Function spaces for decoupling

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.

52 pages. Minor changes to the previous version. To appear in Journal of the London Mathematical Society