Function spaces for decoupling

TL;DR

This work develops a novel family of function spaces $\mathcal{L}_{W,s}^{q,p}(\mathbb{R}^{n})$ that recasts $\ell^{q}L^{p}$ decoupling for the sphere and light cone within a framework compatible with wave-propagator analysis. The spaces interpolate between Hardy-type invariant spaces for Fourier integral operators (when $p=q$) and their generalizations, and they are invariant under the Euclidean half-wave group while generally failing for arbitrary FIOs when $p\neq q$. The authors establish Sobolev-type embeddings, fractional integration improvements, and a precise connection to decoupling inequalities, enabling sharper local smoothing and nonlinear wave estimates. The approach leverages a wave-packet transform to phase space, anisotropic parabolic Hardy spaces, and vector-valued maximal-function techniques, yielding new insights into how decoupling interacts with wave propagation and FIO mappings. Overall, the results broaden the toolkit for analyzing high-frequency, directionally localized data and offer improved regularity results for linear and nonlinear wave equations within a unified analytic framework.

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.

Theorems & Definitions (83)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4