  We study Tikhonov regularization for possibly nonlinear inverse problems with
weighted $\ell^1$-penalization. The forward operator, mapping from a sequence
space to an arbitrary Banach space, typically an $L^2$-space, is assumed to
satisfy a two-sided Lipschitz condition with respect to a weighted
$\ell^2$-norm and the norm of the image space. We show that in this setting
approximation rates of arbitrarily high H\"older-type order in the
regularization parameter can be achieved, and we characterize maximal subspaces
of sequences on which these rates are attained. On these subspaces the method
also converges with optimal rates in terms of the noise level with the
discrepancy principle as parameter choice rule. Our analysis includes the case
that the penalty term is not finite at the exact solution ('oversmoothing'). As
a standard example we discuss wavelet regularization in Besov spaces
$B^r_{1,1}$. In this setting we demonstrate in numerical simulations for a
parameter identification problem in a differential equation that our
theoretical results correctly predict improved rates of convergence for
piecewise smooth unknown coefficients.
