Norm estimates for the $\bar\partial$-equation on a non-reduced space

We study norm-estimates for the $\bar\partial$-equation on non-reduced analytic spaces. Our main result is that on a non-reduced analytic space, which is Cohen-Macaulay and whose underlying reduced space is smooth, the $\bar\partial$-equation for $(0,1)$-forms can be solved with $L^p$-estimates.


Introduction
Various estimates for solutions of the ∂-equation on a smooth complex manifold are known since long ago.The paramount methods are the L 2 -methods, going back to Hörmander, Kohn, and others, and integral representation formulas, first used by Henkin and by Skoda.Starting with [28,29] there has been an increasing interest for L 2 -and L p -estimates for ∂ on nonsmooth reduced analytic spaces in later years, see, e.g., [13,19,23,27,31].In [6,15,16] are results about L 2 -estimates of extensions from non-reduced subvarieties.In this paper we try to initiate the study of L p -estimates for the ∂-equation on a non-reduced analytic space.
Let X be an analytic space of pure dimension n with structure sheaf O X .Locally then we have an embedding i : X → U ⊂ C N and a coherent ideal sheaf J ⊂ O U of pure dimension n such that O X = O Ω /J in X ∩ U .In [7] we introduced a notion of smooth (0, * )-forms on X and proved that if the underlying reduced space X red is smooth and in addition O X is Cohen-Macaulay, then there is a smooth solution to ∂u = φ if φ is smooth and ∂φ = 0.More generally we defined sheaves A q X of (0, q)-forms on X that are closed under multiplication by smooth (0, * )-forms and coincides with E 0,q X where X red is smooth and O X is Cohen-Macaulay, such that 0 → O X → A 0 X → • • • → A n X → 0 is a fine resolution of O X .The solutions to the ∂-equation are obtained by intrinsic integral formulas on X. Variants of the ∂-equation on non-reduced spaces have also been studied by Henkin-Polyakov, [20,21].
In [5] was introduced a pointwise norm |•| on forms φ ∈ E 0, * X .That is, |φ(x)| X is non-negative function on X red which vanishes in a neighborhood of a point x 0 if and only if φ vanishes there.It was proved that O X is complete with respect to the topology of uniform convergence on compacts induced by this norm.In this paper we will only discuss spaces where X red is smooth.In [5] is defined an intrinsic coherent left O X red -module N X of differential operators O X → O X red , and the pointwise norm is defined as where L j is a finite set of local generators for N X .Clearly another set of generators will give rise to an equivalent norm.In particular, it becomes meaningful to say that φ vanishes at a point x ∈ X.The norm extends to smooth (0, * )-forms.By a partition of unity we patch together and define a fixed global | • | X .Let us also choose a volume element dV on X red .We define L p 0, * ;X as the (local) completion of E 0, * X with respect to the L p -norm.In the same way we define C 0, * ;X as the completion with respect to the uniform norm.Our main result is Theorem 1.1.Let X be an analytic space such that O X is Cohen-Macaulay and X red is smooth.Assume that 1 ≤ p < ∞.Given a point x there are neighborhoods V ′ ⊂⊂ V ⊂ X and a constant C p such that if φ ∈ L p 0,1 (V) and ∂φ = 0, then there is ψ ∈ L p 0,0 (V ′ ) such that ∂ψ = φ and Moreover, there is a constant C ∞ such that if φ ∈ C 0,1 (V) and ∂φ = 0, then there is a solution ψ ∈ C 0,0 (V ′ ) such that sup By standard sheaf theory, and the fact that ψ ∈ L p 0,0;X and ∂ψ = 0 implies that ψ ∈ O X , see Lemma 4.8 and (5.1), we get the following corollaries.
Notice that φ defines a Čech cohomology class in H 1 (X, O X ) through (ψ j − ψ k ) j,k , where (ψ j ) are local ∂-solutions on a covering U j of X.
The proof of Theorem 1.1 relies on the integral formulas in [7] in combination with a new notion of sheaves of C 0, * X of (0, * )-currents on X which provide a fine resolution of O X (see Section 4).These sheaves should have an independent interest.In Remark 5.4 we give a heuristic argument for Theorem 1.1 which relies on these sheaves but with no reference to integral formulas.
We first consider a certain kind of "simple" non-reduced space for which we prove these L p -estimates for all (0, q)-forms, see Section 9. We then prove the general case by means of a local embedding X → X, where X is simple.To carry out the proof we need comparison results between the constituents in the integral formulas for the two spaces.One of them is provided by [22], whereas another one, for the so-called Hefer mappings, is new, see Section 8.The proof of Theorem 1.1 is in Section 10.Technical difficulties restrict us, for the moment, to the case with (0, 1)-forms.
We have no idea of whether one could prove Theorem 1.1, e.g., in case p = 2, by L 2 -methods.
The assumption that X red be smooth and X be Cohen-Macaulay is crucial in this paper.In the reduced case considerable difficulties appear already with the presence of an isolated singularity; besides the references already mentioned above, see, e.g., [17,18,25,26,32].In the non-reduced case even when X red is smooth, an isolated non-Cohen-Macaulay point offers new difficulties.We discuss such an example in Section 11.Throughout this paper X is a non-reduced space of pure dimension n and the underlying reduced space Z = X red is smooth, if nothing else is explicitly stated.

