Greedy Bases for Besov Spaces

Abstract

We prove that the Banach space $(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{\ell_{q}}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever 1≤p≤∞ and 1<q<∞. Furthermore, the Banach spaces $(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{1}}$, with 1<p≤∞, and $(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{c_{0}}$, with 1≤p<∞, do not have a greedy basis. We prove as well that the space $(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{q}}$ has a 1-greedy basis if and only if 1≤p=q≤∞.