Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone in R ⁿ and -Γ′ be the antipodes of the dual cone of Γ. Let {Mathematical expression} be a partial differential operator with constant coefficients in R ⁿ, where Q(ζ)≠0 on R ⁿ-iΓ′ and P _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ-iΓ′;P _j(ζ)=0, grad P _j(ζ)≠0} contains some real point on which grad P _j≠0 and grad P _j∉Γ∪(-Γ). Let C be an open cone in R ⁿ-Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. If u∈ℒ′∩L _loc ² (R ⁿ-Γ) and the support of P(-i∂/∂x)u is contained in Γ, then the condition {Mathematical expression} implies that the support of u is contained in Γ.