Sharp ${\ell }^q(L^p)$ decoupling for paraboloids

Abstract

In this short expository note, we prove the following result, which is a special case of the main theorem in arXiv:2011.09451. For each $n\ge 2$ and $p,q\in [2,\infty ]$, we prove upper bounds of ${\ell }^q(L^p)$ decoupling constants for paraboloids in ${\mathbb{R}}^n$, as well as presenting extremisers for each case. Both are sharp up to $\epsilon$-losses.

Added references; made it clear in the abstract that this work is expository