Some preliminaries
Let Y and Y ′ be complex manifolds and f : Y ′ → Y a proper mapping.If τ is a current on Y , then the push-forward, or direct image, f * τ is defined by the relation f * τ.ξ = τ.f* ξ for test forms ξ.If α is a smooth form on Y , then we have the simple but useful relation In [11,9] was introduced the sheaf of pseudomeromorphic currents on Y .Roughly speaking a pseudomeromorphic current is the direct image under a holomorphic mapping of a smooth form times a tensor product of one-variable principal value current 1/z m j and ∂(1/z m ′ k ).This sheaf is closed under ∂ and under multiplication by smooth forms.If a pseudomeromorphic current τ has support on a subvariety V and the holomorphic function h vanishes on V , then hτ = 0 and d h ∧ τ = 0.This leads to the crucial dimension principle Proposition 2.1.Let τ be a pseudomeromorphic current of bidegree ( * , q), and assume that the support of µ is contained in a subvariety of codimension > q.Then τ = 0.
We say that a current a is almost semi-meromorphic in Y if there is a modification π : Y ′ → Y , such that a is the direct image of a form α/f , where α is smooth and f is a holomorphic section of some line bundle on Y ′ .Assume that µ is pseudomeromorphic, a is an almost semimeromorphic current, χ ǫ = χ(|F | 2 /ǫ), where χ is a smooth cut-off function, and F is a tuple of holomorphic functions such that {F = 0} contains the set where a is not smooth.Then the limit U , see [14], consists of all ∂-closed (N, κ)-currents in U with support on Z that are annihilated by JZ , i.e., by all h where h is in U is defined in the same way, but one needs an additional regularity condition, the so-called standard extension property, SEP, see, e.g., [7,Section 2.1].When Z is smooth, the currents in CH Z U (with the definition given here) admit an expansion as in [3, (3.4)], and so the SEP follows.
Let us recall some properties of residue currents associated to a locally free resolution The precise definitions and claimed results can all be found in [10].Let us denote the complex (2.3) by (E, f ).Assume that the vector bundles E k are equipped with hermitian metrics.The corresponding complex of vector bundles is pointwise exact on U \ Z, where Z = Z(J ).There are associated currents U and R. The current U is almost semi-meromorphic on U and smooth on U \ Z, and takes values in Hom (E, E).The current R is a pseudomeromorphic current on U that takes values in Hom (E 0 , E) and has support on Z.One may write R = k R k , where R k is a (0, k)-current that takes values in Hom (E 0 , E k ).They satisfy the relation Here we use the compact notation Moreover, R is annihilated by JZ , and it satisfies the duality principle (2.6) RΦ = 0 if and only if Φ ∈ J .
We will typically assume that the resolution is chosen to be minimal at level 0, i.e, such that is Cohen-Macaulay, then we can choose the resolution so that N 0 = κ.Then it follows that R consists of the only term R κ that takes values in E κ , and from (2.4) that ∂R κ = 0. We conclude that the components µ 1 , .

Pointwise norm on a non-reduced space X
Recall that X is a non-reduced space of pure dimension n with smooth underlying manifold X red = Z.
Consider a local embedding i : X → U ⊂ C N and assume that π : U → Z ∩ U is a submersion.Possibly after shrinking U we can assume that we have coordinates we define N X as the set of all such local operators L obtained from some µ in Hom(O U /J , CH Z U ) and a local submersion.It follows from (2.1) and (3.1) It is coherent, in particular locally finitely generated, and if L j is a set of local generators, then φ = 0 if and only if L j φ = 0 for all j, see [ Let X be the analytic space with structure sheaf O X = O U /I. Consider the tensor product of currents where dτ j /τ is the principal value current.We recall that if where ).For a multiindex m, we will use the short-hand notation Moreover, m ≤ M means that m j ≤ M j for j = 1, . . ., κ.It is readily verified that Let us now return to the setting of a local embedding i : X → U ⊂ C N as above.Notice that if M is large enough in the example, then I ⊂ J .Let µ 1 , . . ., µ ρ be local generators for the coherent O U -module Hom(O U /J , CH Z U ). Then we have a natural mapping It is natural to say that we have an embedding It is well-known, see, e.g., [4,Theorem 1.5] that there are holomorphic functions γ 1 , . . ., γ ρ (possibly after shrinking U ) such that (3.9) µ j = γ j μ, j = 1, . . ., ρ.
From [5,Theorem 1.4] we have that In this way the norm | • | X is thus expressed in terms of the simpler norm | • | X .
3.1.The norm when X is Cohen-Macaulay.So far we have only used the assumption the Z is smooth.Let us now assume in addition that O X is Cohen-Macaulay.Then one can find monomials 1, τ α 1 , . . ., τ α ν−1 such that each φ in O X has a unique representative where φj are in O Z , see, e.g., [7,Corollary 3.3].In this way O X becomes a free O Z -module (in a non-canonical way).Let | • | X,π be the norm obtained from the subsheaf N X,π of N X , consisting of operators L obtained, cf.(3.1), from the submersion π such that (ζ, τ ) → ζ in U .It turns out that Theorem 1.5].By [5,Proposition 3.4] the whole sheaf N X is generated by N X,π for a finite number of generic submersions π ι .It follows that

