A Weakening of the Curvature Condition in \(\mathbb{R}^3\) for \(\ell^p\) Decoupling


Speaker(s): Dominique Kemp (Indiana University)
The celebrated decoupling theorem of Bourgain and Demeter allows for a decomposition in the \(L^p\) norm of functions Fourier supported near curved hypersurfaces \(M \subset \mathbb{R}^n\). In this project, we find that the condition of non-vanishing principal curvatures may be weakened. When \(M \subset \mathbb{R}^3\), we may allow one principal curvature at a time to vanish, and it is assumed additionally that \(M\) is foliated by a canonical choice of curves having nonzero curvature at every point. We find that \(\ell^p\) decoupling over nearly flat subsets of \(M\) holds within this context.