Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter

Abstract:

Let $P(\lambda ,D) = \sum \nolimits _{|\alpha | \leq m} {{a_\alpha }(\lambda ){D^\alpha }}$ be a differential operator with constant coefficients ${a_\alpha }$ depending analytically on a parameter $\lambda$. Assume that each $P(\lambda ,D)$ is hypoelliptic and that the strength of $P(\lambda ,D)$ is independent of $\lambda$. Under this condition we show that there exists a regular fundamental solution of $P(\lambda ,D)$ which also depends analytically on $\lambda$.