Besov regularity of the uniform empirical process

The paths of Brownian motion have been widely studied in the recent years relatively in Besov spaces $B_{p,\mathrm{\infty }}^{\text{a}}$. The results are the same as to the Brownian bridge. In fact these regularities properties are established in some sequence spaces $S_{p,\mathrm{\infty }}^{\text{a}}$ using an isomorphisim between them and $B_{p,\mathrm{\infty }}^{\text{a}}$. In this note, we are concerned with the regularity of the paths of the continuous version of the uniform empirical process in the space $S_{p, \infty}^\a$andinoneofhisseparablesubspace$S_{p, \infty}^{\a, 0}$ for a suitable choice of $\text{a}$ and $p$.