We construct new $3$-dimensional variants of the classical Diederich-Fornaess worm domain. We show that they are smoothly bounded, pseudoconvex, and have nontrivial Nebenh\"{u}lle. We also show that their Bergman projections do not preserve the Sobolev space for sufficiently large Sobolev indices.
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Krantz et al. (2024) studied this question.
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