Basis properties of the Haar system in limiting Besov spaces

TL;DR

This paper investigates the basis properties of the Haar system in limiting Besov spaces, providing a comprehensive analysis of when it forms a Schauder basis and identifying new counterexamples.

Contribution

It offers a complete characterization of the Haar system's basis properties in limiting Besov spaces, including positive results and counterexamples, based on operator norm estimates.

Findings

Positive results for $q\,\leq\,\min\{1,p\}$

Counterexamples in other parameter regimes

Asymptotically optimal growth rates for dyadic averaging operators

Abstract

We study Schauder basis properties for the Haar system in Besov spaces $B^s_{p,q}(\mathbb{R}^d)$. 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 $\mathbb{E}_N$; in particular we find asymptotically optimal growth rates for the norms of these operators in global and local situations.