  We derive multiscale statistics for deconvolution in order to detect
qualitative features of the unknown density. An important example covered
within this framework is to test for local monotonicity on all scales
simultaneously. We investigate the moderately ill-posed setting, where the
Fourier transform of the error density in the deconvolution model is of
polynomial decay. For multiscale testing, we consider a calibration, motivated
by the modulus of continuity of Brownian motion. We investigate the performance
of our results from both the theoretical and simulation based point of view. A
major consequence of our work is that the detection of qualitative features of
a density in a deconvolution problem is a doable task although the minimax
rates for pointwise estimation are very slow.