3.2.
The sheaf E 0, * X of smooth forms on X. Assume that we have a local embedding i : , where J Z is the radical sheaf of Z and we by X reg denote the set of points of X where Z is smooth and O X is Cohen-Macaulay.Remark 3.2.If the underlying reduced space X red is not smooth, or O X is not Cohen-Macaulay, then this definition of Ker i * is not valid.Instead is used as definition that Φ ∧ µ = 0 for all µ in Hom(O U /J , CH Z U ).However, it is true that i * Φ = 0 if i * Φ = 0 where X red is smooth and We define E 0, * X = E 0, * U /Ker i * and have the natural mapping i * : E 0, * U → E 0, * X .By standard arguments one can check that the O X -module E 0, * X so defined does not depend on the choice of local embedding.
If O X is Cohen-Macaulay and i : X → U is a local embedding with coordinates (ζ, τ ) and a monomial basis τ α ℓ , then we have a unique local representation (3.11) of each φ in E 0, * X with φℓ in E 0, * Z , and the other statements in Section 3.2 hold verbatim, with the same proofs, for smooth (0, * )-forms.

Intrinsic currents on X
In the reduced case one can define currents just as dual elements of smooth forms.In the non-reduced case one has to be cautious because there are two natural kinds of currents; suitable limits of smooth forms and dual elements of smooth forms.We have to deal with both kinds.In this paper, the former type appears as (0, * )-currents, while the latter appears as (n, * )-currents.In [8], we study the ∂-equation on a non-reduced space for general (p, q)-forms, and then both type of currents appear in arbitrary bidegrees.
where a α are in C 0, * Z and the sum is finite, and we use the short-hand notation (3.5).Proof.Since τj µ = 0 for all j, µ must have support on Z. Since it is a current, there is a tuple M of positive integers such that τ α ∧ µ is non-zero only when α ≤ M .Let π be the projection (ζ, τ ) → ζ.We claim that (4.1) holds with In fact, given a test form φ with Taylor expansion and using (3.4), we see that It follows from (4.2) that if µ has the expansion (4.1), then ∂µ has an expansion In particular, ∂µ = 0 if and only if each a α (ζ) is ∂-closed.It is also readily verified that a sequence µ k tends to 0 if and only if the associated sums (4.1) have uniformly bounded length and their coefficients a k,α tend to 0 for each fixed α.

The intrinsic sheaf
X are represented by the currents in U that vanish when acting on test forms with a factor in J , JZ , d JZ , which in turn are the currents in U that are annihilated by J , JZ , d JZ , that is, Hom(O U /J , C Z U ). Therefore we have the isomorphism Let ω X be the subspace of ∂-closed elements in C n,0 X .We then obtain the isomorphism In case X is reduced, ω X is the well-known Barlet sheaf of holomorphic n-forms on X, cf.[7, Section 5] and [12].

4.3.
Representations of O X and E 0, * X .Assume that we have a local embedding i : X → Ω.Notice that we have a well-defined mapping It is in fact an isomorphism where X is Cohen-Macaulay (or more generally where X is S 2 ), see [7,Theorem 7.3].
Let µ 1 , . . ., µ ρ be generators for Hom(O U /J , CH Z U ), and consider an element Φ in We can thus (locally) choose a holomorphic matrix A such that is pointwise exact, and holomorphic matrices S and B such that (4.5) In the same way we have a natural mapping If φ is in E 0, * X , then the coefficients in the expansion (4.1) of φ ∧ µ are in E 0, * Z so the image of φ in (4.6) is represented by an element in (E 0, * Z ) r .If X is Cohen-Macaulay we have the unique representation (3.11) with φℓ in E 0, * Z and hence (4.6) defines an E 0, * Z -linear morphism (E 0, * Z ) ν → (E 0, * Z ) r that coincides with T for holomorphic φ.Since (4.4) is pointwise exact, we have the exact complex We now consider what happens with these representations when we change coordinates.
where b ρ are holomorphic.After a preliminary change of coordinates in the ζ-variables, which only affects the coefficients by the factor dζ/dζ ′ , we may assume that Since τ µ = 0 and dτ ∧ µ = 0, and b ρ and c γ,δ only depend on ζ, Note that the expansion of (π ′ ) * ϕ is infinite, but it only runs over γ such that |γ| ≤ |δ|.Since τ δ µ = 0 if |δ| is large enough, the series (4.8) defining a ′ α is thus in fact a finite sum.Thus a ′ α is obtained from a matrix of holomorphic differential operators applied to (a β ).

