  Starting from the partial regularity results for suitable weak solutions to
the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg, as a
corollary, under suitable assumptions of local character on the initial data,
we prove a behavior in time of the $L^\infty_{loc}$-norm of the solution in a
neighborhood of $t=0$. The behavior is the same as for the resolvent operator
associated to the Stokes operator. Besides its own interest, the result is a
main tool to study the spatial decay estimates of a suitable weak solution,
performed in paper F. Crispo and P. Maremonti, On the spatial asymptotic decay
of a suitable weak solution to the Navier-Stokes Cauchy problem (submitted).
