Estimates for the $\bar{\partial}$-equation on canonical surfaces

We study the solvability in $L^p$ of the $\bar\partial$-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case $p=2$ for two natural closed extensions $\bar\partial_s$ and $\bar\partial_w$ of $\bar\partial$. For $\bar\partial_s$ we have solvability, whereas for $\bar\partial_w$ there is solvability if and only if a certain boundary condition $(*)$ is fulfilled at the singularity. Our main tool is certain integral operators for solving $\bar\partial$ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.


Introduction
The classical Dolbeault-Grothendieck lemma states that locally in C n one can solve thē ∂-equation∂u = ϕ if ϕ is a∂-closed (0, r)-form or current. One can obtain a solution u by a Koppelman formula; then u is obtained through multiplication of ϕ with a smooth form followed by convolution with an integrable form, the so-called Bochner-Martinelli form. Thus one even gains some regularity; in particular, one can solve∂ in C ∞ , L p , C α , Sobolev-spaces, etc, see, e.g., [Ra] or [LM]. On singular varieties this is not true in general. There are smooth∂-closed forms which have no local smooth∂-potentials, see, e.g., [R1,Beispiel 1.3.4] and [AS2,Example 1].
Solvability of the∂-equation on singular varieties has been studied in various articles, starting with among others [HP, PS], and in recent years solvability in L 2 has been of particular focus, see, e.g., [FOV,OV,R4]. There are known examples where the∂-equation is not locally solvable in L p , for example when p = 1 or p = 2. On homogeneous varieties, obstructions for solvability in L p have been described explicitly in [R3].
In this paper we study solvability in L p of the∂-equation in a neighborhood of a canonical singularity on a complex surface. On a surface a singularity is canonical if and only if it is a rational double point. Such points are well-studied and have been classified a long time ago as the so-called du Val singularities, see, e.g., the survey [D2]. The possible singularities are of type A n , n ≥ 1, D n , n ≥ 4, E 6 , E 7 and E 8 , and can be realized as isolated hypersurface singularities in C 3 .
Throughout the introduction, we assume that X is a surface with one isolated canonical singularity. We will further assume that X = {f = 0} ⊂ Ω ′ , where Ω ′ ⊂⊂ C 3 is an open pseudoconvex set and f is holomorphic in a neighborhood of Ω ′ and that df = 0 on {f = 0} except at the singular point, which we assume is 0.
Let∂ sm be the∂-operator on smooth (0, r)-forms which have support not intersecting the singularity at the origin. We will consider two extensions of∂ sm as a closed operator on L p (X). One of them is the minimal closed extension, i.e., the strong extension∂ (p) s of∂ sm , which is the graph closure of∂ sm in L p 0,r (X) × L p 0,r+1 (X). That is, ϕ ∈ Dom∂ (p) s ⊂ L p 0,r (X) if and only if there is a sequence of smooth forms ϕ j ∈ L p 0,r (X) with supp ϕ j ∩ {0} = ∅ such that ϕ j → ϕ in L p 0,r (X),∂ϕ j →∂ϕ in L p 0,r+1 (X).
The other extension is the maximal closed extension, i.e., the weak∂-operator∂ (p) w , so that ϕ ∈ Dom∂ (p) w ⊂ L p 0,r (X) if and only if∂ϕ ∈ L p (X) 1 . When it is clear from the context, we will drop the superscript (p) in∂ (p) s and∂ (p) w .
Let ω X be the Poincaré residue of dz 1 ∧dz 2 ∧dz 3 /f . It is an intrinsic∂-closed meromorphic (2, 0)-form on X that is holomorphic outside of 0. We will see below (Proposition 3.3 and Corollary 3.5) that there is a number 2 < q(X) ≤ 4 such that ω X ∈ L q (X) for q < q(X). Let p(X) be the dual exponent of q(X) and let p(X) = 4p(X) 4 − p(X) .
Notice that 4/3 ≤ p(X) < 2 and 2 ≤p(X) < 4. For precise definitions of L p -forms and C α -forms on X, see Section 2.1. In our results, we have the following condition: If ϕ is a (0, 1)-form in Dom∂ (p) w , where p(X) < p ≤ ∞, then it is said to satisfy the condition ( * ) if lim k→∞ X ω X ∧∂χ k ∧ ϕ = 0 ( * ) for some sequence of cut-off functions {χ k } k , where each χ k is 1 in a neighborhood of 0 and the support of χ k approaches {0} when k → ∞. This condition is independent of the sequence of cut-off functions, see Section 4.1, and is thus a kind of boundary condition at {0}. If ϕ is∂-closed, as in the following theorem, by Stokes' theorem the condition ( * ) means that X ω X ∧∂χ ∧ ϕ = 0 (1.1) for some smooth cutoff function χ that is 1 in a neighborhood of 0.
Theorem 1.1. Let X be a surface as above with an isolated canonical singularity at 0.
