This is an announcement for the paper “Building highly conditional quasi-greedy bases in classical Banach spaces” by Fernando Albiachttps://arxiv.org/find/math/1/au:+Albiac_F/0/1/0/all/0/1, José L. Ansorenahttps://arxiv.org/find/math/1/au:+Ansorena_J/0/1/0/all/0/1.

Abstract: It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_m[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_m[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m=\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. However, in the existing literature one finds very few instances of non-superreflexive spaces possessing quasi-greedy basis with conditionality constants as large as possible. Our goal in this article is to fill this gap. To that end we enhance and exploit a combination of techniques developed independently, on the one hand by Garrig'os and Wojtaszczyk in [Conditional quasi-greedy bases in Hilbert and Banach spaces, Indiana Univ. Math. J. 63 (2014), no. 4, 1017-1036] and, on the other hand, by Dilworth et al. in [On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101], and craft a wealth of new examples of non-superreflexive classical Banach spaces having quasi-greedy bases $\mathcal{B}$ with $k_m[\mathcal{B}]=\mathcal{O}(\log m)$.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1712.04004