Removable sets for pseudoconvexity for weakly smooth boundaries

We show that for bounded domains in ${\mathbb{C}}^n$ with ${\mathcal{C}}^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\mathrm{\partial }\mathrm{\Omega }\setminus F$ is ${\mathcal{C}}^2$-smooth and locally pseudoconvex at every point, then $\mathrm{\Omega }$ is globally pseudoconvex. Unlike in the globally ${\mathcal{C}}^2$-smooth case, the condition ``$F$ of (relative) empty interior’’ is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity.