On selection of infinitely differentiable solutions of a class of partially hypoelliptic equations

Abstract

In this paper the existence of a constant κ₀ > 0 is proved such that all solutions of a class of regular partially hypoelliptic (with respect to the hyperplane x'' = (x₂, …, xₙ) = 0 of the space Eⁿ) equations P(D)u = 0 in the strip Ωκ = {(x₁, x'') = (x₁, x₂, …, xₙ) ∈ Eⁿ; |x₁| < κ} are infinitely differentiable when κ ≥ κ₀ and D^α u ∈ L₂(Ωκ) for all multi-indices α = (0, α'') = (0, α₂, …, αₙ) in the Newton polyhedron of the operator P(D).