4.4.
The sheaf C 0, * X of (0, * )-currents.Let us assume now that X is Cohen-Macaulay.We want C 0, * X to be an O X -sheaf extension of E 0, * X so that E 0, * X is dense in a suitable topology.The idea is to define a (0, * )-current φ as something that for each choice of coordinates (ζ, τ ) and basis τ α ℓ as in Section 3.1 has a representation (3.11)where ( φj ) are in (C 0, * Z ) ν , and transform by (4.9).However, to get a more invariant definition we will represent C 0, * X as a subsheaf of the O X -sheaf Let us fix (ζ, τ, τ α ℓ ).Given an expression (3.11), where φ0 , . . .φν−1 are in C 0, * Z , we get a mapping (4.10) CH Z U → C Z U , µ → φ ∧ µ, by expressing µ as in (4.1) and performing the multiplication formally term by term.
Lemma 4.6.The mapping (4.10) defines an element in F that is zero if and only if all φℓ vanish.
All such images in F form a coherent subsheaf F ′ of F that is independent of the local choice (ζ, τ, τ α ℓ ).
Let µ 1 , . . ., µ ρ be generators for Hom(O U /J , CH Z U ). Then the coefficients of φ ∧ µ j , j = 1, . . ., ρ, are given by T ( φℓ ) ∈ (C 0, * Z ) r , where T is the matrix in (4.7).Indeed this holds for the smooth φǫ , and hence for φ.Since T is pointwise injective, the induced mapping is injective as well.If the image of (4.10) vanishes therefore the tuple φℓ vanishes.
For each multiindex γ, for any smooth φ if ψ = ξφ.Moreover, (4.12) ξ( φ ∧ µ) = ψ ∧ µ since both sides are the equal to the current ξφ ∧ µ.If now ( φℓ ) is in (C 0, * Z ) ν and ( ψℓ ) is defined by (4.11), then by a regularization as above we see that still (4.12)holds.Thus the image of (C 0, * Z ) ν is a locally finitely generated O X -module, and hence a coherent subsheaf F ′ of F. It remains to check the independence of the choice of (ζ, τ, τ α ℓ ).Thus assume X has a unique representation (3.11).However, in view of Lemma 4.6, the current φ ∧ µ has an invariant meaning.We have natural mappings ∂ : C 0,q X → C 0,q+1 X , defined by ( φℓ ) → ( ∂ φℓ ).They are well-defined since ∂ commutes with the transition matrices L in the preceding proof.We thus get the complex Proposition 4.8.The sheaf complex (4.13) is exact.
The sequence φ k in C 0, * X converges to φ in U if and only if φℓk → φℓ for each ℓ.
Remark 4.11.From the very definition, cf.Section 3.2, a sequence φ k ∈ E 0, * X tends to 0 at a given point x if and only if given a small local embedding i : . Also the converse is true.In fact, if Φ k are representatives in U and Φ k → 0 in U , then each of the coefficients of Φ k ∧ µ j in the representation (4.1) tend to 0 in E 0, * (Z ∩ U ) for each j.This precisely means that T ( φℓk ) tend to 0 in E 0, * (Z ∩ U ). Since T is pointwise injective this implies that φℓk → 0 in E 0, * (Z ∩ U ) for each ℓ.
Remark 4.12.We only define C 0, * X on the part where Z is smooth, as we there need to embed L p 0, * (X) into a larger space that allows for more flexibility.We do not know what an appropriate definition of C 0, * X would be over the singular part of Z.In [7], we introduce a sheaf W 0, * X of pseudomeromorphic (0, * )-currents on X with the so-called standard extension property, also when Z is singular.On the part where Z is smooth, W 0, * X is a subsheaf of C 0, * X , and consists of currents which admit a representation (3.7),where the ψm are in W 0, * Z ⊆ C 0, * Z .
Remark 4.13.We do not know if the embedding C 0, * X → F is an isomorphism, i.e., if F ′ = F.For any h in F that can be approximated by smooth forms h ǫ in F, it follows as above that h is in F ′ , but it is not clear that this is possible for an arbitrary h in F.An analogous statement for the subsheaf W 0, * X is indeed true, see [7, Lemma 7.5], but the proof relies on the fact that elements in W 0, * Z are in a suitable sense generically smooth, and does not generalize to C 0, * X .

