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

Abstract

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 \(\varepsilon\)-losses.