\(L^p\) estimates for the Cauchy-Riemann operator on \(q\)-convex intersections in \(\mathbb{C}^n\) (Q5928169)

A smooth bounded domain in \(\mathbb{C}^n\) is called \(q\)-convex if it admits a defining function \(\rho\) such that at every boundary point the Levi-form of \(\rho\) has at least \(n-q\) nonnegative eigenvalues. The authors construct a new solution operator for \(\overline\partial\) on piecewise smooth intersections of strictly \(q\)-convex domains. This involves quite delicate kernel constructions of Berndtsson-Anderson type, with multiple weights. For the resulting solution operators, the authors are able to show \({\mathcal L}^p\)-estimates, (Theorem 1). In fact, more precise estimates, from \({\mathcal L}^p\) to \({\mathcal L}^r\), are given that depend on the configuration of the intersecting domains that define the given domain (Theorem 2).