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

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 $F_{p,q}^s$ and $B_{p,q}^s$ are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that $1\mathrm{/}p<s<1$ and $f\in B_{p,q}^s$, we actually prove that the usual Haar coefficient norm, $\mathrm{\parallel }\{2^j⟨f,h_{j,\mu }⟩{\}}_{j,\mu }{\mathrm{\parallel }}_{b_{p,q}^s}$ remains equivalent to $\mathrm{\parallel }f{\mathrm{\parallel }}_{B_{p,q}^s}$, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case $s=1$ and $q=\mathrm{\infty }$, we show that such an expression gives an equivalent norm for the Sobolev space $W_p^1(\mathbb{R})$, $1<p<\mathrm{\infty }$, which is related to a classical result by Bočkarev. Finally, in several endpoint cases we clarify the relation between dyadic and standard Besov and Triebel-Lizorkin spaces.