  We show that the Hardy spaces for Fourier integral operators form natural
spaces of initial data when applying $\ell^{p}$-decoupling inequalities to
local smoothing for the wave equation. This yields new local smoothing
estimates which, in a quantified manner, improve the bounds in the local
smoothing conjecture on $\mathbb{R}^{n}$ for $p\geq 2(n+1)/(n-1)$, and
complement them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under
application of Fourier integral operators, and they are essentially sharp.
