  The global existence issue for the isentropic compressible Navier-Stokes
equations in the critical regularity framework has been addressed in [7] more
than fifteen years ago. However, whether (optimal) time-decay rates could be
shown in general critical spaces and any dimension d $\ge$ 2 has remained an
open question. Here we give a positive answer to that issue not only in the L 2
critical framework of [7] but also in the more general L p critical framework
of [3, 6, 14]. More precisely, we show that under a mild additional decay
assumption that is satisfied if the low frequencies of the initial data are in
e.g. L p/2 (R d), the L p norm (the slightly stronger $\dot B^0_{p,1}$ norm in
fact) of the critical global solutions decays like t --d(1 p -- 1 4) for t
$\rightarrow$ +$\infty$, exactly as firstly observed by A. Matsumura and T.
Nishida in [23] in the case p = 2 and d = 3, for solutions with high Sobolev
regularity. Our method relies on refined time weighted inequalities in the
Fourier space, and is likely to be effective for other hyperbolic/parabolic
systems that are encountered in fluid mechanics or mathematical physics.
