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.

翻译：本文研究了在具有齐次权重的加权$L_q$空间中从含噪信息恢复算子的问题。证明了一系列一般性定理，并将其应用于寻找含多个权重的多维Carlson型不等式的精确常数，以及从含噪傅里叶变换恢复微分算子的相关问题。特别地，在$L_p$度量下，获得了从含噪傅里叶变换恢复广义拉普拉斯算子幂次的最优方法。