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

We prove that the sequence spaces lp ⊕ lq and the spaces of infinite matrices lp(lq ), lq l(p) and ( ∞ n=1 n lp)lq , 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 (ml/q )∞ m=l.Open Access funding provided by Universidad Pública de Navarra. F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces

On the quasi-greedy property and uniformly bounded orthonormal systems

Technical report, Aalborg Univ., Dept of Math.We derive a necessary condition for a uniformly bounded orthonormal basis for L2(\Omega), \Omega a probability space, to be quasi-greedy in Lp(\Omega), p \neq 2, and then use this condition to prove that many classical systems, such as the trigonometric system and Walsh system, fail to be quasi-greedy in Lp, p \neq 2, i.e., thresholding is not well-behaved in Lp, p \neq 2, for such systems