On the singularities of solutions of partial differential equations with constant coefficients

LetP be a differential operator with constant coefficients in ℝ ⁿ . Ifu is a distribution, the singular support ofu is the complement of the largest set whereu ∈C ^∞. Necessary and sufficient conditinos are obtained for a closed convex set Γ to be equal to the singular support ofu for someu withPu ∈C ^∞ or, equivalently, for Γ to contain the singular support ofu for someu withPu ∈C ^∞ butu ∉C ^∞. Related local uniqueness theorems analogous to the Holmgren theorem with supports replaced by singular supports are also given, as well as applications concerningP-convexity with respect to singular supports.