ≤ C ( w ) ǫ − n − k , for some constant C ( w ) > ǫ < d ( z, ∂ Ω), we take the limit as ǫ → d ( z, ∂ Ω) andwe get;(14) | w ( z ) | ≤ C ( w ) [ d ( z, ∂ Ω)] − l with l = n + k and then:(15) Z Ω | w | [ d ( z, ∂ Ω)] l dλ < ∞ . If we now assume that w ∈ H − s (IR n ) for a given s >
0, equation (10)becomes:(16) | w ( z ) | = | < w ( ζ ) , ρ ǫ ( z − ζ ) > ζ | ≤ || w || H − s ( IR n ) || ρ ǫ ( z − ζ ) || H s ( IR n ) . Let k be the integer defined by s ≤ k < s + 1 so that (denoting as usual byˆ φ the Fourier transform of φ ):(17) || φ || H s ( IR n ) = Z IR n (1+ | ξ | ) s | ˆ φ ( ξ ) | dλ ( ξ ) ≤ Z IR n (1+ | ξ | ) k | ˆ φ ( ξ ) | dλ ( ξ ) = || φ || H k ( IR n ) As k is an integer, the norm || φ || H k ( IR n ) is equivalent to the sum of the L norms of the derivatives of φ of order less or equal to k , we have:(18) || φ || H k ( IR n ) ≤ C ( k ) Z IR n X | α |≤ k | D α φ | dλ
6e replace φ by φ ǫ ( ζ ) := ǫ n φ ( ζǫ ) (with ǫ ≤
1) so that we get:(19) || φ ǫ || H k ( IR n ) ≤ C ( k ) ǫ − k − n [ Z IR n X | α |≤ k | D α φ | dλ ]Using (17), (18) and (19) with φ ( ζ ) = ρ ( z − ζ ) (for a fixed z ∈ Ω with ǫ < d ( z, ∂ Ω) ≤
1) we finally obtain:(20) | w ( z ) | ≤ C ( k, n ) || w || H − s ( IR n ) ǫ − k − n and when ǫ → d ( z, ∂ Ω):(21) | w ( z ) | ≤ C ( k, n ) || w || H − s ( IR n ) [ d ( z, ∂ Ω)] − k − n (22) Z Ω | w ( z ) | [ d ( z, ∂ Ω)] k + n dλ ( z ) < + ∞ . Remark 1.
Instead of using mean properties of harmonic functions, onecan also use the elementary solution of ∆ in IR n as in [KS1996] proposition10.1., p. 53.We also need the following theorem of L. Schwartz (in his book on dis-tribution theory [Schw1950]). We can also directly use theory of Sobolevspaces. We say that a measure µ defined on an open bounded subset Ω ofIR n , is of polynomial growth at most l in Ω, if R Ω d ( z, ∂ Ω) l d | µ | ( z ) < + ∞ . Theorem 6.
A measure of polynomial growth l defined on an open boundedsubset Ω of IR n can be extended as a distribution on IR n of order at most l .Moreover if w ∈ L (Ω , loc ) verifies the estimate: R Ω | w ( z ) | [ d ( z, ∂ Ω)] l dλ ( z ) < + ∞ , with l ∈ IN, then for every r > l , w can be extended as a distributionin H − r − n ( IR n ) (particularly in H − l − n − ( IR n )) . Remark 2. If R Ω | w ( z ) | [ d ( z, ∂ Ω)] l dλ ( z ) < + ∞ , let us observe that theextension ˜ w of w depends ( a priori ) on the choice of r > l . We will use theresults ot theorem 6 in the case of IC n = IR so that ˜ w can be constructedin H − l − n − . Remark 3. If R Ω | w ( z ) | [ d ( z, ∂ Ω)] l dλ ( z ) < + ∞ , as Ω is bounded, Schwarzinequality implies that R Ω d ( z, ∂ Ω) l | w ( z ) | dλ ( z ) < + ∞ . w defines on Ω ameasure of polynomial growth at most l . Hence the first part of theorem 6implies that w can be extended to IR n as a distribution of order at most l .7 roof. In L. Schwartz’s book there is no proof and no references so that wegive the following proof. We consider the subspace F ⊂ D (IC n ) of functionsthe derivatives of which vanish at the order ≤ l − ζ ∈ ∂ Ω.For a given z ∈ Ω, we choose a point ζ ∈ ∂ Ω such that | z − ζ | = d ( z, ∂ Ω)and we apply Taylor’s formula at the point ζ ∈ ∂ Ω, at the order l − φ ∈ F restricted to the real interval { tz + (1 − t ) ζ | t ∈ IR , ≤ t ≤ } linking in Ω the point ζ ∈ ∂ Ω to z ∈ Ω, so that we obtain:(23) φ ( z ) = l Z (1 − t ) l − (cid:20) Σ | α | = l D α φ ( ζ + t ( z − ζ )) ( z − ζ ) α α ! (cid:21) dt, and then:(24) | φ ( z ) | ≤ C ( l, n ) [ d ( z, ∂ Ω)] l max ξ ∈ ¯Ω [Σ | α | = l | D αξ φ ( ξ ) | ] . For all functions φ ∈ F and all measures µ on Ω of polynomial growth l , i.e R Ω d ( z, ∂ Ω) l d | µ | ( z ) < + ∞ , (using (24)) we have:(25) | Z Ω φ dµ | ≤ C ( l, n ) [ Z Ω d ( z, ∂ Ω) l d | µ | ] max ξ ∈ ¯Ω [Σ | α |≤ l | D αξ φ ( ξ ) | ] . For a given measure µ of polynomial growth l , we consider the space E l (IC n ) of functions of class C l on IC n . We apply Hahn Banach theorem tothe linear form φ → R Ω φ dµ defined on the subspace F ⊂ D (IC n ) ⊂ E l (IC n )and continuous for the seminorm max ξ ∈ ¯Ω [Σ | α |≤ l | D αξ φ ( ξ ) | ]. This linear formcan be extended in a continuous linear form T on E l (IC n ), such that:(26) | < T, φ > | ≤ C ( l, n ) [ Z Ω d ( z, ∂ Ω) l d | µ | ] max ξ ∈ ¯Ω [Σ | α |≤ l | D αξ φ ( ξ ) | ] . for all φ ∈ E l (IC n ), i.e. a distribution of order l on IC n (with compact support).Let us now assume that w ∈ L (Ω , loc ) verifies the estimate:(27) I l ( w ) := Z Ω | w ( z ) | [ d ( z, ∂ Ω)] l dλ ( z ) < + ∞ . for some integer