We prove a modified form of the classical Morrey-Kohn-Hörmander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in \({{\mathbb {C}}}^n\), where the inner domain has \({\mathcal {C}}^{1,1}\) boundary, we show that the \(L^2\) Dolbeault cohomology group in bidegree (p, q) vanishes if \(1\le q\le n-2\) and is Hausdorff and infinite-dimensional if \(q=n-1\), so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the \(L^2\) Sobolev space \(W^1\) on any pseudoconvex domain with \({\mathcal {C}}^{1,1}\) boundary. We also generalize our results to annuli between domains which are weakly q-convex in the sense of Ho for appropriate values of q.

我们证明了经典Morrey-Kohn-Hörmander身份的一种修改形式，适用于伪凹边界。将此结果应用于\（{{\ mathbb {C}}} ^ n \）中两个有界伪凸域之间的环，其中内部域具有\（{\ mathcal {C}} ^ {1,1} \）边界，我们证明如果（（1 \ le q \ le n-2 \）为零，则双度（p，  q）中的\（L ^ 2 \） Dolbeault同调群消失，如果\（q = n-1 \），因此Cauchy-Riemann算子在每个双学位中都有一个封闭范围。作为对偶结果，我们证明Cauchy-Riemann算子在\（L ^ 2 \） Sobolev空间\（W ^ 1 \）中是可解的在具有\（{\ mathcal {C}} ^ {1,1} \）边界的任何伪凸域上。我们还推广了我们的成果，这是弱域之间的环q -凸中豪感为适当的值q。