(i) Assume that p(X) < p ≤ 4. If ϕ is a∂ s -closed (0, r)-form in L p (X), r = 1, 2, then there is u in the domain of∂ (p) s in a neighborhood of 0 such that∂ s u = ϕ. (ii) Assume thatp(X) < p ≤ ∞. If ϕ is a∂ w -closed (0, 1)-form in L p (X), then there is a solution in L p to∂ w u = ϕ in a neighborhood of 0. If p = ∞, then one can choose u in C α for α < 4/p(X) − 2. If ϕ is a (0, 2)-form the same holds for p(X) < p ≤ ∞.
(iii) Assume that p(X) < p ≤p(X). If ϕ is a∂ w -closed (0, 1)-form in L p (X), then there is a solution in L p to∂ w u = ϕ in a neighborhood of 0 if and only if ϕ satisfies the condition ( * ).
Notice that if∂ w u = ϕ, then (1.1) follows from Stokes' theorem since ω X ∧∂χ is a∂closed smooth form with compact support. Thus the condition ( * ) is necessary in the theorem. It turns out that ( * ) is automatically fulfilled whenp(X) < p ≤ ∞, see the comment after the proof of Theorem 1.5. In Section 5 we study the condition ( * ) explicitly for the various types of canonical singularities. Theorem 5.1 asserts that in the case of a singularity of type A n , n ≥ 1, any form ϕ ∈ Dom∂ w ⊂ L 2 0,r (X) satisfies ( * ). For each of the other singularities, that is, of type D n , n ≥ 4, E 6 , E 7 and E 8 , however, there is a (0, 1)-form ϕ ∈ ker∂ w ⊂ L 2 (X) such that the equation∂ w u = ϕ has no solution in a neighborhood of 0, see Theorem 5.6. It follows that for these ϕ the condition ( * ) is not satisfied.
1 This is what we take as definition of∂ (p) w on X. However, to be precise, this definition only coincides with the maximal closed extension of∂sm for p ≥ 4/3, which is the only case of interest to us. In general, that ϕ lies in the domain of the maximal closed extension of∂sm means that∂ϕ|X reg ∈ L p (Xreg). When p ≥ 4/3, it then follows that∂ϕ ∈ L p (X), see [R2,Satz 4.3.3].
To the best of our knowledge, the only known cases of Theorem 1.1 for general surfaces with canonical singularities are the following: Part (i) for p = 2 was proven in [R5,Corollary 1.3]. Part (ii) for p = 2 and (0, 2)-forms was proven in [OR,Theorem 4.3], which builds on the vanishing result from [S]. Some weaker versions of part (ii) are known as well. For ϕ with compact support, it was proven that one can find solutions in L p (for arbitrary p) or with C α -estimates in [RZ]. Moreover, for continuous (0, 1)-forms ϕ with compact support, C α -estimates for solutions were obtained in [AZ1,AZ2].
Various results are known for the A 1 -singularity, as is detailed in the introduction of [LR1]. That there are obstructions to solving∂ w in L 2 on the D 4 -singularity was proven in [P,Proposition 4.13].
As mentioned above, a large part of the study of the∂-equation on singular varieties has been restricted to L 2 -spaces. Integral formulas open up for new results about solvability in L p -spaces for p = 2, as well as other norms. For the proof of Theorem 1.1 our main tool is an integral operator introduced in [AS1, AS2]. Keeping the notation above, let Ω ⊂⊂ Ω ′ be an open set containing 0 and let D = X ∩ Ω. There is an operator K : on D \{0}. The operator is given by an intrinsic integral kernel K(ζ, z) on X × D \{0} that contains the Poincaré residue ω X as a factor in the first variable. In [AS2] it was proved that K and (1.2) can be extended to certain fine sheaves A r X of currents defined across 0 and coinciding with C ∞ 0,r outside 0, so that∂u = ϕ is solvable in A X as soon as∂ϕ = 0. In order to prove Theorem 1.1 we have to extend K and (1.2) to L p . To this end we first consider mapping properties of K.
Theorem 1.2. The integral operator K extends to compact operators Since the sheaves A r X are quite implicitly defined and its sections must have singularities at X sing in general, it is interesting to note the following consequence of (1.4). Corollary 1.3. For X as above we have that In order to obtain solutions to the∂ s -equation in L p we extend (1.2) by approximating ϕ by smooth forms with support away from 0. If ϕ is in the domain of∂ (p) s , it follows that (1.2) holds, so if∂ϕ = 0 we get the solution u = Kϕ to∂u = ϕ. The problem is to see that u is in the domain of∂ (p) s . This is "harder" for large p and our upper bound is 4. (1.5) In case of∂ w we have basically the opposite problem. Since a priori we have no approximation by smooth forms with support away from the origin it is "harder" to obtain the extension of (1.2) for small p, while it then directly follows from Theorem 1.2 that the solution is in the domain of∂ w .