l ≥ φ ∈ F , Cauchy-Schwarz inequality gives:(28) | < w, φ > | = | Z Ω wφ dλ | ≤ I l ( w ) Z Ω | φ ( z ) | [ d ( z, ∂ Ω)] − l dλ ( z ) . Using inequality(24), (28) becomes:(29) | < w, φ > | ≤ C ( l, n, Ω) I l ( w ) [max ξ ∈ ¯Ω Σ | α | = l | D αξ φ ( ξ ) | ] . C ( l, n, Ω) := [ C ( l, n )] R Ω dλ .For every r > l , we use classical Sobolev inequality:(30) max ξ ∈ IR n Σ | α |≤ l | D αξ φ ( ξ ) | ≤ C ( r ) || φ || H r + n and inequality (29), so that we obtain:(31) | < w, φ > | ≤ C ( l, n, r, Ω) [ I l ( w )] || φ || H r + n . with C ( l, n, r, Ω) := C ( r ) [ C ( l, n, Ω)] .Using still Hahn-Banach Theorem for the linear form φ → < w, φ > de-fined on the subspace F of H r + n (IR n ) and continuous for the norm of H r + n , we extend w as a distribution T ∈ H − r − n such that: || T || H − r − n ≤ C ( l, n, r, Ω) [ I l ( w )] (of course we can also do this extension by using or-thogonal projection on the closed subspace ¯ F in the Hilbert space H r + n (IR n )).We can now prove theorem 1. We follow P. Dolbeault’s proof of the Dolbeault-Grothendieck lemma. A.Grothendieck’s proof was different, (in some sense) more elementar thanP. Dolbeault’s proof but not useful for our present purpose. Of course wecan suppose (w.l.o.g.) that f has compact support in IC n (using a cutofffunction in D (IR n ) equal to 1 in a neighborhood of ¯Ω). Let us remind thatIC n being equipped with its usual flat Hermitian metric, the Laplacian actingon differential forms and currents is defined on IC n by:(32) 12 ∆ = 12 ( dd ⋆ + d ⋆ d ) = ¯ ∂ ¯ ∂ ⋆ + ¯ ∂ ⋆ ¯ ∂ = ∂∂ ⋆ + ∂ ⋆ ∂, so that ∆ f is the usual Laplacian on IC n acting on each coefficient of thecurrent f . ¯ ∂ ⋆ (resp. ∂ ⋆ ) (resp. d ⋆ ) is the adjoint of ¯ ∂ (resp. ∂ ) (resp. d := ∂ + ¯ ∂ ) for the same constant metric on IC n (there is no weight function).At first we solve in IC n the Laplacian equation:(33) 12 ∆ v := ( ¯ ∂ ¯ ∂ ⋆ + ¯ ∂ ⋆ ¯ ∂ ) v = ¯ ∂ ⋆ f. ( v and ¯ ∂ ⋆ f are of bidegree ( p, q ).)If we write: f = P ′ | I | = p, | J | = q +1 f I,J dz I ∧ d ¯ z J (Σ ′ means that we only sum onstrictly increasing multi-indices I and J ), we have (cf. [H¨or1966] paragraph4.1, p. 82 or 85 or [Dem2012] Chapter 6, paragraph 6.1) :(34) ¯ ∂ ⋆ f = ( − p − Σ ′ | I | = p, | K | = q (cid:20) Σ j = nj =1 ∂∂z j ( f I,jK ) (cid:21) dz I ∧ d ¯ z K . f is of bidegree (0 , f = Σ j = nj =1 f j d ¯ z j , ¯ ∂ ⋆ f = − Σ j = nj =1 ∂f j ∂z j and (33) is the Laplace equation in IC n : Σ j = nj =1 ∂ ∂z j ∂ ¯ z j v = Σ j = nj =1 ∂f j ∂z j . v is obtained by convolution of each coefficient ∂f I,jK ∂z j of ¯ ∂ ⋆ f in (34) withthe elementar solution E of the usual Laplacian in IC n .We set:(35) g := f − ¯ ∂v As ¯ ∂ = 0, we have (by usual computation):(36) 12 ∆( ¯ ∂v ) = ( ¯ ∂ ¯ ∂ ⋆ + ¯ ∂ ⋆ ¯ ∂ ) ¯ ∂v = ¯ ∂ ¯ ∂ ⋆ ¯ ∂v = ¯ ∂ ( ¯ ∂ ¯ ∂ ⋆ + ¯ ∂ ⋆ ¯ ∂ ) v = ¯ ∂ ( 12 ∆ v )i.e. ¯ ∂ commute with ∆ on IC n . Using (33), we have: ∆( ¯ ∂v ) = ¯ ∂ ¯ ∂ ⋆ f , and:(37) 12 ∆ g = 12 ∆ f −
12 ∆( ¯ ∂v ) = ( ¯ ∂ ¯ ∂ ⋆ + ¯ ∂ ⋆ ¯ ∂ ) f − ¯ ∂ ¯ ∂ ⋆ f = ¯ ∂ ⋆ ¯ ∂ f. Hence (as ¯ ∂f = 0 on Ω), g is harmonic on Ω:(38) ∆ g = 0 .f being of order k with compact support, ¯ ∂ ⋆ f is of order k + 1 with the samesupport (the coefficients of ¯ ∂ ⋆ f are linear combination of derivatives ∂∂z j ofthe coefficients of f ). Therefore the solution v of the Laplacian equation(33) is of order at most k . Indeed it is obtained by convolution: E ⋆ ∂f I,jK ∂z j = ∂E∂z j ⋆f I,jK of each coefficient ∂f I,jK ∂z j of ¯ ∂ ⋆ f in(34) with the elementary solution E := − C n | z | − n +2 of ∆, the first derivatives ∂E∂z j of E are O ( | z | − n +1 ), thenin L (IC n , loc ) and the convolution ∂E∂z j ⋆ f I,jK of a function in L (IR n , loc )with a distribution of order k and compact support is still of order at most k . Hence ¯ ∂v is of order at most k + 1 and g = f − ¯ ∂v is too of order at most k + 1.We write: g = P | I | = p, | J | = q +1 g I,J dz I ∧ d ¯ z J with strictly increasing multi-indices I and J . Let g I,J be a coefficient of g . As g I,J is harmonic in Ω, wecan apply lemma 5 in IR n to g I,J which is a distribution of order at most k + 1, we get an inequality:(39) | g I,J ( z ) | ≤ C (Ω , g I,J ) [ d ( z, ∂ Ω)] − n − k − . Hence :(40) | g ( z ) | ≤ C (Ω , g ) [ d ( z, ∂ Ω)] − l for some constant C (Ω , g ) > l := 2 n + k + 1 (and z ∈ Ω) and then:1041) Z Ω | g | [ d ( z, ∂ Ω)] l dλ < ∞ where dλ is the Lebesgue measure on IC n .L. H¨ormander’s L estimates for ¯ ∂ (corollary 4) provide a solution u inΩ of the equation:(42) ¯ ∂u = g with