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