Greedy bases for Besov spaces

We prove thatthe Banach space $({\oplus }_{n=1}^{\mathrm{\infty }}{\mathrm{\ell }}_p^n{)}_{{\mathrm{\ell }}_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\leq p \leq\infty$and$1<q<\infty$\mathrm{.}Furthermore,theBanachspaces$(\oplus_{n=1}^\infty \ell_p^n){\ell_1}$,with$1<p\le \infty$,and$(\oplus{n=1}^\infty \ell_p^n){c_0}$,with$1\le p<\infty$ do not have a greedy bases. We prove as well that the space $(\oplus{n=1}^\infty \ell_p^n)_{\ell_q}$hasa1-greedybasisifandonlyif$1\leq p=q\le \infty$.