Haar frame characterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterparts
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We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin spaces on 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∈ B^s_p,q, we actually prove that the usual Haar coefficient norm, {2^j⟨ f, h_j,μ⟩}_j,μ_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=∞, we show that such an expression gives an equivalent norm for the Sobolev space W^1_p(ℝ), 1<p<∞, 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.
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