  This paper deals with Tikhonov regularization for linear and nonlinear
ill-posed operator equations with wavelet Besov norm penalties. We focus on
$B^0_{p,1}$ penalty terms which yield estimators that are sparse with respect
to a wavelet frame. Our framework includes among others, the Radon transform
and some nonlinear inverse problems in differential equations with distributed
measurements. Using variational source conditions it is shown that such
estimators achieve minimax-optimal rates of convergence for finitely smoothing
operators in certain Besov balls both for deterministic and for statistical
noise models.
