Haar wavelet characterization of Besov-type spaces with optimal indices and its application to pointwise multipliers

Abstract

In this article, we give a sufficient and necessary condition on the indices of Besov-type spaces on $\mathbb{R}$ under which the characterization of these spaces in terms of the Haar system holds. This confirms that the sufficient condition obtained by H. Triebel is optimal. Moreover, a special case of our results also negatively answers a question, asked by H. Triebel, about the characterization of lifted local bounded mean oscillation spaces $J^s\mathrm{bmo}$ on $\mathbb{R}$ with $s\in (-1,-1/2]$ in terms of the Haar system. As an application, we prove that characteristic functions of intervals are pointwise multipliers of Besov-type spaces with the aforementioned optimal indices.