Abstract. Let X be a pure n-dimensional complex analytic set in C N with an isolated singularity at 0. We obtain some L 2 -existence and regularity results for the ∂-operator on a deleted neighborhood of the singular point 0 in X. IntroductionLet X be a pure n-dimensional complex analytic set in C N with an isolated singularity at 0 and letand identify i ∂∂ z 2 with the euclidean metric in C N . Then X * inherits a Kähler metric from its embedding in C N , which we call the ambient metric. Due to the incompleteness of the metric, there are many possible closed L 2 -extensions of the ∂-operator originally acting on smooth forms on X * . We consider the maximal (distributional) ∂-operator. The pointwise norm of a form f defined in X * with respect to the ambient metric will be denoted by |f |, its L 2 -norm by f and the volume element by dV . Let R be a positive number. We set for 0 < r < R, B r := {z ∈ C N ; z < r}, B r := {z ∈ C N ; z ≤ r}, X r := X ∩ B r , X * r := X * ∩ B r and X r * := X * ∩ B r . We shall choose R small enough, so that bB r intersects X transversally for all 0 < r < R.In this paper we address the question of whether it is possible to solve the equation ∂u = f in X * r with L 2 -estimates for f ∈ L 2 p,q (X r ), ∂f = 0 in X * r . This problem was studied in [5] for conic singularities and ∂-closed, square integrable (0, 1) forms and later on in [4], for generic surfaces with an isolated singularity at 0 and for the same type of forms. Global analogues of this question for projective surfaces with isolated singularities were studied by Haskell [9], Nagase [12], Pardon [15], Pardon and Stern [16], and for compact Kähler spaces with isolated singularities by Ohsawa [13,14].There seems to be a "dichotomy of results" for our problem, depending on whether p + q < n or p + q > n. More precisely we have: Theorem 1.1. Let p + q < n, q > 0. There exists a closed subspace H of finite codimension in

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Abstract. We obtain some L 2 -results for the ∂ operator on forms that vanish to high order on the singular set of a complex space.

We construct a new solution operator for ∂ on certain piecewise smooth q-convex intersections. L p estimates are obtained for the solution operators of ∂-closed forms on such domains.

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