L p -spaces
Assume that X is Cohen-Macaulay and that the underlying manifold Z = X red is smooth.Recall that we have chosen a Hermitian metric on Z and let dV be the associated volume form.Assume 1 ≤ p < ∞.If K ⊂ X is a compact subset and φ is in E 0, * (X) then is finite and defines a semi-norm on E 0, * (X).We define the sheaf L p loc;0, * as the completion of E 0, * X with respect to these semi-norms.In particular we get the spaces L p 0, * (K) for any compact subset K ⊂ X.For a relatively compact open subset V ⊂⊂ X we let L p 0, * (V) = L p 0, * (V).Clearly these spaces are independent of the choice of | • | X and Hermitian structure on Z.In the same way we define the sheaf C 0, * X as the completion of E 0, * X with respect to the semi-norms sup K |φ| X .
Example 5.2.Let X be the space in Example 3.1 and let V = U ′ ∩ X, where U ′ is a relatively compact subset of U .Let L j,p ( Vred ) be the Sobolev space of all (0, * )-currents whose holomorphic derivates up to order j are in L p (Z).It follows from (11.5) that L p ( V) can be realized as all expressions of the form (3.7), where ψ m ∈ L |M −m|,p 0, * ( Vred ).
For a general Cohen-Macaulay space X there is no such simple way to describe L p (X) locally in terms of a single choice of (ζ, τ, τ α ℓ ).
Notice that if we have an embedding ι : X → X as in Section 3, then, with the notation used there, φ L p (X∩U ) ∼ j γ j φ L p ( X∩U ) .
Remark 5.4.Here is a heuristic proof of Theorem 1.1.For simplicity, let us assume that only two submersions π 1 and π 2 are needed in (3.13).Assume that φ is in L p 0,1 (V).Then we can find a solution u ι in C 0,0 (V) to ∂u ι = φ so that the coefficients with respect to (ζ ι , τ ι , (τ ι ) α ι ℓ ) of u ι are in L p (V red ).This means that |u ι | X,π ι is in L p (V red ) for each ι.Now h = u 2 − u 1 is ∂-closed, thus holomorphic, and hence bounded.It follows that also In view of (3.13) one might conclude that u 2 actually is in L p 0,0 (V) if we disregard the problem pointed out in Remark 5.3.Clearly, this argument breaks down if φ has bidegree (0, q + 1), q ≥ 1.
It is not clear to us if it is possible to make this outline into a strict argument.In any case, we will prove Theorem 1.1 by means of an integral formula from [7].Besides being a closed formula for a solution, it also makes sense at non-Cohen-Macaulay points, and offers a possibility to obtain a priori estimates, cf.Section 11.Hopefully it could lead to results for general (0, q)-forms.
6. Koppelman formulas on X 6.1.Koppelman formulas in C N .Let U ⊂ C N be a domain, and let U ′ ⊂⊂ U .Moreover, let δ η be contraction by the vector field is a smooth form such that g k,k has bidegree (k, k) and only contains holomorphic differentials with respect to ζ.We say that g is a weight in U with respect to U ′ if ∇ η g = 0 and g 0,0 is 1 when ζ = z.Notice that if g and g ′ are weights, then g ′ ∧ g is again a weight.The basic observation is that if g is a weight, then (6.1) we can find a weight g, with respect to U ′ , with compact support in U , such that g depends holomorphically on z and has no antiholomorphic differentials with respect to z.For our purpose we can assume that these domains are balls with center at 0 ∈ U .Then we can take (6.2) Here χ is a cutoff function in U that is 1 in a neighborhood of U ′ .It is convenient to choose it of the form χ = χ(|ζ| 2 ) where χ(t) is identically 1 close to 0 and 0 when t is large.Elaborating this construction one can obtain Koppelman formulas for ∂.Let defines integral operators E 0, * +1 (U ) → E 0, * (U ′ ) such that φ = ∂Kφ + K( ∂φ) in U ′ .The integral in (6.4) is, by definition, the pushforward π * (g ∧ B ∧ φ), where π is the natural projection U × U ′ → U ′ .
6.2.Hefer morphisms.Let (E, f ) be a locally free resolution as in (2.3).As in [2] and elsewhere, we equip E := ⊕E k with a superstructure, by splitting into the part ⊕E 2k of even degree and the part ⊕E 2k+1 of odd degree.An endomorphism α ∈ End(E) is even if it preserves the degree, and odd if it switches the degree.The total degree deg α of a form-valued morphism α is the sum of the endomorphism degree and the form degree of α.
For instance, f is an odd endomorphism.The contraction by δ η is a derivation (and has odd degree) that takes the total degree into account, so if α and β are two morphisms, then In order to construct division-interpolation formulas with respect to (E, f ), in [2] was introduced the notion of an associated family H = (H ℓ k ) of Hefer morphisms.Here H ℓ k are holomorphic (k − ℓ)-forms with values in Hom (E ζ,k , E z,ℓ ) so they are even.They are connected in the following way: To begin with, H ℓ k = 0 if k − ℓ < 0, and Let R and U be the associated currents, see Section 2. The basic observation is that g ′ = f 1 (z)H 1 U + H 0 R is a kind of non-smooth weight so that if Φ is holomorphic, then When defining these integral operators, we tacitly understand that only components of the integrands that contribute to the integral should be taken into account.

6.3.
Local Koppelman formulas on X.Now assume that our non-reduced space X is locally embedded in a pseudoconvex domain U .Let V = X ∩ U and V ′ = X ∩ U ′ ⊂⊂ V. Let (E, f ) be a locally free resolution of O X as in (2.3).Then RΦ = 0 if Φ = 0, cf.(2.6), and hence (6.6) is an intrinsic representation formula for φ ∈ O(V ′ ).Following [9] and [7], one can define operators (6.7) mapping (0, * + 1)-forms in V to (0, * )-forms in V ′ .However, not even in 'good' cases the formula (6.7), as it stands, produces a form that is smooth in U ′ , cf. [7, Remark 10.4], so the precise definition of Kφ is somewhat more involved, cf.[7, Section 9]: 2).The equality (6.7) is to be interpreted as the fact that there is a unique pseudomeromorphic current u = Kφ in V ′ such that for all µ ∈ Hom(O U /J , CH Z ) in U ′ .By [7, Theorem 9.1] the operators so defined satisfy the Koppelman formula (6.8) Remark 6.1.In general, Kφ is not necessarily smooth in V ′ , so one has to replace E 0, * X by the sheaves A 0, * X , cf.Introduction, [7] and Section 11.Let us now assume that Z = X red is smooth.By shrinking U we can assume that we have coordinates (ζ, τ ) in U as usual, and we let (z.w) be the corresponding 'output' coordinates in U ′ .If in addition X is Cohen-Macaulay we can choose (E, f ) so that the associated free resolution (2.3) of O U /J has length κ = N − n.Then R has just one component R κ .For a smooth (0, * + 1)-form φ in V, then (6.9) where B is the Bochner-Martinelli form with respect to (ζ, τ ; z, w), and ( ) n denotes the component of bidegree (n, n − * − 1) in (ζ, τ ).

