  This paper addresses the regularization by sparsity constraints by means of
weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special
attention is payed to convergence rates in norm and to source conditions. As
main result it is proven that one gets a convergence rate in norm of
$\sqrt{\delta}$ for $1\leq p\leq 2$ as soon as the unknown solution is sparse.
The case $p=1$ needs a special technique where not only Bregman distances but
also a so-called Bregman-Taylor distance has to be employed.
  For $p<1$ only preliminary results are shown. These results indicate that,
different from $p\geq 1$, the regularizing properties depend on the interplay
of the operator and the basis of sparsity. A counterexample for $p=0$ shows
that regularization need not to happen.
