Function spaces for decoupling

We introduce new function spaces ${\mathcal{L}}_{W,s}^{q,p}({\mathbb{R}}^n)$ that yield a natural reformulation of the ${\mathrm{\ell }}^q{L}^p$ 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.