Let \(D \subset \mathbb C^n\) and \(G \subset \mathbb C^m\) be pseudoconvex domains, let \(A\) (resp. \(B\)) be an open subset of the boundary \(\partial D\) (resp. \(\partial G\)) and let \(X\) be the 2-fold cross \(((D\cup A)\times B)\cup(A\times (B\cup G))\). Suppose in addition that the domain \(D\) (resp. \(G\)) is locally \(\mathcal C^2\) smooth on \(A\) (resp. \(B\)).
The authors determine the “envelope of holomorphy” \(\widehat X\) of \(X\) in the sense that any function continuous on \(X\) and separately holomorphic on (\(A\times G)\cup (D\times B)\) extends to a function continuous on \(\widehat X\) and holomorphic on the interior of \(\widehat X\). A mixed cross theorem and two quantitative cross theorems are established. A generalization to \(N\)-fold crosses is also given: Theorem. {Let \(D_j \subset \mathbb C^{n_j}\) be a pseudoconvex domain and let \(\varnothing\neq A_j\) be an open subset of \(\partial D_j\), \(j = 1,\dots,N\). Suppose in addition that each domain \(D_j\) is locally \(\mathcal C^2\) smooth on \(A_j\), \(j = 1,\dots,N\). Then \(X \subset\widehat X\) and for any \(f\in\mathcal C(X)\cap\mathcal O_s(X^\circ)\), there is a unique \(\widehat f\in\mathcal C({\widehat X})\cap\mathcal O({\widehat X}^\circ)\) such that \(\widehat f = f\) on X. Moreover, if \(| f| _X <\infty\) then} \[ | \widehat f(z)| \leq | f| _A^{1-\omega(z)}| f| _X^{\omega(z)},\quad z\in\widehat X. \] Some estimates for the plurisubharmonic measures, that are crucial for the proof of this theorem are established. The case of a 2-fold cross is considered separately.