We study the behavior of Haar coefficients in Besov and Triebel–Lizorkin spaces on R\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb R}$$\end{document}, 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\documentclass[12pt]{minimal}
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\begin{document}$$s<1$$\end{document}, in which the spaces Fp,qs\documentclass[12pt]{minimal}
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\begin{document}$$F^s_{p,q}$$\end{document} and Bp,qs\documentclass[12pt]{minimal}
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\begin{document}$$B^s_{p,q}$$\end{document} are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that 1/p<s<1\documentclass[12pt]{minimal}
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\begin{document}$$1/p<s<1$$\end{document} and f∈Bp,qs\documentclass[12pt]{minimal}
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\begin{document}$$f\in B^s_{p,q}$$\end{document}, we actually prove that the usual Haar coefficient norm, ‖{2j⟨f,hj,μ⟩}j,μ‖bp,qs\documentclass[12pt]{minimal}
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\begin{document}$$\Vert \{2^j\langle f, h_{j,\mu }\rangle \}_{j,\mu }\Vert _{b^s_{p,q}}$$\end{document} remains equivalent to ‖f‖Bp,qs\documentclass[12pt]{minimal}
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\begin{document}$$\Vert f\Vert _{B^s_{p,q}}$$\end{document}, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case s=1\documentclass[12pt]{minimal}
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\begin{document}$$s=1$$\end{document} and q=∞\documentclass[12pt]{minimal}
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\begin{document}$$q=\infty $$\end{document}, we show that such an expression gives an equivalent norm for the Sobolev space Wp1(R)\documentclass[12pt]{minimal}
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\begin{document}$$W^{1}_p({\mathbb R})$$\end{document}, 1<p<∞\documentclass[12pt]{minimal}
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\begin{document}$$1<p<\infty $$\end{document}, which is related to a classical result by Bočkarev. Finally, in several endpoint cases we give optimal inclusions between Bp,qs\documentclass[12pt]{minimal}
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\begin{document}$$B^s_{p,q}$$\end{document}, Fp,qs\documentclass[12pt]{minimal}
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\begin{document}$$F^s_{p,q}$$\end{document}, and their dyadic counterparts.
