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

TL;DR

This paper establishes sharp upper bounds for $ ext{ell}^q(L^p)$ decoupling constants for paraboloids in high-dimensional spaces, providing extremisers and extending previous results in harmonic analysis.

Contribution

It proves sharp decoupling bounds for paraboloids in $ ext{ell}^q(L^p)$ spaces, including extremisers, generalizing earlier work and offering precise estimates.

Findings

Decoupling constants are bounded sharply up to epsilon-losses.

Extremisers for the decoupling inequalities are identified.

Results are valid for all dimensions $n \\ge 2$ and $p, q \\in [2, \\infty]$.

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 $\varepsilon$-losses.