On subspaces of Cesàro spaces

We obtain a characterization of subspaces of $L_p$, with $1<p<\infty$, on which the $L_p$-norm is equivalent to the norm of the Cesàro space $\operatorname{Ces}_p$. Also, we show that $\operatorname{Ces}_p$ has a complemented copy of the Cesàro sequence space $\operatorname{ces}_p$.