Analytic discs and extension of CR functions (Q2747672)

The authors introduce a notion of weak \(q\)-pseudoconvexity at a boundary point \(z_0\) of a \(\mathcal C^2\)-smooth domain \(\Omega\) of \(\mathbb C^N\); namely they assume that in a suitable \(\mathcal C^2\) orthogonal frame at \(z_0\), the complex Hessian \((1/i)\partial\bar\partial\rho\) of a defining function \(\rho\) of \(\Omega\) is block diagonal with a \(\leq 0\) block of size \(q\) and a \(\geq 0\) block of size \((N-q)\). Under this assumption they prove that the equation \(\bar\partial u=f\) for forms \(f\) of degree \(\geq (q+1)\) that are smooth and satisfy \(\bar\partial f=0\) on \(U\cap\overline{\Omega}\), (for a fundamental system of open neighborhoods \(U\) of \(z_0\)) is solvable with \(u\) smooth in \(U'\cap\overline{\Omega}\) provided \(U'\Subset U\).