  Spatially inhomogeneous functions, which may be smooth in some regions and
rough in other regions, are modelled naturally in a Bayesian manner using
so-called Besov priors which are given by random wavelet expansions with
Laplace-distributed coefficients. This paper studies theoretical guarantees for
such prior measures - specifically, we examine their frequentist posterior
contraction rates in the setting of non-linear inverse problems with Gaussian
white noise. Our results are first derived under a general local Lipschitz
assumption on the forward map. We then verify the assumption for two non-linear
inverse problems arising from elliptic partial differential equations, the
Darcy flow model from geophysics as well as a model for the Schr\"odinger
equation appearing in tomography. In the course of the proofs, we also obtain
novel concentration inequalities for penalized least squares estimators with
$\ell^1$ wavelet penalty, which have a natural interpretation as maximum a
posteriori (MAP) estimators. The true parameter is assumed to belong to some
spatially inhomogeneous Besov class $B^{\alpha}_{11}$, $\alpha>0$.
  In a setting with direct observations, we complement these upper bounds with
a lower bound on the rate of contraction for arbitrary Gaussian priors. An
immediate consequence of our results is that while Laplace priors can achieve
minimax-optimal rates over $B^{\alpha}_{11}$-classes, Gaussian priors are
limited to a (by a polynomial factor) slower contraction rate. This gives
information-theoretical justification for the intuition that Laplace priors are
more compatible with $\ell^1$ regularity structure in the underlying parameter.
