  We obtain lower norm bounds for multivariate functions in weighted Lebesgue
spaces on unbounded domains in finite Euclidean space. These are identified by
the solution of a system of nonlinear integral equations, with linear
constrains on one-dimensional marginals. We recognize the importance of the
Smirnov condition for the corresponding weights, which guarantees uniqueness of
solutions of the system. Sufficient conditions on weights are formulated so to
satisfy the Smirnov property, and counterexamples are constructed, when the
property is violated.
