Basis properties of the Haar system in limiting Besov spaces

We study Schauder basis properties for the Haar system in Besov spaces $B_{p,q}^s({\mathbb{R}}^d)$. We give a complete description of the limiting cases, obtaining various positive results for $q\le \min \{1,p\}$, and providing new counterexamples in other situations. The study is based on suitable estimates of the dyadic averaging operators ${\mathbb{E}}_N$; in particular we find asymptotically optimal growth rates for the norms of these operators in global and local situations.