the L estimate:(43) Z Ω | u | [ d ( z, ∂ Ω)] l dλ < ∞ As Ω is bounded, Cauchy-Schwarz inequality gives the following L es-timate:(44) Z Ω | u | [ d ( z, ∂ Ω)] l dλ < ∞ Therefore a coefficient u I,J of u defines a measure of polynomial growth l on Ω. Using L. Schwartz’s theorem 6, such a measure (of polynomial growth l ) defined on Ω can be extended as a distribution on IC n (of order at most l )so that u can be extended as a current on IC n of order at most l . Then u + v is a current on IC n verifying:(45) ¯ ∂ ( u + v ) = f on Ω. Moreover u + v has order at most l = k + 2 n + 1.We now consider the case of a given f ∈ H − s ( p,q +1) (IC n ) for some s ≥ ∂ ⋆ f ∈ H − s − p,q ) (IC n ). Classicaly we can find a solution v of the Laplaceequation (33) in H − s +1( p,q ) (IC n ) so that g = f − ¯ ∂v is also in H − s ( p,q +1) (IC n ). Weapply lemma 5 to every coefficient g I,J of g in IC n = IR so that | g ( z ) | ≤ C [ d ( z, IC n \ Ω )] − k − n where k is the integer such that s ≤ k < s + 1 andtherefore:(46) Z Ω | g | [ d ( z, ∂ Ω)] k +2 n dλ < + ∞ . Corollary 4 implies we can solve ¯ ∂u = g = f − ¯ ∂v with the estimate:(47) Z Ω | u | [ d ( z, ∂ Ω)] k +2 n dλ < + ∞ . We now apply theorem 6 in IC n = IR to every coefficient of u with l = k + n ( l + n = k + 2 n ) so that for every r > k , u can be extended as acurrent in IC n in H − r − n ( p,q ) (IC n ). As v ∈ H − s +1( p,q ) (IC n ), u + v is too in H − r − n ( p,q ) (IC n )and verifies ¯ ∂ ( u + v ) = f in Ω. 11 emark 4. We have a little more precise result: f = ¯ ∂ ( u + v ) in Ω with u ∈ L ,k + n ( p,q ) (Ω) ∩ H − r − n ( p,q ) (IC n ) (for every r > k particularly for r = k + 1)and v ∈ H − s +1( p,q ) (IC n ). We will use the fact that v ∈ H − s +1( p,q ) (IC n ) has a betterregularity than f ∈ H − s ( p,q ) (IC n ) in section 4. We will now see that theorem 1 remains true for a relatively compact openStein subset Ω of a given Stein manifold X . We can essentially use the samereasoning as in IC n . But we need much stronger technical results.Let us recall that a complex manifold X is Stein if, by definition, globalholomophic functions O ( X ) separate the points of X , give local holomor-phic coordinates on X and if X is holomorphically convex (for all compact K in X the holomorphic hull ˆ K of K is compact with ˆ K := { x ∈ X | | f ( x ) | ≤ max ξ ∈ K | f ( ξ ) |} )). Let us also remind the two following other characteri-zations of a Stein manifold X of complex dimension n . The first one, acomplex holomorphic manifold X is Stein if and only if it can be imbeddedas a closed complex submanifold of IC + . The second one, X is Stein ifand only if there exists a strictly plurisubharmonic exhaustive function ψ on X of class C (if X is a closed submanifold of IC + , we can take for ψ the restriction to X of the function || x || defined on IC + ).Hence X is a K¨ahlerian manifold [Weil1958] (taking, for instance, the K¨ahlermetric associated with the closed K¨ahler form i∂ ¯ ∂ψ ).It is proved in [Ele75] that if we consider a relatively compact Stein open sub-set Ω of X and the geodesic distance associated with a given K¨ahlerian met-ric on X , then the function: − log d ( z, ∂ Ω) + C (Ω , ψ ) ψ , is strictly plurisub-harmonic in Ω for a constant C (Ω , ψ ) large enough.Therefore using [Dem2012] Chapter VIII, paragraph 6, Theorem 6.1 p. 376and 6.5, p. 378 or [Dem1982] the following result (similar to corollary 4)still holds on a Stein manifold: Theorem 7.
Let Ω be a relatively compact open Stein subset of the Steinmanifold X . We consider on X a given K¨ahler form ω , the geodesic distanceon X associated with ω and for z ∈ Ω the corresponding distance d ( z, ∂ Ω) tothe boundary of Ω . Let us consider a holomorphic Hermitian vector bundle F of rank r on X and currents with values in F . Let k ≥ be a givenreal number. Then for every g ∈ L p,q +1) (Ω , F, loc ) with ¯ ∂g = 0 such that: R Ω | g | [ d ( z, ∂ Ω)] k dλ < + ∞ , there exists u ∈ L p,q ) (Ω , F, loc ) such that: (48) ¯ ∂u = g n Ω and: (49) Z Ω | u | [ d ( z, ∂ Ω)] k dλ ≤ C (Ω , F, k ) Z Ω | g | [ d ( z, ∂ Ω)] k dλ, where dλ = ω n n ! is the positive measure on X defined by the ( n, n ) form ω n n ! ( C (Ω , F, k ) is a constant > only depending on Ω , F and k ). Let us give more details about how to deduce theorem 7 from theorems6.1 and 6.5 in [Dem2012]. At first let us remind Demailly’s theorem 6.5 (forthe sake of simplicity we state it with a little more restrictive assumption):
Theorem 8.
