Many-dimensional operators of convolution type in spaces of weight-integrable functions

We show that a convolution operator in the weight space $L_p^{\langle b\rangle}$ is similar to a generalized convolution operator in $L_p$. We obtain necessary and sufficient conditions for an operator of convolution type, acting in a weight space, to have the Noether property in a cone. These conditions say, in effect, that the operator symbol must not degenerate on the hull of some tubular domain associated with the weight and the cone.