A Laurent expansion for solutions to elliptic equations

Abstract:

Let $P(\xi )$ be a homogeneous elliptic polynomial of degree $m$. Let $E$ be a fundamental solution for the partial differential operator $P(D)$. Suppose $\Omega$ is a neighborhood of 0 in ${{\mathbf {R}}^n}$. Suppose $f \in {C^\infty }(\Omega \sim \{ 0\} )$ satisfies $P(D)f = 0$ in $\Omega \sim \{ 0\}$. It is shown that there is a differential operator $H(D)$ (perhaps of infinite order) and a function $g \in {C^\infty }(\Omega )$ satisfying $P(D)g = 0$ in $\Omega$, such that $f = H(D)E + g$ in $\Omega \sim \{ 0\}$. This analog of the Laurent expansion for $f$ is made unique by requiring that the Cauchy principal value of $H(D)E$ be equal to $H(D)E$.