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

#### October 20, 3:15pm - October 20, 4:15pm

##### 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.