  The paper concerns problems of the recovery of operators from noisy
information in weighted $L_q$-spaces with homogeneous weights. A number of
general theorems are proved and applied to finding exact constants in
multidimensional Carlson type inequalities with several weights and problems of
the recovery of differential operators from a noisy Fourier transform. In
particular, optimal methods are obtained for the recovery of powers of
generalized Laplace operators from a noisy Fourier transform in the
$L_p$-metric.
