Asymptotic analysis of fundamental solutions of hypoelliptic operators

Abstract

Abstract Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator 𝐏 ⁢ ( i ⁢ ∂ x ) = ( P 1 ⁢ ( i ⁢ ∂ x ) ) m 1 ⁢ ⋯ ⁢ ( P l ⁢ ( i ⁢ ∂ x ) ) m l \mathbf{P}(i\partial_{x})=(P_{1}(i\partial_{x}))^{m_{1}}\cdots(P_{l}(i\partial% _{x}))^{m_{l}} with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation 𝐏 ⁢ ( i ⁢ ∂ x ) ⁢ u = f {\mathbf{P}(i\partial_{x})u=f} in ℝ n {\mathbb{R}^{n}} . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation.