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

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 to compute the trace of Besov spaces on the boundary $\Gamma$ of bounded Lipschitz domains $\Omega$ with smoothness $s$ restricted to $0<s<1$ and no further restrictions on the parameters $p,q$. We conclude with some more applications in terms of pointwise multipliers.