Let ( X, ω ) be a Stein manifold X of complex dimension n witha given K¨ahler metric ω . Let us consider a holomorphic Hermitian vectorbundle F of rank r on X and a IC ∞ function φ on X such that ic ( F )+ i∂ ¯ ∂φ ≥ µ ω where c ( F ) is the curvature form of F and µ > a given constant. Thenfor every g ∈ L n,q +1) ( X, F, loc ) with ¯ ∂g = 0 such that: R X | g | e − φ dV < + ∞ ,there exists u ∈ L n,q ) ( X, F, loc ) such that: (50) ¯ ∂u = g in X and: (51) Z X | u | e − φ dV ≤ µ Z X | g | e − φ dV. where dV = ω n n ! is the positive measure on X defined by the ( n, n ) form ω n n ! . In theorem 6.5 in [Dem2012], F is a line bundle but the result is still validfor a vector bundle : you only need to consider the positivity of the curvatureform ic ( F ) of F in the strong sense of Nakano as explained in [Dem2012](theorem 6.1). For ( p, q )-form (with p = n and values in F ) we consider(n,q)-forms with values in the new vector bundle F N ∧ p T ⋆ ( X ) N ∧ n T ( X ).If now Ω is a relatively compact open Stein subset of X , we can choosea constant C (Ω , F ) such that ic ( F ) + C (Ω , F, ) i∂ ¯ ∂ψ ≥ ω on ¯Ω (in thestrong sense of Nakano). For every C ∞ plurisubharmonic function φ on Ωwe can apply theorem 8 restricted to the Stein manifold Ω and the function C (Ω , F ) ψ + φ so that (as ψ is bounded on Ω) we get an estimate:(52) Z Ω | u | e − φ dV ≤ C (Ω , F ) Z Ω | g | e − φ dV. We can now take φ = − k log d ( z, ∂ Ω) + k C (Ω , ψ ) ψ , (for some k ≥
0) so that(as ψ is bounded on Ω) we get the estimate (49) of theorem 7 (for (p,q) formswith values in F ). The function φ := − k log d ( z, ∂ Ω) + k C (Ω , ψ ) ψ is onlycontinuous on Ω but as φ is stritly plurisubharmonic on the Stein manifoldΩ it can be closely approximated by a family ( φ ǫ ) (0 < ǫ < ǫ ) of C ∞ strictly13lurisubharmonic functions on Ω as explained in [Dem2012] chapter 1, para-graph 5.E. page 42 (Richberg theorem (5.21)) such that φ ≤ φ ǫ ≤ φ + ǫ .At first we obtain the estimate (52) for the functions φ ǫ and a solution u ǫ of (50). Taking the limit as ǫ goes to 0 and using the weak compacity ofthe closed ball of L ,k ( p,q ) ( X, F ), we get (52) for φ and a weak limit u of asubsequence of the family ( u ǫ ).We will only use lemma 5 in a local chart of X (i.e. in IC n ) : we do’ntneed to extend this lemma to the Riemannian Laplacian operator on X withvariable coefficients. Replacing IC n = IR by a complex Riemannian mani-fold X , extension theorem 6 is still valid as it is a local result (alongside theboundary of Ω) using a partition of unity of class C ∞ on X .We will give an appropriate simple extension of theorem 1 in IC n . (proposi-tion 10) which will be enough to be able to work on Stein manifold X andwith an arbitrary holomorphic vector bundle on X . Applying and iteratingthis last result in local charts of X we will reduce the problem to J-P. De-mailly’s estimates for ¯ ∂ (theorem 7).To make short, we set : d Ω ( z ) := d ( z, ∂ Ω). Let us remind that for a given k ∈ IR, we denote by L ,k (Ω) = L ,k (0 , (Ω) the space of functions u ∈ L (Ω , loc )such that R Ω | u | [ d Ω ( z )] k dλ < + ∞ and we set: || u || k := R Ω | u | [ d Ω ( z )] k dλ .We need the following preliminary lemma. Lemma 9.
Let Ω be a bounded open subset of IR n and k ∈ IN be given.Then for every w ∈ L ,k (Ω) there exists a solution v ∈ L ,k + n − (Ω) of theequation: ∆ v = ∂∂x w or of the equation : ∆ v = w such that : || v || k + n − ≤ C (Ω , k, n ) || w || k . We will apply lemma 9 in IC n = IR , so that v ∈ L ,k +2 n − (Ω) and wewill consider the equations : ∆ v = ∂∂x j w , 1 ≤ j ≤ n (we only refer to theequation ∆ v = w for completeness). Proof.
We can suppose that
Diam Ω <
1. According to theorem 6 remark 3, w can be extended to IR n as a distribution of order at most k with compactsupport that we denote by ˜ w . We suppose that the support of ˜ w is a subsetof an open ball B of IR n and we set L := ¯ B .The solution v is given by the convolution with the elementary solution E = − c n | x | − n +2 of the Laplacian in IR n , v := E ⋆ ( ∂∂x ˜ w ) = ( ∂∂x E ) ⋆ ˜ w (resp. v := E ⋆ ˜ w ). Let us set K := ∂∂x E = ( n − c n | x | − n +1 (resp. K = E ).We will need the fact that K ∈ L loc (IR n ) and is of class C ∞ outside 0. Weonly have to prove that v | Ω verifies the right estimate : v ∈ L ,k + n − (Ω).Let ψ be a cutoff function in D (IR n ) such that 0 ≤ ψ ≤ ψ = 1 in aneighborhood of the closed ball ¯ B (0 , B (0 ,
2) and such that ψ ( x ) = ψ ( − x ) for all x ∈ IR n . We can split v into v := K ⋆ ˜ w = ( ψK ) ⋆ ˜ w + [(1 − ψ ) K ] ⋆ ˜ w . (1 − ψ ) K ∈ C ∞ (IR n ) and ˜ w
14s a distribution with compact support in IR n , then [(1 − ψ ) K ] ⋆ ˜ w is inC ∞ (IR n ) and is bounded on the bounded subset ¯Ω. Therefore we only haveto verify that v ′ := ( ψK ) ⋆ ˜ w is in L ,k + n − (Ω). Henceforth we replace K by ˜ K := ψK . ˜ K has its support in the ball B (0 , K ( x ) = ˜ K ( − x ) for all x ∈ IR n and we only have to consider v ′ := ˜ K ⋆ ˜ w instead of v .The idea of the proof is to locally use in Ω the classical L inequality forconvolution || ˜ K ⋆ w || L ≤ || ˜ K || L || w || L in IR n (of course after truncating w by an appropriate cutoff function) (cf. [H¨or1983] Corollary 4.5.2. p. 117).Close to the boundary ∂ Ω we get another estimate, using the extension ˜ w of w as a distribution in IR n . It will give us a polynomial gap [ d Ω ( z )] − n +1 . Wehave to prove the following inequality (using duality between L ,k + n − (Ω)and L , − k − n +1 (Ω)):(53) | < ˜ K ⋆ ˜ w, φ > | ≤ C (Ω , ˜ K, n ) || w || k || φ || − k − n +1 for all φ ∈ D (Ω) or equivalently:(54) | < ˜ K ⋆ φ, ˜ w > | ≤ C (Ω , ˜ K, n ) || w || k || φ || − k − n +1 for all φ ∈ D (Ω) (we have < T ⋆ S, φ > = < ˇ T ⋆ φ, S > for S , T in E ′ (IR n ),ˇ T ( x ) := T ( − x ), we take T = ˜ K which is symmetric and S = ˜ w ).For a given ǫ >
0, let us define Ω ǫ := { x ∈ Ω | d Ω ( x ) > ǫ } , K ǫ := { x ∈ Ω | d Ω ( x ) ≥ ǫ } , V ǫ := { x ∈ Ω | ǫ < d Ω ( x ) < ǫ } so that for all x and y in V ǫ we have the inequality:(55) 12 d Ω ( y ) ≤ d Ω ( x ) ≤ d Ω ( y ) . Of course we will later take ǫ = 2 − j , j ∈ IN and fulfill Ω with the exhaustivefamily of sets V ǫ . For a given subset A of IR n we denote by χ A be thecharacteristic function of A : χ A ( x ) = 1 if x ∈ A , χ A ( x ) = 0 if x / ∈ A .We define ψ ǫ = ρ ǫ ⋆ χ K ǫ . ψ ǫ ∈ D (Ω) is a function such that 0 ≤ ψ ǫ ≤ ψ ǫ = 1 on a neighborhood of K ǫ := { x ∈ Ω | d Ω ( x ) ≥ ǫ } , Supp ψ ǫ ⊂ Ω ǫ and | D α ψ ǫ | ≤ C ( α ) ǫ −| α | for all multi-indices α ∈ N n where C ( α ) is aconstant only dependent on α . Indeed we have : ψ ǫ ( y ) = R K ǫ ρ ǫ ( y − x ) dλ ( x )with ρ ǫ ( x ) = ǫ n ρ ( xǫ ) so that D α ψ ǫ ( y ) = ǫ n + | α | R K ǫ ( D α ρ )( y − xǫ ) dλ ( x ) and | D α ψ ǫ ( y ) | ≤ ǫ | α | R R n | D α ρ | ( x ) dλ ( x ).We also set : ψ ′ ǫ = 1 − ψ ǫ so that ψ ǫ + ψ ′ ǫ = 1 on Ω. Supp ψ ′ ǫ ⊂ { x ∈ Ω | d Ω ( x ) < ǫ } so that Supp ψ ′ ǫ ∩ K ǫ = ∅ . ǫ being fixed, let us assume at first that Supp φ ⊂ V ǫ . We set: I := < ˜ K ⋆ φ, ψ ǫ ˜ w > and I := < ˜ K ⋆ φ, ψ ′ ǫ ˜ w > so that we have:(56) | < ˜ K ⋆ φ, ˜ w > | = | I + I | ≤ | I | + 2 | I | with I := < ˜ K ⋆ φ, ψ ǫ ˜ w > = < ψ ǫ ( y ) R x ∈ Suppφ ˜ K ( y − x ) φ ( x ) dλ ( x ) , ˜ w > y and I := < ˜ K ⋆ φ, ψ ′ ǫ ˜ w > = < ψ ′ ǫ ( y ) R x ∈ Suppφ ˜ K ( y − x ) φ ( x ) dλ ( x ) , ˜ w > y .15s Supp ψ ǫ ⊂ Ω we have ψ ǫ ˜ w = ψ ǫ w ( w is in L ,k (Ω) and ˜ w is an extensionof w as a distribution in D ′ (IR n )),and then I := < ˜ K ⋆ φ, ψ ǫ ˜ w > = < ˜ K ⋆ ( ψ ǫ w ) , φ > . We use Cauchy-Schwarzinequality and convolution inequality : | I | ≤ || ˜ K ⋆ ( ψ ǫ w ) || || φ || ≤ || ˜ K || || ψ ǫ w || || φ || . | I | ≤ || ˜ K || || ψ ǫ w || || φ || ≤ || ˜ K || ( ǫ ) − k || ψ ǫ w || ,k (2 ǫ ) k + n − || φ || , − k − n +1 .Indeed as d Ω ( x ) ≥ ǫ on the support of ψ ǫ , we have: ( ǫ ) k || ψ ǫ w || ≤|| ψ ǫ w || ,k and as d Ω ( x ) ≤ ǫ on V ǫ which contains the support of φ , wehave: (2 ǫ ) − k + n − || φ || ≤ || φ || , − k − n +1 .Finally we get:(57) | I | ≤ k +2 n − ǫ n − || ˜ K || || w || ,k || φ || , − k − n +1 for every φ with Supp φ ⊂ V ǫ .We now consider the term I . As ˜ w is a distribution of order at most k and has in support in the ball L , ˜ w satisfies an inequality :(58) | < ˜ w, ψ > | ≤ C ( ˜ w ) sup y ∈ L Σ | α |≤ k | D αy ψ ( y ) | , for all ψ ∈ C l (IR n ) with C ( ˜ w ) := C (Ω , L ) || w || k using theorem 6 remark3 (replacing L by a compact neighborhood of L if necessary).We apply inequality (58) to the function ψ ( y ) := R x ∈ Supp φ ψ ′ ǫ ( y ) K ( y − x ) φ ( x ) dλ ( x ) and we take the derivative in the variable y under the sym-bol R so that we get :(59) | I ) | ≤ C ( ˜ w ) max y ∈ L \ K ǫ Σ | α |≤ k | Z x ∈ Supp φ D αy [ ψ ′ ǫ ( y ) ˜ K ( y − x )] φ ( x ) dλ ( x ) | Let us observe that
