Existence of Almost Greedy Bases in Mixed-Norm Sequence and Matrix Spaces, Including Besov Spaces - Albiac, Fernando; Ansorena, José L.; Bello, Glenier; Wojtaszczyk, Przemysław (Constructive approximation, 2024)
We prove that the sequence spaces ℓ p ⊕ ℓ q and the spaces of infinite matrices ℓ p ( ℓ q ) , ℓ q ( ℓ p ) and ( ⨁ n = 1 ∞ ℓ p n ) ℓ q , which are isomorphic to certain Besov spaces, have an almost greedy basis whenever 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 ∞ .