# Function spaces for decoupling

## TL;DR

This paper introduces new function spaces that reformulate decoupling inequalities and improve classical estimates, with invariance properties under specific operators.

## Contribution

The authors define novel function spaces $\\mathcal{L}_{W,s}^{q,p}$ that connect decoupling inequalities with Hardy spaces for Fourier integral operators.

## Key findings

- Spaces are invariant under Euclidean half-wave propagators.
- Achieve improvements in fractional integration theorems.
- Enhance local smoothing estimates.

## Abstract

We introduce new function spaces $\mathcal{L}_{W,s}^{q,p}(\mathbb{R}^{n})$ that yield a natural reformulation of the $\ell^{q}L^{p}$ decoupling inequalities for the sphere and the light cone. These spaces are invariant under the Euclidean half-wave propagators, but not under all Fourier integral operators unless $p=q$, in which case they coincide with the Hardy spaces for Fourier integral operators. We use these spaces to obtain improvements of the classical fractional integration theorem and local smoothing estimates.