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Besov priors are nonparametric priors that can model spatially inhomogeneousfunctions. They are routinely used in inverse problems and imaging, where theyexhibit attractive sparsity-promoting and edge-preserving features. A recentline of work has initiated the study of their asymptotic frequentistconvergence properties. In the present paper, we consider the theoreticalrecovery performance of the posterior distributions associated to Besov-Laplacepriors in the density estimation model, under the assumption that theobservations are generated by a possibly spatially inhomogeneous true densitybelonging to a Besov space. We improve on existing results and show thatcarefully tuned Besov-Laplace priors attain optimal posterior contractionrates. Furthermore, we show that hierarchical procedures involving ahyper-prior on the regularity parameter lead to adaptation to any smoothnesslevel.

Besov-Laplace priors in density estimation: optimal posterior contraction rates and adaptation