  The objective of the present paper is to introduce the concept of a spatially
inhomogeneous linear inverse problem where the degree of ill-posedness of
operator $Q$ depends not only on the scale but also on location. In this case,
the rates of convergence are determined by the interaction of four parameters,
the smoothness and spatial homogeneity of the unknown function $f$ and degrees
of ill-posedness and spatial inhomogeneity of operator $Q$. Estimators obtained
in the paper are based either on wavelet-vaguelette decomposition (if the norms
of all vaguelettes are finite) or on a hybrid of wavelet-vaguelette
decomposition and Galerkin method (if vaguelettes in the neighborhood of the
singularity point have infinite norms). The hybrid estimator is a combination
of a linear part in the vicinity of the singularity point and the nonlinear
block thresholding wavelet estimator elsewhere. To attain adaptivity, an
optimal resolution level for the linear, singularity affected, portion of the
estimator is obtained using Lepski [Theory Probab. Appl. 35 (1990) 454-466 and
36 (1991) 682-697] method and is used subsequently as the lowest resolution
level for the nonlinear wavelet estimator. We show that convergence rates of
the hybrid estimator lie within a logarithmic factor of the optimal minimax
convergence rates. The theory presented in the paper is supplemented by
examples of deconvolution with a spatially inhomogeneous kernel and
deconvolution in the presence of locally extreme noise or extremely
inhomogeneous design. The first two problems are examined via a limited
simulation study which demonstrates advantages of the hybrid estimator when the
degree of spatial inhomogeneity is high. In addition, we apply the technique to
recovery of a convolution signal transmitted via amplitude modulation.
