Wavelets and bidemocratic pairs in weighted norm spaces (Q1731492)

Let $m\ge 2$ be a fixed integer. The authors characterize the weight functions $\omega$ such that a given $m$th rank Haar system $H(m)$ is a greedy basis for $L^p(\mathbb{R}, \omega)$, with $1 < p < \infty$, provided that $H(m)$ is an unconditional basis for $L^p(\mathbb{R}, \omega)$, see [\textit{K. S. Kazarian} et al., Tohoku Math. J. (2) 70, No. 4, 567--605 (2018; Zbl 1414.42043)]. It is shown that the class of weight functions $\omega$ for which $H(m)$ is an unconditional basis for $L^p(\mathbb{R}, \omega)$ coincides with the class of weight functions for which $\big( H(m), H^{\ast}(m)\big)$ is a bidemocratic pair for $L^p(\mathbb{R}, \omega)$. Here $ H^{\ast}(m)$ denotes the biorthogonal system to $H(m)$.