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Different Besov-Norm Definitions

equivalent to the following but I'm not sure (especially about whether I want to allow negative $j$ and whether I have to use $\langle\rangle$ or not) and I'm failing to prove the equivalence of the Norms
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Different Besov-Norm Definitions

In this paper http://arxiv.org/abs/1007.3418 one can find definitions for both inhomogenous and homogenous besov norms. … The second definitions are the ones for the inhomogenous besov norms if you take away the ⟨⟩ brackets. …
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