Greedy bases for Besov spaces

TL;DR

This paper investigates the existence of greedy bases in Banach spaces related to Besov spaces, establishing conditions under which such bases exist or do not, and characterizing when a 1-greedy basis is present.

Contribution

It proves the presence of greedy bases in certain Besov space isomorphic Banach spaces and characterizes when 1-greedy bases occur, extending the understanding of basis properties in these spaces.

Findings

Spaces with $1 extless q extless \infty$ have greedy bases.

Spaces with $q=1$ or $q= ext{c}_0$ do not have greedy bases.

A 1-greedy basis exists if and only if $p=q$ in the space.

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 $1\leq p \leq\infty$ 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 $1\leq p=q\le \infty$.