Haar frame characterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterparts

TL;DR

This paper characterizes Besov and Triebel-Lizorkin spaces on the real line using Haar coefficients, extends the parameter range for these characterizations, and explores optimal embeddings into dyadic spaces, including endpoint cases.

Contribution

It provides new Haar frame characterizations for Besov and Triebel-Lizorkin spaces, extending known results up to smoothness s<1, and clarifies embeddings and equivalences at endpoint cases.

Findings

Haar frame characterizations extend to s<1.

Classical Besov spaces are closed in their dyadic counterparts for 1/p<s<1.

Equivalent norms for Sobolev spaces at s=1, q=∞.

Abstract

We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin spaces on $\mathbb{R}$, for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness $s<1$, in which the spaces $F^s_{p,q}$ and $B^s_{p,q}$ are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that $1/p<s<1$ and $f\in B^s_{p,q}$, we actually prove that the usual Haar coefficient norm, $\|\{2^j\langle f, h_{j,\mu}\rangle\}_{j,\mu}\|_{b^s_{p,q}}$ remains equivalent to $\|f\|_{B^s_{p,q}}$, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case $s=1$ and $q=\infty$, we show that such an expression gives an equivalent norm for the Sobolev space $W^{1}_p(\mathbb{R})$, $1<p<\infty$, which is related to a classical result by Bo\v{c}karev. Finally, in several endpoint cases we clarify the relation between dyadic and standard Besov and Triebel-Lizorkin spaces.