  In this work we investigate some regularization properties of the
incompressible Euler equations and of the fractional Navier-Stokes equations
where the dissipative term is given by $(-\Delta)^\alpha$, for a suitable power
$\alpha \in (0,\frac{1}{2})$ (the only meaningful range for this result).
Assuming that the solution $u \in L^\infty _t(C^\theta_x)$ for some $\theta \in
(0,1)$ we prove that $u \in C^\theta_{t,x}$, the pressure $p\in
C^{2\theta-}_{t,x}$ and the kinetic energy $e \in
C^{\frac{2\theta}{1-\theta}}_t$. This result was obtained for the Euler
equations in [Is13] with completely different arguments and we believe that our
proof, based on a regularization and a commutator estimate, gives a simpler
insight on the result.