Supp ψ ′ ǫ ⊂ IR n \ K ǫ .For x ∈ Supp φ ⊂ V ǫ and y ∈ IR n \ K ǫ we have | x − y | ≥ ǫ > d Ω ( x ) (thereexists z ∈ ∂ Ω such that d Ω ( y ) = | y − z | < ǫ , then | x − z | ≥ d Ω ( x ) > ǫ sothat | x − y | ≥ | x − z | − | y − z | > ǫ − ǫ = ǫ ) and then | D βy ˜ K ( y − x ) | = O ( | x − y | −| β |− n +1 ) = O ([ d Ω ( x )] −| β |− n +1 ). We also have | D γy ψ ′ ǫ ( y ) | = O ( ǫ −| γ | ) = O ([ d Ω ( x )] −| γ | ) so that (as we have | β | + | γ | = | α | ) :(60) | D αy [ ψ ′ ǫ ( y ) ˜ K ( y − x )] | ≤ C (Ω)[ d Ω ( x )] −| α |− n +1 (59) and (60) imply:(61) | I ) | ≤ C (Ω , ˜ w ) Z x ∈ Supp φ [ d Ω ( x )] − k − n +1 | φ ( x ) | dλ ( x )Using the Cauchy-Schwarz inequality we have:(62) | I | ≤ [ C (Ω , ˜ w )] ( λ ( Supp φ )) Z x ∈ Supp φ [ d Ω ( x )] − k − n +2 | φ ( x ) | dλ ( x )16nd then:(63) | I | ≤ [ C (Ω , ˜ w )] ( λ ( V ǫ )) || φ || , − k − n +1 (as Supp φ ⊂ V ǫ and as || φ || , − k − n +1 := R x ∈ V ǫ [ d Ω ( x )] − k − n +2 | φ ( x ) | dλ ( x )).Using inequalities (56), (57) (for I ) and (63) (for I ) we get:(64) | < K⋆ ˜ w, φ > | ≤ C (Ω , ˜ w )[( λ ( V ǫ )) + ǫ n − ] Z x ∈ V ǫ [ d Ω ( x )] − k − n +2 | φ ( x ) | dλ ( x )for all φ with Supp φ ⊂ V ǫ and therefore:(65) Z V ǫ | K ⋆ ˜ w | [ d Ω ( x )] k +2 n − dλ ≤ C (Ω , ˜ w )[( λ ( V ǫ )) + ǫ n − ]with C (Ω , ˜ w ) := 2[ C (Ω , ˜ w )] + 22 k +2 n − || ˜ K || k || w || ,k ≤ C ( k, n, Ω) || w || ,k .As [ λ ( V ǫ )] ≤ λ ( V ǫ )], taking ǫ = 2 − j , j ∈ IN, summing on j and settingΩ ′ := S j ∈ IN V − j we have:(66) Z Ω ′ | K ⋆ ˜ w | [ d Ω ( x )] k +2 n − dλ ≤ C (Ω , ˜ w )[ λ (Ω ′ ) + 2] , Taking ǫ = − j , j ∈ IN, summing on j and setting Ω ′′ := S j ∈ IN V − j wehave:(67) Z Ω ′′ | K ⋆ ˜ w | [ d Ω ( x )] k +2 n − dλ ≤ C (Ω , ˜ w )[ λ (Ω ′′ ) + 2] , Giving j ∈ IN, the set K − j \ Ω − j := { x ∈ (Ω) | d Ω ( x ) = 2 − j } is a subset ofΩ ′′ so that Ω ⊂ Ω ′ ∪ Ω ′′ . Summing (66) and (67) we finally have :(68) Z Ω | K ⋆ ˜ w | [ d Ω ( x )] k +2 n − dλ ≤ C (Ω , ˜ w )[ λ (Ω) + 2] , i.e. || v ′ || k + n − ≤ C (Ω , ˜ w )[ λ (Ω) + 2].We can now prove the following slight extension of theorem 1: the aimof which is to iterate the construction made in IC n . Proposition 10.
Let Ω be a bounded Stein open subset of IC n and f bea given current of bidegree ( p, q + 1) on IC n with compact support which is ¯ ∂ -closed on Ω and can be written f = g + h with g | Ω ∈ L ,k ( p,q +1) (Ω) and h ∈ H − s ( p,q +1) ( IC n ) for some s ≤ k . Then there exists a current w = u + v ofbidegree ( p, q ) (with compact support) in IC n such that: (69) ¯ ∂w = f,in Ω ,with w = u + v ∈ H − k − n − p,q ) ( IC n ) , u | Ω ∈ L ,k +4 n +1( p,q ) (Ω) and v ∈ H − s +1( p,q ) ( IC n ) ). v has a better regularity than h . u has still a poly-nomial L growth in Ω as g (even if it is larger than that of g ). Proof.
Using extension theorem 6 (remark 2), we can assume that g andtherefore f are in H − k − n − p,q +1) (IC n ). At first we have to solve:(70) 12 ∆ v = ¯ ∂ ⋆ f = ¯ ∂ ⋆ g + ¯ ∂ ⋆ h. We solve separately the equations ∆ v = ¯ ∂ ⋆ g in Ω and ∆ v == ¯ ∂ ⋆ h in IC n . Using lemma 9 (in IR n ) we can at first find v ∈ L ,k +2 n − p,q ) (Ω)and then using theorem 6 (remark 2) we can find an extension of v toIC n which is a distribution in H − k − n (IC n ). We set v := v + v . As h ∈ H − s (IC n ), v is in H − s +1 (IC n ) so that v := v + v ∈ H − k − n ( p,q ) (IC n ). f − ¯ ∂v ∈ H − k − n − p,q +1) (IC n ) is harmonic in Ω and therefore (using lemma 5) f − ¯ ∂v is in L ,k +4 n +1( p,q +1) (Ω). Finally corollary 4 gives a solution u ∈ L ,k +4 n +1( p,q +1) (Ω) of theequation ¯ ∂u = f − ¯ ∂v in Ω. Setting w = u + v , we get ¯ ∂w = f in Ω. Usingstill theorem 6 (remark 2) we get an extension of u in H − k − n − p,q ) (IC n ) so that w ∈ H − k − n − p,q ) (IC n ).We need the following lemma making comparisons between the severaldistances to boundary we have to consider. Lemma 11.