Notice that Theorem 1.1 follows from Theorems 1.2, 1.4 and 1.5 and the discussion about the necessity of the condition ( * ) after the theorem.
Notice that if ϕ is a∂-closed (0, 1)-form with compact support then it automatically satisfies ( * ), and so we can solve∂ w u = ϕ in L p if p(X) < p ≤ ∞. By means of a slight variation of the operator K, introduced in [AS1], one can even get a solution with compact support. In case ϕ is a (0, 2)-form in L p (X) with compact support andp(X) < p ≤ ∞, then there is a solution with compact support if and only if X ϕ∧ hω X = 0 for all h ∈ O(X), (1.7) see Theorem 4.2 below.
Our interest in canonical singularities is partly motivated by the earlier works [LR1,LR2], where similar results as above are studied for affine cones over projective complete intersections. The results about solvability in L p obtained in these articles are in case the degree of these homogeneous varieties is small enough. Here, it is interesting to note that the degree is small if the singularities are mild in the sense of the minimal model program. It turned out that positive results about solvability in L 2 hold precisely for the varieties with canonical singularities, see [LR2].
The results in this article overlap with results from [LR1,LR2] only in the case of the A 1 -singularity, where in [LR1,LR2], it was shown that the∂ w -and∂ s -equations are locally solvable in L p unconditionally for p in certain intervals. On a general canonical surface, as studied in this article, solvability depends on the condition ( * ). The main novelty is the understanding of this condition and a quite sharp non-trivial estimate of the integral kernels from [AS2] on such a surface. The final estimate of the integral operators is done along the same lines as in [LR1,LR2].
We now consider the case of functions. There is an integral operator P : C ∞ 0,0 (X) → O(D) in [AS1,AS2] such that ϕ = K∂ϕ + Pϕ (1.8) on D \ {0}. In order to formulate the following result about extension of (1.8) to L p we need a condition ( * ) for functions ϕ that is explained in Section 4.2 below.
The present paper is organised as follows. After providing some preliminaries in Section 2, in Section 3 we recall the integral formulas from [AS1,AS2], analyse their integral kernels and prove Theorem 1.2 and its corollary. Section 4 is devoted to∂-homotopy formulas and proofs of Theorems 1.4, 1.5 and 1.6 and also to a discussion of condition ( * ). We also include a discussion of the domain of the∂ X -operator from [AS2] and prove that Kϕ ∈ Dom∂ X for certain ϕ ∈ Dom∂ s , see Theorem 4.3. In Section 5 we characterize the du Val singularities with respect to ( * ). Finally we recall some integral estimates on singular varieties from [LR2] in an appendix, Section 6.

Preliminaries
In this section we specify the spaces of differential forms that we consider and explain some basic tools. Throughout the section i : X ֒→ Ω ′ ⊂ C N is an analytic variety of pure dimension n, and D ⊂⊂ X is an open subset of X.
2.1. C α -and L p -forms on an analytic variety. Let 1 ≤ p ≤ ∞. Since D * := D ∩ X reg is a submanifold of some open subset of C N , it inherits a Hermitian metric from C N . We say that a (0, r)-form ϕ is in L p 0,r (D) if ϕ| D * is in L p 0,r (D * ) with respect to the induced volume form dV X . When it is clear from the context, we will drop the subscript in L p 0,r (D). It will be convenient to represent (0, r)-forms on X in a certain "minimal" manner: Any (0, r)-form ϕ on D * can be written (uniquely) in the form at each point z ∈ D * . In fact, one starts with any representation and then at each point takes the orthogonal projection of the form onto Λ 0,r T * D * , see, e.g., [R2, Lemma 2.2.1].
In particular, then ϕ ∈ L p 0,r (D) if and only if ϕ I ∈ L p (D) for all I. If one has an arbitrary representation of ϕ of the form (2.1), then , there exists a representation (2.1) such that all the coefficients (locally) admit C k -extensions to a neighborhood of D. For 0 ≤ α < 1, we say that a (0, r)-form ϕ on D is C α if locally on D there is a representation (2.1) such that all the coefficients ϕ I are C α , i.e., Hölder continuous with exponent α, on D. It is well-known, that a function that is C α on D has a C α -extension to ambient space, see, e.g., [M]. Thus a form ϕ on D is in C α if and only if it is the pull-back to D of a C α -form in ambient space. Notice that C 1 (D) ⊂ C α (D) for all α < 1. For α = 1, we denote the Lipschitz continuous functions by C 0,1 (D) in order to avoid conflict of notation with continuously differentiable functions.
It is not hard to check that these definitions are independent of the choice of embedding of X, and hence are intrinsic notions on X. Fix an embedding D → Ω ⊂ C N . We can then define the Hölder-norm of a form ϕ on D, where the infimum runs over all representations (2.1) of ϕ in ambient space, and the norms on the right hand side of (2.4) are over D. This norm is, up to constants, independent of the embedding.
