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Applying this result to an annulus between two bounded pseudoconvex domains in $\\mathbb{C}^n$, where the inner domain has\n  $\\mathcal{C}^{1,1}$ boundary, we show that the $L^2$ Dolbeault cohomology group in bidegree $(p,q)$ vanishes if $1\\leq q\\leq n-2$ and is Hausdorff and infinite-dimensional if $q=n-1$, so that the Cauchy-Riemann operator has closed range in each bidegree. 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