Basis properties of the Haar system in limiting Besov spaces

Gustavo Garrigós; Andreas Seeger; Tino Ullrich

doi:10.1007/978-3-030-72058-2\_11

paper

Basis properties of the Haar system in limiting Besov spaces

arXiv:1901.09117 · doi:10.1007/978-3-030-72058-2\_11

Abstract

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