Extension of Koppelman formulas to currents
We keep the notation from the preceding section.
The proposition gives a new proof of the exactness of (4.13).
By assumption B is of the form (6.3), where Take µ = µ(z, w) ∈ Hom(O U /J , CH Z U ). Since R is annihilated by τ and dτ , and µ is annihilated by w and d w, see Section 2, we have that where B(ζ, z) is the Bochner-Martinelli kernel with respect to the variables ζ, z, and g(ζ, z) only depends on ζ and z (provided that it is chosen as in (6.2), but for (ζ, τ ) and (z, w), however, this special choice of g is not important).More precisely, in view of the representation (4.1) of R κ , its action involves holomorphic derivatives with respect to τ followed by evaluation at τ = 0, cf.(3.4).Therefore all terms involving τ can be cancelled without affecting the integral.
For the same reason all terms involving w disappear.Therefore H is the only factor in the integral that depends on w.Using the expansions of the form (3.7) of φ together with the fact that R κ is annihilated by J , and the expansion (4.1) of R κ , and evaluating the τ -integral in the right hand side of (7.2) we get , for appropriate holomorphic functions h ℓ ′ .If we express each occurrence of w in the basis w α ℓ as in (3.11) modulo J (with w instead of τ ) and using that µ is annihilated by J , we get where h ℓ ′ ,ℓ are polynomials in ζ, z.Thus where the K ℓ ( φǫ ) is the result of multiplying the tuple ( φǫ ℓ ′ ) by a matrix of smooth forms in ζ, z followed by convolution by the Bochner-Martinelli form B(ζ). Therefore, each limit lim ǫ→0 K ℓ ( φǫ ) =: K ℓ (φ) exists in the sense of currents on Z and is independent of the regularization φǫ , and we see that K(φ) = ℓ K ℓ (φ)w α ℓ = lim K(φ ǫ ) is well-defined.Since the Koppelman formula holds for φ ǫ , it follows that it also holds for φ by letting ǫ → 0.

Comparison of Hefer mappings
We will use an instance of the following general result.Lemma 8.1.Let a : ( Ê, f ) → (E, f ) be a morphism of complexes, and let Ĥ and H denote holomorphic Hefer mappings associated to ( Ê, f ) and (E, f ), respectively.Then (locally) there exist holomorphic (k − ℓ + 1)-forms C ℓ k with values in Hom ( Êζ,k , E z,ℓ ) such that Here, just as in [22], we consider a as a morphism in End(⊕( Êk ⊕ E k )), and thus a is a morphism of even degree, cf.Section 6.2.
Proof.Since H ℓ ℓ and Ĥℓ ℓ are the identity mappings on E ℓ,z and Êℓ,z , respectively, when ζ = z, one can solve the equation (8.2) by [2,Lemma 5.2].We now proceed by induction over k − ℓ.We know the lemma holds if k − ℓ ≤ 0 so let us assume that it is proved for k − ℓ ≤ m and assume k − ℓ = m + 1.By [2, Lemma 5.2], it is then enough to see that the right hand side of (8.3) is δ η -closed.To simplify notation we suppress indices and variables.By (6.5), δH = Hf − f H and δ Ĥ = Ĥ f − f Ĥ.In addition, f a = a f and since f is of odd degree, while a is of even degree, δf = −f δ and δa = aδ.We then have, using that f f = 0 and f f = 0, and using the relations above it is readily verified that the right hand side vanishes.