Remark 2.1. Regularity properties of ϕ like smoothness, Hölder continuity etc, will be reflected by the coefficients on D * in the minimal representation (2.2) above. However, even if ϕ is smooth across the singularity, the coefficients in the minimal representation may be discontinuous there.
2.2. Cut-off functions. We will use the following cut-off functions to approximate forms by forms with support away from isolated singularities. As in the proof of Proposition 3.3 in [PS], let ρ k : R → [0, 1], k ≥ 1, be smooth cut-off functions satisfying and |ρ ′ k | ≤ 2. Moreover, let r : R + → [0, 1/2] be a smooth increasing function such that and |r ′ | ≤ 1. As cut-off functions we will use µ k (ζ) := ρ k log(− log r(|ζ|)) on X if X has an isolated singularity at 0. Note that there is a constant C such that On the domain of the∂ s -operator.
Lemma 2.4. [LR2,Lemma 5.2] Assume that X has an isolated singularity at 0 ∈ D and that D has smooth boundary. Let 1 ≤ p ≤ 2n and let ϕ ∈ L p 0,r (D) such that ϕ ∈ Dom∂ 3. Integral operators on surfaces with canonical singularities 3.1. The Koppelman integral kernels for a hypersurface. Let us recall the definition of the Koppelman integral operators from [AS2] in the situation of a hypersurface i : Let Ω ⊂⊂ Ω ′ be an open set and let D := X ∩ Ω.
Let ω X be the Poincaré residue of the meromorphic form dζ 1 ∧. . .∧dζ n+1 /f . This means that ω X is the unique meromorphic (n, 0)-form on X such that df ∧ ω X = 2πidζ 1 ∧ · · · ∧ dζ n+1 . (3.1) In [AS2, Section 3] so-called structure forms were introduced as generalizations of the Poincaré residue for more general X; we will therefore refer to ω X as the structure form on X. Recall that 1/f and ω X define principal value currents on Ω ′ and X, respectively. Identifying these with their respective currents, ω X can be defined as the unique current such that For coordinates ζ = (ζ 1 , . . . , ζ n+1 ) such that ∂f /∂ζ 1 is generically non-vanishing on X reg , ω X is the pull-back of to X. Alternatively, letting where f ′ ℓ = ∂f /∂ζ ℓ , we have that ω X is realised as the pull-back to X of ϑ dζ 1 ∧ · · · ∧ dζ n+1 . Here, the norm |∂f | is computed in C n+1 , i.e., |∂f | 2 = |f ′ l | 2 . Let η j = ζ j − z j and let δ η be interior multiplication by 2πi η j ∂/∂ζ j . We will consider forms with anti-holomorphic differentials of both ζ and z but only holomorphic differentials with respect to ζ. The (full) Bochner-Martinelli form is B : in Ω ′ × Ω ′ , here lower indices denote bidegree, is a weight with respect to Ω if (δ η −∂)g = 0 and g 0,0 (z, z) = 1 for z ∈ Ω. We say that g is holomorphic with respect to z if the coefficients are holomorphic in z and there are no anti-holomorphic differentials with respect to z.
Let h be such a Hefer form and let g be a weight as in Example 3.1. We can then define an integral operator K that acts on forms on X and produces forms on D = X ∩ Ω ′ in the following way: We let where the kernel has the form K(ζ, z) = ω X (ζ) ∧K(ζ, z), (3.7) dζ 1 ∧ · · · ∧ dζ n+1 ∧K(ζ, z) = h ∧ (g ∧ B) n , and (g ∧ B) n denotes the components of g ∧ B of bidegree (n, * ), cf. [AS2,Section 8]. It follows that K(ζ, z) = ϑ h ∧ (g ∧ B) n and so, in view of (3.3), (3.4), and (3.5) we get that where the c i and the a ijk are smooth (n, * )-forms and the b ij are smooth (0, * )-forms. We have thus shown Proposition 3.2. We can write K = K 1 + K 2 , where K 1 and K 2 are defined by integral kernels k 1 and k 2 , respectively, that are sums of terms of the form respectively, where a(ζ, z) and b(ζ, z) are smooth on X × D.
We also need to consider the projection operator P, which is defined by where the integral kernel P (ζ, z) is defined in a similar way to (3.7). Namely, cf. [AS2,(5.5)]. Notice that since h and g are smooth,P is smooth, and so |P (ζ, z)| |ω X (ζ)|. If X has an isolated singularity in Ω and we choose g according to Example 3.1, then for each z, ζ → g n,n (ζ, z) is supported away from X sing and the corresponding P is then smooth in ζ and holomorphic in z.
3.2. L 2+ -property of the structure form for a canonical hypersurface.
Proposition 3.3. Let i : Y → Ω ⊂ C n+1 be a hypersurface with canonical singularities and X ⊂⊂ Y . Then there exists a real number q(X) > 2 such that ω Y ∈ L q (X) for 1 ≤ q < q(X), where ω Y is the structure form of Y .
