Greedy bases for Besov spaces

Abstract

We prove thatthe Banach space $({\oplus }_{n=1}^{\infty }{\ell }_p^n{)}_{{\ell }_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\le p\le \infty$ and $1<q<\infty$. Furthermore, the Banach spaces $({\oplus }_{n=1}^{\infty }{\ell }_p^n{)}_{{\ell }_1}$, 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 $({\oplus }_{n=1}^{\infty }{\ell }_p^n{)}_{{\ell }_q}$ has a 1-greedy basis if and only if $1\le p=q\le \infty$.