Let Ω and Ω j be two bounded open subsets of the complexRiemannian manifold X such that Ω ∩ Ω j = ∅ and u ∈ L ,k ( p,q ) (Ω) . Then: (71) Z Ω ∩ Ω j | u | [ d Ω ∩ Ω j ( z )] k dλ ≤ Z Ω | u | [ d Ω ( z )] k dλ. If ψ j ∈ D (Ω j ) , and u ∈ L ,k ( p,q ) (Ω ∩ Ω j ) then: (72) Z Ω | ψ j u | [ d Ω ( z )] k dλ ≤ ǫ − kj Z Ω ∩ Ω j | ψ j u | [ d Ω ∩ Ω j ( z )] k dλ, and (73) Z Ω | ¯ ∂ψ j ∧ u | [ d Ω ( z )] k dλ ≤ ǫ − kj Z Ω ∩ Ω j | ¯ ∂ψ j ∧ u | [ d Ω ∩ Ω j ( z )] k dλ, with ǫ j := min z ∈ Supp ψ j d Ω j ( z ) . Let us observe that
Supp ( ψ j u ) ⊂ Ω ∩ Ω j and Supp ( ¯ ∂ψ j ∧ u ) ⊂ Ω ∩ Ω j so that integrating ψ j u (resp. ¯ ∂ψ j ∧ u ) on Ω or on Ω ∩ Ω j gives the sameresult but we will need to consider later this extension of ψ j u to Ω (by zerooutside its support). 18 roof. For z ∈ Ω ∩ Ω j , we have: d Ω ∩ Ω j ( z ) = min( d Ω ( z ) , d Ω j ( z )) ≤ d Ω ( z ) andtherefore:(74) Z Ω ∩ Ω i | u | [ d Ω ∩ Ω i ( z )] k dλ ≤ Z Ω | u | [ d Ω ( z )] k dλ On the other hand, we define ǫ j := min z ∈ Supp ψ j d Ω j ( z ) = min z ∈ Supp ψ j d ( z, ∂ Ω j )(0 < ǫ j ≤
1) so that for z ∈ Supp ψ j , we have: d Ω ∩ Ω j ( z ) = min( d Ω ( z ) , d Ω j ( z )) ≥ min( d Ω ( z ) , ǫ j ) ≥ ǫ j d Ω ( z ) (as d Ω ( z ) ≤
1) and then:(75) Z Ω ∩ Ω j | ψ j u | [ d Ω ( z )] k dλ ≤ ǫ − kj Z Ω ∩ Ω j | ψ j u | [ d Ω ∩ Ω j ( z )] k dλ and :(76) Z Ω ∩ Ω j | ¯ ∂ψ j ∧ u | [ d Ω ( z )] k dλ ≤ ǫ − kj Z Ω ∩ Ω j | ¯ ∂ψ j ∧ u | [ d Ω ∩ Ω j ( z )] k dλ. We can now prove the following result by the same reasoning as in thecase of IC n . Moreover we consider currents in D ′ ( p,q ) ( X, F ) with values in agiven holomorphic vector bundle F (to simplify we only consider current in H − k ( X, F ), k ∈ IN).
Theorem 12.
Let Ω be a relatively compact open Stein subset of a Steinmanifold X and F be a given Hermitian holomorphic vector bundle on X .Then for every current f of bidegree ( p, q + 1) on X with values in F (andwith compact support in X ) which is ¯ ∂ -closed on Ω , there exists a current w of bidegree ( p, q ) on X with values in F (with compact support) such that: (77) ¯ ∂w = f, in Ω . Moreover if f is in H − k ( p,q +1) ( X, F ) for some k ∈ IN, we can find asolution w in H − n − − r ( p,q ) ( X, F ) with r = k (4 n + 2) − n − , more precisely w = u + v , u | Ω ∈ L ,r ( p,q ) (Ω , F ) , u ∈ H − n − − r ( p,q ) ( X, F ) and v ∈ H − k +1( p,q ) ( X, F ) .Proof. Let us assume that f ∈ H − k ( p,q +1) ( X, F, loc ). We will prove that wecan find a solution u ∈ H − n − − r ( p,q ) ( X, F, loc ).By considering local charts of X , we will locally reduce the problem to thecase of IC n and using a partition of the unity, we will patch together theselocal solutions to obtain a first approximate global solution ˜ w such that f := f − ¯ ∂ ˜ w has in some sense better regularity than f and then weiterate the construction (replacing f by f ) until f becomes enough regularso that we can use J-P. Demailly’s L -estimate for ¯ ∂ (i.e. until we havefound in the temperate cohomology class of f a new current f such that19 | Ω ∈ L ,k + r ( p,q +1) (Ω , F ) for some r large enough).Using local charts on X and Borel-Lebesgue lemma, we can find a finiteopen covering of the compact set ¯Ω by relatively compact open subsets Ω j of X , 1 ≤ j ≤ N such that every ¯Ω j is contained in a geodesic chart forthe given Riemannian metric and every Ω j is biholomorphic to a boundedopen ball U j := B j ( z j , r j ) of IC n , by a local biholomorphic map φ j definedon a neighborhood of ¯Ω j ⊂ X and taking its values into IC n ( z j ∈ φ j ( ¯Ω), r j > j ) thatthe exponential map sending the tangent space T z j X (of X at z j ) into X isa diffeomorphism of a open ball in T z j X onto a geodesic open ball of center z j containing ¯Ω j so that the geodesic distance and the Euclidian distancecoming from IC n (by means of φ j ) are equivalent on a neighborhood of ¯Ω j and so that the spaces L ,k ( p,q ) (Ω j ∩ Ω , F ) ( k ∈ IN) associated with the geodesicdistance to ∂ (Ω j ∩ Ω) or with the Euclidian distance to ∂ (Ω j ∩ Ω) comingfrom IC n (by means of φ j ) are the same. Finally we can also suppose thatthe given holomorphic vector bundle F is trivial on a neighborhood of each¯Ω j .For every bounded Stein open subset φ j (Ω ∩ Ω j ) ⊂ φ j (Ω j ) =: U j ⊂ IC n weuse the construction made in the case of IC n (i.e. remark 4, F is trivial on aneighborhood of ¯Ω j ) so that we can construct u j ∈ H − k − n − p,q ) (Ω j , F ) with u j | Ω ∩ Ω j ∈ L ,k + n ( p,q ) (Ω j ∩ Ω , F ) and (it is the key point) v j ∈ H − k +1( p,q ) (Ω j , F )such that w j := u j + v j is a solution of f = ¯ ∂w j = ¯ ∂ ( u j + v j ) in Ω ∩ Ω j and w j ∈ H − k − n − p,q ) (Ω j , F ).Let ψ j ∈ D (Ω j )) ≥ P j = Nj =1 ψ j = 1 on aneighborhood of ¯Ω. Then ψ j w j ∈ H − k − n − p,q ) ( X, F ) and a key point is that ψ j u j | Ω ∈ L ,k + n ( p,q ) (Ω , F ) for the distance d Ω ( z ) using lemma 11 inequality (72)(we apply with u = u j ) and then for i = j , ψ j u j | Ω ∩ Ω i ∈ L ,k + n ( p,q ) (Ω ∩ Ω i , F ) forthe distance d Ω ∩ Ω i ( z ) using lemma 11 inequality (71) (we apply with u = ψ j u j and to Ω i instead of Ω j ). Particularly ( P j = Nj =1 ψ j u j ) | Ω ∈ L ,k + n ( p,q ) (Ω , F )for the distance d Ω ( z ) and for all 1 ≤ i ≤ N , ( P j = Nj =1 ψ j u j ) | Ω ∩ Ω i ∈ L ,k + n ( p,q ) (Ω ∩ Ω i , F ) for the distance d Ω ∩ Ω i ( z ).Gluing together the local solutions w j , we define ˜ w = P j = Nj =1 ψ j w j =˜ u +˜ v , with ˜ u = P j = Nj =1 ψ j u j and ˜ v = P j = Nj =1 ψ j v j so that ˜ w ∈ H − k − n − p,q ) ( X, F ),˜ u | Ω ∈ L ,k + n ( p,q ) (Ω , F ), ˜ v ∈ H − k +1( p,q ) ( X, F ) and we obtain:(78) ¯ ∂ ˜ w = j = N X j =1 ψ j ¯ ∂w j + j = N X j =1 ( ¯ ∂ψ j ) ∧ w j . We have: ¯ ∂w j = f in Ω ∩ Ω j and Supp ψ j ⊂ Ω j so that equation (78) implies20n Ω :(79) ¯ ∂ ˜ w = ( j = N X j =1 ψ j ) f + j = N X j =1 ¯ ∂ψ j ∧ w j , or (in Ω) :(80) ¯ ∂ ˜ w = f + j = N X j =1 ¯ ∂ψ j ∧ w j . We set: f := − P j = Nj =1 ¯ ∂ψ j ∧ w j , g := − P j = Nj =1 ¯ ∂ψ j ∧ u j and h := − P j = Nj =1 ¯ ∂ψ j ∧ v j , so that we obtain:(81) f = f − ¯ ∂ ˜ w = g + h .f is ¯ ∂ -closed on Ω and we have f := g + h with g ∈ H − k − n − p,q +1) ( X, F )and h ∈ H − k +1( p,q +1) ( X, F ) (as each v j ∈ H − k +1( p,q ) (Ω j , F )). Moreover accordingto lemma 11, g | Ω ∈ L ,k + n ( p,q +1) (Ω , F ) for the distance d Ω ( z ) (inequality (73))and for all i , g | Ω ∩ Ω i ∈ L ,k + n ( p,q +1) (Ω ∩ Ω i , F ), for the distance d Ω ∩ Ω i ( z ) (in-equality (71) applied in Ω i to each form ¯ ∂ψ j ∧ u j of bidegre (p,q+1) with j = i ). Let us observe that g or h are not necessarly ¯ ∂ -closed (that isthe difficulty). The key point is that f is better than f in the sense that h ∈ H − k +1( p,q +1) ( X, F ) has a best regularity than f ∈ H − k ( p,q +1) ( X, F ) and g has a L polynomial growth in Ω.We have built a first approximate global solution ˜ w such that f := f − ¯ ∂ ˜ w is better than f . Applying proposition 10 and lemma 11 to each Stein opensubset φ j (Ω ∩ Ω j ) ⊂ U j ⊂ IC n , we can iterate this construction and constructa finite sequence of currents on X : f l +1 = g l +1 + h l +1 = f l − ¯ ∂ ˜ w l +1 , l ∈ IN, f := f , such that h l +1 ∈ H − k + l +1( p,q +1) ( X, F ) has a strictly better regularitythan h l ∈ H − k + l ( p,q +1) ( X, F ) and g l +1 | Ω ∈ L ,k + n + l (4 n +1)( p,q +1) (Ω , F ) has still a L polynomial growth in Ω, moreover g l +1 ∈ H − k − n − − l (4 n +1)( p,q +1) ( X, F ). Given f l = g l + h l , we apply proposition 10 to each φ j (Ω ∩ Ω j ) ⊂ IC n , we constructa solution w l +1 ,j = u l +1 ,j + v l +1 ,j of ¯ ∂w l +1 ,j = f l in Ω ∩ Ω j ⊂ X (Ω ∩ Ω j isbiholomorphic to φ j (Ω ∩ Ω j )).We set ˜ w l +1 := P j = Nj =1 ψ j w l +1 ,j = ˜ u l +1 + ˜ v l +1 with ˜ u l +1 := P j = Nj =1 ψ j u l +1 ,j ,˜ v l +1 := P j = Nj =1 ψ j v l +1 ,j and then f l +1 := f l − ¯ ∂ ˜ w l +1 = − P j = Nj =1 ¯ ∂ψ j ∧ w l +1 ,j = g l +1 + h l +1 , with g l +1 := − P j = Nj =1 ¯ ∂ψ j ∧ u l +1 ,j and h l +1 := − P j = Nj =1 ¯ ∂ψ j ∧ v l +1 ,j .The estimates we need to iterate are straightforward consequences of propo-sition 10 (we use replacing k by k + n + ( l − n + 1) and s by k − l atthe step l ≥
1) and lemma 11. Of course the polynomial L growth of g l +1
21s larger than that of g l but it does’nt matter as we are able to solve the ¯ ∂ equation in Ω for any L polynomial growth.Finally for l = k (it is the last key point) h k ∈ H p,q +1) ( X, F ) = L , p,q +1) ( X, F ),particularly h k | Ω ∈ L , p,q +1) (Ω , F ), so that f k verifies the L -estimate weneed: f k = g k + h k has polynomial L growth in Ω : R Ω | f k | [ d Ω ( z )] r dλ < + ∞ with r = k + n +( k − n +1) = k (4 n +2) − n −
1. We can now use J-P.Demailly’s theorem 7 to solve in Ω the equation ¯ ∂ ˜ w k +1 = f k with ˜ w k +1 ∈ L ,r ( p,q ) (Ω , F ) so that we obtain: f = ¯ ∂ ˜ w with ˜ w := ˜ w k +1 + Σ l = kl =1 ˜ w l = u + v , u := ˜ w k +1 + Σ l = kl =1 ˜ u l and v := Σ l = kl =1 ˜ v l , u ∈ L ,r ( p,q ) (Ω , F ), v ∈ H − s +1( p,q ) ( X, F ).Using theorem 6 we get an extension of u in H − n − − r ( p,q ) ( X, F ) so that ˜ w ∈ H − n − − r ( p,q ) ( X, F ).As explained in P. Schapira’s article [Scha2020] (remark 2.3.4), theorem12 implies the following result (theorem 2.3.3 in [Scha2020]). We refer to[Scha2020] for the definitions of the (derived) sheave O tp X sa of temperateholomorphic functions (defined on the subanalytic site X sa ) and of otherobjects associated with. Theorem 13. (P. Schapira) Let X be a complex Stein manifold and let Ωbe a subanalytic relatively compact Stein open subset of X contained in aStein compact subset K of X . Let F be a coherent O X -module defined ona neighborhood of K . Then RΓ(Ω; F tp ) is concentrated in degree 0.[Dem1982] Demailly, J.P., Estimations L pour l’op´erateur ¯ ∂ d’un fibr´evectoriel holomorphe semi-positif au-dessus d’une vari´et´e k¨ahl´erienne compl`ete in Annales scientifiques de l’ ´Ecole Normale Sup´erieure , 457-511, (1982).[Dem2012] Demailly, J.P., Complex Analytic and Differential Geometry.
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