Proof. We denote by ω n Y Grothendieck's dualizing sheaf (sometimes also called the sheaf of Barlet-Henkin-Passare holomorphic n-forms on Y ). As Y is a hypersurface, in particular Cohen-Macaulay, ω n Y is a locally free O Y -module of rank one, and the structure form ω Y is a generator of ω n Y , see, e.g., [AS2] and (3.2). Let π : M → Y be a resolution of singularities such that the exceptional divisor has only normal crossings. Since Y has canonical singularities, π * ω Y extends across π −1 Y sing to a holomorphic n-form on M . Pick any Hermitian metric on M and let dV M be the corresponding volume form. Then i n 2 π * ω Y ∧ π * ω Y = AdV M for some smooth non-negative function A on M .
Let s be local coordinates on M and let dV s = (i/2) n ds 1 ∧ds 1 ∧ . . . ∧ds n ∧ds n . Then dV s = BdV M for some smooth positive function B. Let ̟ = i • π, where i is the inclusion Y ֒→ Ω ⊂ C n+1 . Then, on M \ π −1 Y sing , s → ̟(s) is a local parametrization of Y reg ⊂ Ω and it is well-known that π * dV Y = det HdV s , where H = t Jac ̟ · Jac ̟ ≥ 0 and Jac ̟ = ∂̟ ν /∂s µ ν,µ is the Jacobian matrix of ̟. Notice that det H is a non-negative real-analytic function that vanishes precisely on π −1 Y sing . It follows that (det H) −ǫ/2 is locally integrable with respect to dV M for some ǫ > 0. We now get Thus C is a globally defined function and each point in Y has a neighborhood where C −ǫ/2 is integrable for some ǫ > 0. Since π −1 X ⊂⊂ M , there is an ǫ(X) > 0 such that C −ǫ/2 is integrable on π −1 X for all ǫ < ǫ(X). Recall as long as q − 2 < ǫ(X), and so we may take q(X) = 2 + ǫ(X).
Lemma 3.4. Let Y ⊂ Ω ⊂ C 3 be a hypersurface with an isolated canonical singularity, and let X and q(X) be as in Proposition 3.3. Then q(X) ≤ 2 + 2 m , where m is the maximum of the multiplicities of the divisors in the unreduced exceptional divisor in a minimal resolution of singularities of Y .
Proof. Assume that Y = {f = 0} ⊂ Ω ⊂ C 3 , and that Y has an isolated singularity at z = 0. Then we claim that on Y reg for some constant c, where as above the norm |ω Y | is with respect to the norm on Y reg induced by the norm on C 3 , while |∂f | is with respect to the norm on C 3 . Indeed, for any (2, 0)-form α on Y reg , one has the formula |α| Yreg = |α ∧ ∂f | C 3 |∂f | C 3 , and thus (3.12) follows from (3.1).
Let A and C be as in the proof of Proposition 3.3. Let π : M → Y be a minimal resolution of singularities of Y . This resolution is crepant, i.e., π * ω 2 Y = ω 2 M , see for example [I,Theorem 7.5.1]. Thus, the function A is strictly positive.
Since Y has an isolated singularity at 0, |∂f | |z|, so by (3.12), |ω Y | 1/|z|. Since A is strictly positive, π * |ω Y | ∼ C −1/2 , and it thus follows from (3.11) that for q ≥ 2, If Z i is an irreducible component of the unreduced exceptional divisor Z, and Z i has multiplicity m i , then π * |z| 2 vanishes to order 2m i along Z i , and thus, in order for the integral on the right-hand side to be finite, we must have that m i (q − 2) < 2 for all m i .
In combination with a calculation of the multiplicities as in for example [I,Example 7.2.5] or [BPV,Proposition 3.8], we obtain the following corollary.
In particular, we always have that q(X) ≤ 4, so p(X) ≥ 4/3 for all surfaces with canonical singularities.

Mapping properties of K.
Proof of the L p mapping properties in Theorem 1.2. By Proposition 3.2 we have the decomposition K(ζ, z) = k 1 (ζ, z) + k 2 (ζ, z), where where b ′ (ζ, z) is bounded. By Lemma 6.3, k 1 is uniformly integrable over X in ζ as well as in z, and so K 1 maps L p (X) → L p (D) continuously for all 1 ≤ p ≤ ∞ by the generalized Young inequality, [Ra, Appendix B] and Lemma 2.2. Note that we can then decompose K 2 into the consecutive application of two operators To analyse this chain, choose 2 < q < q(X) so that ω X ∈ L q (X). By Hölder's inequality, the operator ϕ → ϕ ∧ ω X maps L p (X) → L a (X) continuously for 1 ≤ a ≤ ∞ defined by 1/a = 1/p + 1/q (for p so that 1/p + 1/q ≤ 1). The second operator in (3.13) can again be analysed by the generalised Young inequality. By Lemma 6.3, |ζ − z| −2 ∈ L s (X) in ζ and in z for all s < 2, in particular for s defined by 1/s + 1/q = 1, since q > 2. Then, since 1/p = 1/a − 1/q = 1/a + 1/s − 1, it follows from the generalised Young inequality, [Ra, Appendix B], that maps L a (X) → L p (D) continuously. Combining, we see that the composed operator (3.13) given by the kernel k 2 is a bounded mapping L p (X) → L p (D) for any p such that 1/p + 1/q ≤ 1. Thus K is a bounded mapping L p (X) → L p (D) for all p(X) < p ≤ ∞. The kernel k 1 is integrable in both variables, and by truncating it, we get a bounded kernel corresponding to a compact operator; by standard arguments, cf., for example [Ra, Appendix C], this converges to K 1 , and it is thus a compact operator. If we decompose the operator K 2 as in (3.13), the same holds for the right-most operator, and thus also K 2 is compact.
