Existence of Almost Greedy Bases in Mixed-Norm Sequence and Matrix Spaces, Including Besov Spaces

Abstract

AbstractWe prove that the sequence spaces $$\ell _p\oplus \ell _q$$ ℓ p ⊕ ℓ q and the spaces of infinite matrices $$\ell _p(\ell _q)$$ ℓ p ( ℓ q ) , $$\ell _q(\ell _p)$$ ℓ q ( ℓ p ) and $$(\bigoplus _{n=1}^\infty \ell _p^n)_{\ell _q}$$ ( ⨁ n = 1 ∞ ℓ p n ) ℓ q , which are isomorphic to certain Besov spaces, have an almost greedy basis whenever $$0<p<1<q<\infty $$ 0 < p < 1 < q < ∞ . 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 Dilworth et al. (Stud Math 159(1):67–101, 2003), 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 $$(m^{1/q})_{m=1}^\infty $$ ( m 1 / q ) m = 1 ∞ .