  We consider the inverse problem of recovering an unknown functional parameter
$u$ in a separable Banach space, from a noisy observation $y$ of its image
through a known possibly non-linear ill-posed map ${\mathcal G}$. The data $y$
is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach
to the problem and consider Besov space priors (see Lassas et al. 2009), which
are well-known for their edge-preserving and sparsity-promoting properties and
have recently attracted wide attention especially in the medical imaging
community.
  Our key result is to show that in this non-parametric setup the maximum a
posteriori (MAP) estimates are characterized by the minimizers of a generalized
Onsager--Machlup functional of the posterior. This is done independently for
the so-called weak and strong MAP estimates, which as we show coincide in our
context. In addition, we prove a form of weak consistency for the MAP
estimators in the infinitely informative data limit. Our results are remarkable
for two reasons: first, the prior distribution is non-Gaussian and does not
meet the smoothness conditions required in previous research on non-parametric
MAP estimates. Second, the result analytically justifies existing uses of the
MAP estimate in finite but high dimensional discretizations of Bayesian inverse
problems with the considered Besov priors.