Proof of the C α mapping properties in Theorem 1.2. Let us first consider the operator K. Note that for ν = 1, 2, k ν (ζ, z)ϕ(ζ) is a sum of terms of the form k (3.14) for each k ′ ν . For ν = 1, we may assume that k 1 is of the form (3.8). Then k ′ 1 is a sum of functions of the form (3.8) with a(ζ, z) replaced by one of its coefficients a ′ (ζ, z). We may assume that k ′ 1 is one such function; then Since a(ζ, z) depends smoothly on z, we may assume that |a ′ (ζ, z) − a ′ (ζ, w)| |z − w|, and since the remaining integrand in I 1 (z, w) is integrable in ζ by Lemma 6.3, I 1 (z, w) |z−w|. The integrand in I 2 (z, w) is bounded by a constant times and by the same argument as for the Bochner-Martinelli kernel on C 2 , see, e.g., [LT,Proposition III.2.1], and using Lemma 6.3, one obtains that I 2 (z, w) |z − w| α for any α < 1, and thus K 1 is C α for any α < 1.
For the integral on D 3 , we use the following inequality, see the proof of [LT,Lemma III.2.2]. It follows that (3.15) e.g., by assuming that |ζ −z| ≤ |ζ −w| and adding and subtracting (ζ i − w i )(ζ j −z j )/|ζ −w| 4 inside the absolute value sign on the left-hand side. Using Hölder's inequality as above, we get
Since (3.14) holds uniformly for z, w ∈ D, if {ϕ j } j and thus {ϕ ′ j } j are bounded sequences in L ∞ (X), then {(Kϕ j ) ′ } j are equicontinuous in the C α (D)-norm and thus K is compact by the Arzelà-Ascoli theorem.
Proof of Corollary 1.3. The stalk of A X at the singular point is a finite sum of currents of the form where each K i is an integral operator as in Theorem 1.2, mapping forms on D i := Ω i ∩ X to forms on D i+1 , where Ω = Ω ν+1 ⊂⊂ Ω ν ⊂⊂ · · · ⊂⊂ Ω 1 ⊂⊂ C 3 are pseudoconvex domains, and ξ i are smooth forms on D i . The corollary now follows from Theorem 1.2.
3.4. The operatorsK andP on forms with compact support. Let H ⊂ X be a compact Stein subset such that D is relatively compact in the interior of H. In [AS1] are constructed integral operators, that we here denote byK andP, which map smooth forms with compact support in D to smooth forms in X \ {0} that vanish outside H, such that (3.16) In fact,P maps forms with support in D to smooth forms. Moreover,Pϕ = 0 unless r = 2. The kernels for these operators are obtained by choosing the weight g differently; with notation as in Example 3.1, we let χ = χ(z) and we interchange the roles of ζ and z in the functions s i (ζ, z). The resulting weight is then holomorphic in ζ and has compact support H in z.
Since the proof of the mapping properties above essentially only uses that g is smooth, it follows that an analogue of Theorem 1.2 holds also for these operators. The subscript c denotes forms with compact support.
Note that the operators in fact map to forms with support in the fixed compact set H.
It follows from the proof of Theorem 1.5 that if ϕ ∈ Dom∂ (p) w , with p >p(X), then ( * ) is automatically fulfilled for ϕ, since if µ k is as in Section 2.2, then X ω X ∧∂µ k ∧ ϕ → 0 by Hölder's inequality.
It is worth noting that since the condition ( * ) does not depend on p, we have the following consequence of Theorem 1.5: Corollary 4.1. If the∂ w -equation is locally solvable on a canonical surface for some p 0 > p(X), then is is locally solvable for all p ≥ p 0 .
Morally this means that the number of obstructions to solving the∂ w -equation in the L p -sense is decreasing in p. Theorem 1.1 in [R3] shows that the same kind of phenomenon holds for homogeneous varieties with an isolated singularity.
