  In a convolution model, we observe random variables whose distribution is the
convolution of some unknown density f and some known or partially known noise
density g. In this paper, we focus on statistical procedures, which are
adaptive with respect to the smoothness parameter tau of unknown density f, and
also (in some cases) to some unknown parameter of the noise density g. In a
first part, we assume that g is known and polynomially smooth. We provide
goodness-of-fit procedures for the test H_0:f=f_0, where the alternative H_1 is
expressed with respect to L_2-norm. Our adaptive (w.r.t tau) procedure behaves
differently according to whether f_0 is polynomially or exponentially smooth. A
payment for adaptation is noted in both cases and for computing this, we
provide a non-uniform Berry-Esseen type theorem for degenerate U-statistics. In
the first case we prove that the payment for adaptation is optimal (thus
unavoidable). In a second part, we study a wider framework: a semiparametric
model, where g is exponentially smooth and stable, and its self-similarity
index s is unknown. In order to ensure identifiability, we restrict our
attention to polynomially smooth, Sobolev-type densities f. In this context, we
provide a consistent estimation procedure for s. This estimator is then
plugged-into three different procedures: estimation of the unknown density f,
of the functional \int f^2 and test of the hypothesis H_0. These procedures are
adaptive with respect to both s and tau and attain the rates which are known
optimal for known values of s and tau. As a by-product, when the noise is known
and exponentially smooth our testing procedure is adaptive for testing
Sobolev-type densities.