L p -estimates in special cases
In this section we consider the space X, O X = O U /I, in Example 3.1 where, in a local embedding and suitable coordinates (ζ, τ ) in U , I = τ M +1 .
Since I is a complete intersection, the Koszul complex provides a resolution of O U /I.That is, if e 1 , . . ., e κ is a nonsense basis for the trivial vector bundle Ê1 ≃ C κ ×U , then the resolution is generated by ( Ê, f ), where Êk = Λ k Ê1 , each fk is contraction by and e * j is the dual basis.The associated residue current is then it is readily checked that a choice of Hefer forms Ĥℓ k is given by contraction by ∧ k−ℓ h.In particular, where we use the multiindex notation In particular, with the notation (3.5), and the formula (3.6), Using the notation from Section 6.3 and Section 7, we consider the operators As was noted in Section 8, only the parts of B and g depending on z, ζ are relevant.In view of (3.4) we therefore get Since B(ζ, z) only depends on ζ − z, by a change of variables, we see that for appropriate constants c β ′ ,β ′′ .Since B(ζ, z) is uniformly integrable in ζ and z, and g is smooth, it follows by, e.g., [30,Appendix B], that . From (9.3) and (3.8) it follows that there is a constant C p such that (9.4) Example 9.1.Let X = C n × X 0 be an analytic space which is the product of C n with a space X 0 whose underlying reduced space is a single point 0 ∈ C κ , i.e., such that , where J = π * J 0 , and J 0 ⊂ O C κ τ is an ideal such that Z(J 0 ) = 0 and π is the projection π(ζ, τ ) = τ .Note in particular that this includes the basic examples X as in Example 3.1.
The same statements hold for C 0,q instead of L p 0,q .
In particular, if ψ ∈ L p 0,q+1 (X ∩ U ) and ∂ψ = 0, then ∂Kψ = ψ in X ∩ U ′ by (6.8).Thus Theorem 1.1 holds for all q when X is of the form as in Example 9.1.
Proof.If ψ ∈ L p 0,q+1 (X ∩ U ), then by definition there is a sequence ψ k ∈ E 0,q+1 (X ∩ U ) such that ψ − ψ k L p (X∩U ) → 0. It follows from (9.4) that Kψ k is a Cauchy sequence in L p 0,q (X ∩ U ′ ) and hence has a limit Kψ.Clearly this limit satisfies (9.6).Moreover, it is in C 0,q (X ∩U ′ ).Thus these extended operators satisfy the Koppelman formula, see Proposition 7.1.The statements about C 0,q follow in exactly the same way.Remark 9.3.We use the intrinsic integral formulas on X ∩ U here for future reference.To obtain the theorem one can just as well solve the ∂-equation with relevant L p -Sobolev norms in X ∩ U for each coefficient in the expansion (3.7).However, this is naturally done by an integral formula on Z ∩ U , and the required computations are basically the same.
We finish this section with an example showing that the spaces in Example 9.1 may not necessarily be written in the simple form as in Example 3.1 after a change of coordinates, even if J is a complete intersection.
Example 9.4.Let J be generated by (w 3  1 , w 2 1 + w 3 2 ).Then we claim that one cannot find local coordinates τ 1 , τ 2 near 0 such that J is generated by (τ ℓ 1 , τ m 2 ).Indeed, since the multiplicity of J is 9, ℓm = 9.The assumptions imply that where the a jk and b jk are holomorphic.One may exclude the case ℓ = m = 3 since the above equations would imply that w 2 2 belongs to the ideal generated by (w 1 , w 2 ) 3 .The case ℓ = 1, m = 9 may be excluded as that would imply that 2 ) for some holomorphic functions c j , which would contradict the fact that τ 1 is part of a coordinate system near 0.

L p -estimates at Cohen-Macaulay points
Assume that we have a local embedding X → U where Z ∩ U is smooth and X is Cohen-Macaulay.Moreover, assume that we have coordinates , and a basis τ α ℓ for O X over O Z .We may also assume that we have a Hermitian resolution (E, f ) of O X = O U /J of minimal length, so that its associated residue current is R = R κ .
In general, if X is Cohen-Macaulay, and the underlying space Z is smooth, it is not possible to choose coordinates so that X becomes a product space as in Example 9.1, even if the space is defined by a complete intersection.
, and O X = O/J .Then Z(J ) = {w = 0}, so J is a complete intersection ideal, and X is Cohen-Macaulay.We claim that one cannot choose new local coordinates (ζ, τ 1 , τ 2 ) near 0 such that J = π * J 0 , where τ is an ideal such that Z(J 0 ) = {τ = 0} and π(ζ, τ ) = τ .Indeed, assume that there are such coordinates.First of all, from any set of generators of an ideal, one may select among them a minimal subset of generators, and the number is independent of the choice of generators.Thus, one may assume that J is generated by f 1 (τ ), f 2 (τ ).Since f and g generate J , there is an invertible matrix A of holomorphic functions such that f = Ag and g = A −1 f .Note that if m is the maximal ideal of functions vanishing at {z = w = 0}, then g belongs to mJ Z .Since f = A −1 g, the same must hold for f .Since {τ = 0} = {w = 0}, one may write τ = Bw for some holomorphic matrix B. Note also that since f only depends on τ , f = Cτ mod J 2 Z for some constant matrix C. Since f belongs to mJ Z , we must have that C = 0, i.e., f = 0 mod J 2 Z .Thus, also g = 0 mod J 2 Z , which yields a contradiction.
Let us assume that we have coordinates (ζ, τ ) in U and choose a simple ideal I as in Section 9, such that I ⊂ J , and hence, as in Section 3, get the embedding where O X = O U /I. Let V = X ∩ U and V ′ = X ∩ U ′ as before and let V = X ∩ U and V′ = X ∩ U ′ .Here is our principal result.
Proposition 10.2.Let V and V ′ be as above and K as in Section 6.3.
Proof.Choose an embedding (10.1) as above.Since the proposition is local we can assume that we have a basis τ α ℓ in U .Let φ be a smooth (0, * )-form in V.As in Section 9, let ( Ê, f ) be the Koszul complex of I = τ M +1 in U .Let us choose a morphism a : ( Ê, f ) → (E, f ) of complexes that extends the natural surjection O U /I → O U /J and such that a 0 is the identity morphism Ê0 ≃ E 0 , see, e.g., [22,Proposition 3.1].By (3.10), we are to estimate the L p ( V′ )-norm of where γ is any of the functions in (3.9).(By the way, one can choose γ j as the components of a κ , cf. [7, Example 6.9]).
Lemma 10.3.We have that Proof.Recall from Section 6.3 that μ ∧ γKφ is defined as the limit of where χ is a cut-off function and . By [22,Theorem 4.1], R κ a 0 = a κ Rκ .Using Lemma 8.1, the fact that a 0 is the identity, and that fκ Rκ = 0 by (2.5), we get (10.5) Since γJ ⊆ I and μ is annihilated by I we have that γ(z)f 1 (z, w)μ = 0 so by (10.5), (10.4) is equal to Taking the limit as ǫ → 0, we obtain (10.3).
By the dimension principle, Proposition 2.1, therefore the limit of each such term is 0 since k + 2κ < n + 2κ.Thus the claim holds.
The main term T 11 φ is precisely K(γφ), so from (9.4) and (3.10), as desired.The remaining two terms T 121 φ and T 21 Φ in (10.9) are simpler since their integrands do not contain the factor B. We now use that Φ has the form (3.11) and Rκ only depends on τ .Integrating with respect to τ therefore does not give rise to any derivates with respect to ζ.Thus, the L p (V ′ )-norms of these two terms are bounded by integrals of the form , where ξ j (ζ, z) are smooth forms with compact support in Z ∩ U .It follows from (3.12) and (3.13) that these terms are φ L p (V) .Thus part (i) is proved.
We now consider part (ii), so assume that φ ∈ L p 0,1 (V), p < ∞ and ∂φ = 0. We cannot deduce (ii) directly from (i).The problem is that we do not know whether it is possible to regularize φ so that the smooth approximands are ∂-closed, cf.Remarks 5.3 and 5.4.By Proposition 7.1 we know that ∂Kφ = φ in the current sense.We must show that actually Kφ is in L p (V ′ ) and that (10.2) holds.Let φ k be a sequence of smooth (0, 1)-forms in V that converge to φ in L p (V) and let Φ k denote the representatives in U given by (3.11).Since T 123 φ k and T 23 φ k vanish for degree reasons, we have (10.10) where for φ in E 0,1 (V).We conclude that GΦ k has a limit GΦ in L p ( V′ ) and that (10.11) so arguing as in the proof of Proposition 7.1 the claim follows, since ∂ φk,ℓ → 0 for each ℓ.
Note that if we drop the assumption that φ be a (0, 1)-form, then the terms T 123 φ and T 23 φ no longer vanish, and it is not clear to us how to estimate them.It is also not clear to us whether the estimate (10.2) holds if φ is not ∂-closed.
In the case of product spaces as in Example 9.1, then one may choose C 0 κ , Ĥ0 κ and Γ such that they only contain holomorphic differentials dτ .In that case, all terms but T 11 φ vanish for any (0, q)-form φ, since all the other terms involve integrals of forms of degree κ + 1 in dτ , which thus vanish for degree reasons.Thus, one in fact has that γKφ = T 11 φ = K(γφ), cf. the proof of Proposition 9.2.

