The null space of the \(\overline \partial \)-Neumann operator. (Q1774064)

In a first part, the author proves results concerning closed range of \(\overline{\partial}\) and invertibility of \(\square\) (and concerning its null space on \((n-1)\)-forms) on spherical shells, by systematic use of spherical harmonics. Some of these results are known, but the author's methods yield more precise constants. In section 3, the author proves a theorem that precises the notion that when the Levi form has the critical signature at some point of the boundary (\((n-q-1,q)\)), then the \(\overline{\partial}\)-Neumann problem is flawed on the domain at the level of \((0,q)\)-forms: the dimension of the null space of \(\square_{q}\) is infinite, or \(\overline{\partial}\) does not have closed range on \(L^{2}_{(0,q)}\) (possibly both). One of the results in section 2 is that the null space of \(\square_{n-1}\) has \(n\) independent multipliers. In section 4, it is shown that this property characterizes shells bounded by two confocal ellipsoids. A description of the null space of \(\square_{n-1}\) is obtained for these domains that allows to discuss the boundary behavior of the kernel of the orthogonal projection on this null space when the range of \(\square_{n-1}\) is closed, at a boundary point where the Levi form has signature \((0,n-1)\).