Let ϕ ∈ L p 0,1 (X), where p(X) < p ≤ 4. Assume that ϕ ∈ Dom∂ s . Then by Theorem 1.4, ϕ =∂ s Kϕ which implies particularly that ϕ =∂ w Kϕ. Hence, ϕ satisfies ( * ). It would be interesting to know whether the converse is also true, i.e., if ϕ satisfies ( * ), does it follow that ϕ ∈ Dom∂ s ? 4.2. Proof of Theorem 1.6. As explained after Proposition 3.2, the operator P is defined by an integral kernel P (ζ, z) that is smooth with compact support in ζ, and holomorphic in z. Therefore P extends to a compact operator P : L p (X) → O(D), cf. the proof of Theorem 1.2.
The formula (1.8) for ϕ ∈ Dom∂ (p) s and p(X) < p ≤ 4 is proved in the same way as Theorem 1.4 above, using that (1.8) holds for the smooth functions ϕ j , and that Pϕ j → Pϕ. Now assume that ϕ is a function in Dom∂ for any smooth∂-closed (0, 1)-form α and sequence χ k as in Section 4.1. In particular, if ϕ is∂-closed, i.e., holomorphic on the regular part of X, then as X is a canonical surface, X sing has codimension 2 and thus ϕ is bounded in a neighborhood of the singularity at the origin. Therefore ϕ ∈ L p for any p ≥ 1 and it follows as for (0, 1)-forms above that ( * ) is satisfied. If ϕ ∈ Dom∂ (p) w and p >p(X), then (1.8) can be verified in the same way as Theorem 1.5 above. If instead p(X) < p ≤p(X) and ϕ satisfies (4.6), one just needs to make minor modifications. Namely, at the point where one considers γ(ζ)−γ(0), then γ is a (0, 1)-form, and one then writes γ = γ i dζ i , decomposes γ i (ζ) = γ i (0) + (γ i (ζ) − γ i (0)) and proceeds as in the proof above. The condition (4.6) is then finally applied with α = γ i (0)dζ i .

4.3.
Homotopy formulas with compact support. We get versions of Theorems 1.4 and 1.5 also for the operators in Theorem 3.6.
Theorem 4.2. Assume we are in the situation of Theorem 1.2.
These statements are proved essentially by the same arguments as in the proofs of Theorems 1.4 and 1.5. For (4.7), notice that as ϕ has compact support in D, when choosing the approximating sequences {ϕ j } j we may, in addition, assume that the ϕ j have compact support in D as well.
For the last statement, notice that the kernel forP has the form hω X with respect to ζ in a Stein neighborhood of the support of ϕ. Since X is Stein, we can assume that h is holomorphic on X and soPϕ = 0 if (1.7) holds. Conversely, ifPϕ = 0, then u =Kϕ is a solution to∂u = ϕ with support on the compact set H, see Section 3.4. It then follows that (1.7) holds if and only if u satisfies ( * ), which as we saw in Section 4.1 is automatically satisfied for p >p(X).
Note that if ϕ is a (0, 1)-form in L p c (X) and∂ϕ = 0, then it automatically satisfies ( * ), so if p > p(X), then u =Kϕ is a solution with compact support to∂u = ϕ.

4.4.
On the domain of the∂ X -operator. The setting in [AS2] is rather different compared to this article. Here we are mainly concerned with forms on X with coefficients in L p , while in [AS2], the type of forms/currents considered, denoted W r X , are "generically" smooth, see [AW]. They include principal value currents α/f , where f is holomorphic and α is smooth, and direct images of such currents, but with no growth restrictions on the singularities. For the precise definition of the sheaf W r X we refer to [AS2,AW]. Thē ∂-operator considered in [AS2] is somewhat different from∂ s and∂ w considered here. For currents in W r X , one can define the product with the structure form ω X associated to the variety. A current µ ∈ W r X lies in Dom∂ X if∂µ ∈ W r+1 X and∂(µ ∧ ω X ) =∂µ ∧ ω X . Combining our results about K and the∂ w -and∂ s -operator with some properties about the W X -sheaves, we obtain a result for the∂ X -operator, providing a partial answer to a question in [AS2], cf. the paragraph at the end of page 288 in [AS2].
Since Kϕ ∈ Dom∂ (p) s , there is a sequence of smooth forms ψ j with support away from the singularity such that ψ j → Kϕ and∂ψ j →∂Kϕ in L p (D). Since ω X ∈ L q (X) for each q < q(X), by Hölder's inequality, ψ j ∧ ω X and∂ψ j ∧ ω X converge in L 1 (D) to Kϕ ∧ ω X and∂Kϕ ∧ ω, respectively. Hencē We conclude that Kϕ ∈ Dom∂ X .

Examples and counterexamples
In this section, we study the condition ( * ) and∂ w -Koppelman formulas for all types of canonical surface singularities: A n , n ≥ 1, D n , n ≥ 4, E 6 , E 7 and E 8 . We focus on the important case of L 2 -cohomology, i.e., p = 2. However, we also get some statements for p ≥ 2. All in all we obtain a complete picture about the solvability of the∂ w -equation in the L 2 -sense at canonical surface singularities.
5.1. The A n -singularities. Recall that the A n -singularity for n ≥ 1 is the variety X Theorem 5.1. Let X be (a neighborhood of the origin of ) the A n singularity {ζ 1 ζ 2 = ζ n+1 3 } ⊂ C 3 , let p ≥ 2, and let ϕ ∈ Dom∂ (p) w ⊂ L p 0,r (X). Then ϕ satisfies the condition ( * ). In combination with Theorem 1.1 we get the following.
Corollary 5.2. The∂ w -equation is solvable at the A n -singularity in the L p -sense for p ≥ 2.

5.2.
On the Euler characteristics of the structure sheaf. As a preparation for the proof of the existence of obstructions for solvability of the∂ w -equation at canonical singularities in the L 2 -sense, we need some observations on the behaviour of the Euler characteristics of the structure sheaf under resolution of singularities. Let F → X be a coherent analytic sheaf over a compact complex space X of pure dimension n, and let χ(F) be the Euler characteristic of F, If D is a divisor on X, associated to a line bundle L → X, then χ O X (D) is the holomorphic Euler characteristic of L. Proof. Since X is a normal space, π * O M = O X . Moreover, canonical singularities are rational so that R k π * O M = 0 for k > 0. Hence, the Leray spectral sequence gives H k (X, O X ) ∼ = H k (M, O M ) for k ≥ 0.
If we assume that the∂ w -equation is locally solvable in the L 2 -sense, then we obtain another representation of χ(O X ) for arbitrary normal complex surfaces.
Theorem 5.4. Let X be a compact normal complex surface, π : M → X a resolution of singularities with only normal crossings, Z := π −1 (X sing ) the unreduced exceptional divisor and E := |Z| the exceptional divisor. If the∂ w -equation is locally solvable in the L 2 -sense for (0, 1)-forms, then Proof. Following [R4, Section 2.1], let C 0,r denote the fine sheaves L 2,loc 0,r ∩ Dom∂ w and consider the sheaf complex 0 → O X −→ C 0,0∂ w −→ C 0,1∂ w −→ C 0,2 → 0. (5.5) It is easy to see that (5.5) is exact at C 0,0 because X is normal; a germ f ∈ ker∂ w ⊂ C 0,0 is a holomorphic function on the regular locus of X, and so it is also strongly holomorphic by normality. Moreover, (5.5) is exact at C 0,2 , see [OR,Theorem 4.3]. (It is usually not difficult to solve∂-equations in the highest degree, see also [S].) In general, (5.5) is not necessarily exact at C 0,1 , but here, we assume that this is the case. Thus (5.5) is a fine resolution of O X ; in particular H k (X, O X ) = H k (Γ(X, C 0,• )). By [R4,Theorem 1.13] H k (Γ(X, C 0,• )) = H k (M, O M (Z − E)), and so which proves (5.4).
Combining Proposition 5.3 and Theorem 5.4 we get: Corollary 5.5. Let X be a compact complex surface with at most canonical singularities. If the∂ w -equation is locally solvable in the L 2 -sense for (0, 1)-forms on X, then for any resolution of singularities π : M → X with only normal crossings.
So, if we are looking for obstructions to solvability of the∂ w -equation in the L 2 -sense at canonical singularities, we just need to find configurations violating (5.6).
Theorem 5.6. There exist obstructions to local solvability of the∂ w -equation in the L 2sense for (0, 1)-forms at singularities of type D n , n ≥ 4, E 6 , E 7 and E 8 .
Hence, ( * ) does not hold for all ϕ ∈ ker∂ w ⊂ L 2 0,1 at such singularities. Proof. Let X be a projective variety with a single singularity of one of the types above, and π : M → X a resolution of singularities with only normal crossings. In view of the discussion above it suffices to show that (5.6) does not hold. For the D n -singularities, n ≥ 4, this was proved in the proof of Theorem 4.8 in [P] using the Riemann-Roch formula for regular complex surfaces where K is the canonical divisor on M . Since O(K) is trivial on a neighborhood of the exceptional set, Z j · K = 0 for any irreducible component Z j of the exceptional set, cf. [D2, page 135], and thus (Z − E) · K = 0. Pardon proved that (Z − E) · (Z − E) = −2 so that and in particular (5.6) does not hold. For the remaining singularities, E 6 , E 7 and E 8 , we proceed analogously to [P] and show that (5.7) holds also for these singularities. Now let π : M → X be the minimal resolution of X. Then the exceptional divisor Z has normal crossings and the irreducible components E j have self-intersection −2 and pairwise intersections according to the Dynkin diagrams of E 6 , E 7 or E 8 , see, e.g., [D2]. The labels of the nodes in the following diagrams are the multiplicities of the corresponding divisors in the unreduced fundamental cycle Z, cf., e.g., [I,Example 7.2.5] and [BPV,Proposition 3.8