Existence of almost greedy bases in mixed-norm sequence and matrix spaces, including Besov spaces

TL;DR

This paper demonstrates the existence of almost greedy bases in certain mixed-norm sequence and matrix spaces, including Besov spaces, by extending existing methods to nonlocally convex components and analyzing their fundamental functions.

Contribution

It extends the Dilworth-Kalton-Kutzarova method to mixed-norm spaces with nonlocally convex parts, establishing the presence of almost greedy bases in these spaces.

Findings

Almost greedy bases exist in $oldsymbol{ ext{ell}_p ext{+} ext{ell}_q}$ and matrix spaces for $0<p<1<q< ext{infinity}.

Constructed bases have Lebesgue parameters growing in a prescribed manner.

Fundamental functions of these bases grow as $(m^{1/q})_{m=1}^ ext{infinity}$.

Abstract

We prove that the sequence spaces $\ell_p\oplus\ell_q$ and the spaces of infinite matrices $\ell_p(\ell_q)$, $\ell_q(\ell_p)$ and $(\bigoplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which are isomorphic to certain Besov spaces, have an almost greedy basis whenever $0<p<1<q<\infty$. More precisely, we custom-build almost greedy bases in such a way that the Lebesgue parameters grow in a prescribed manner. Our arguments critically depend on the extension of the Dilworth-Kalton-Kutzarova method from [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101], which was originally designed for constructing almost greedy bases in Banach spaces, to make it valid for direct sums of mixed-normed spaces with nonlocally convex components. Additionally, we prove that the fundamental functions of all almost greedy bases of these spaces grow as $(m^{1/q})_{m=1}^\infty$.