An example where X is not Cohen-Macaulay
In this section we consider an example where Z = X red is smooth but X is not Cohen-Macaulay.Since X red is smooth, it is still possible to define L p loc (X) as in Section 5.However, our solutions Kφ are not smooth at the non-Cohen-Macaulay point.In view of works on L p -estimates on non-smooth reduced spaces it therefore might be natural to define L p (X) as the completion of the space of smooth forms with support on the Cohen-Macaulay-part of X.
In any case we do not pursue this question here, but just discuss an a priori estimate of the solutions.
Let Ω = C 4 z,w and J = J (w 2 1 , w 1 w 2 , w 2 2 , z 2 w 1 − z 1 w 2 ), and let X have the structure sheaf O Ω /J .Then Z = C 2 z , and X has the single non-Cohen-Macaulay point (0, 0).Outside that point X is locally of the form discussed in Section 9 so that we have local L p -estimates for ∂ for all (0, * )-forms there.Thus the crucial question is what happens at (0, 0).The structure sheaf O X has the free resolution (E, f ) We equip the vector spaces E k with the trivial metrics.Consider also the Koszul complex (F, δ w 2 ) generated by w 2 := (w 2 1 , w 2 2 ), which is a free resolution of O/I, where I = w 2 1 , w 2 2 .If X has structure sheaf O X = O/I we thus have an embedding ι : X → X.
We take the morphism of complexes a : Let R and R be the residue associated with (E, f ) and (F, δ w 2 ) , respectively.It is wellknown, see, e.g., [7], that R = R2 is equal to the Coleff-Herrera product 11.1.The current R. In [7, Example 6.9] we found that where is the minimal left-inverse to f 3 .Since µ 0 is pseudomeromorphic with support on {w = 0}, wi µ 0 = 0, and therefore Since X has pure dimension R 3 = ∂σ 3 ∧ R 2 , where the left hand side is the product of the almost semi-meromorphic current ∂σ 3 and the pseudomeromorphic current R 2 , cf.

4. 1 .
The sheaf of currents C Z U .Let U ⊂ C N be an open subset and Z a submanifold as before.Let C Z U be the O U -sheaf of all (N, * )-currents in U that are annihilated by JZ and d JZ .Clearly these currents have support on Z. Lemma 4.1.If (ζ, τ ) are local coordinates in U so that Z = {τ = 0}, then each current µ in C Z U has a unique representation

Lemma 4 . 4 .
Let (ζ, τ ) and (ζ ′ , τ ′ ) be two coordinate systems in U as before.There is a matrix L of holomorphic differential operators such that if µ ∈ Hom(O U /J , C Z U ), and (a α ) and (a ′ α ) are the coefficients in the associated expansions (4.1), then (a