Embedding of $H^s$ into $L^p$ for $p$ sufficiently big

Question

I am reading a book about Euler equation and at some point we aim to prove the existence of global solution in 2D and I struggle with a point in the proof. We have a strong solution of the Euler equation $u \in L^\infty([0,T], H^s)$ with $\nabla \cdot u = 0$ for $s > d/2 + 1 = 2$ (here $d$ is the dimension). We can see that $\omega \in H^{s-1}$ as $$|\hat \omega(t, \xi)| \le C|\xi||\hat u(t, \xi)|.$$ Now the author claims that the $L^p$ norm of $\omega $ for every $p$ sufficiently big vanishes, i.e. $$\frac{d}{dt} \| \omega(t) \|_{L^p} = 0.$$ However I don't really get why we can even consider this $L^p$ norm, a priori $\omega$ is not even in this space. Is there any embedding of $H^s$ into $L^p$ for $p$ sufficiently big that I am missing?