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We provide non-smooth atomic decompositions for Besov spaces $\Bd(\rn)$,$s>0$, $0<p,q\leq \infty$, defined via differences. The results are used tocompute the trace of Besov spaces on the boundary $\Gamma$ of bounded Lipschitzdomains $\Omega$ with smoothness $s$ restricted to $0<s<1$ and no furtherrestrictions on the parameters $p,q$. We conclude with some more applicationsin terms of pointwise multipliers.

Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces