We give a characterization of the two-microlocal Besov spaces in terms of the local Besov type conditions. As an easy consequence, we obtain the inclusions between the two-microlocal Besov spaces and the local Besov spaces. These results are natural extensions of those obtained by Jaffard and Meyer, who treated the pointwise Hölder regularity in terms of two-microlocal estimates. The Daubechies wavelets play a key role throughout the paper.
We give a further decay estimate for the Dziubański–Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley–Wiener theorem for ultradifferentiable functions.
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