Bernstein-Sato-Polynome und Faltungsgruppen zu Differentialoperatoren

Abstract

This work relies on Bernstein’s method of analytic continuation of the distribution-valued function $P^1$ ($P$ a polynomial) with respect to the complex exponent 2. In the case of a homogeneous, elliptic partial differential operator $P(\mathrm{i}\partial /2\pi )$, the convolvability and the validity of the relation $T_{\lambda }\cdot T_{\mu }=T_{\lambda +\mu }$ in the “convolution group” $\{T_{\lambda }\}$, which corresponds to $P^1$ through Fourier transform, is characterized by a condition on the indices $\lambda ,\mu$. In this way, we generalize the known convolution properties of the elliptic Riesz kernels $R_{\lambda }$, which represent the convolution group of the Laplacean operator. In a second part, Bernstein’s process of analytic continuation is carried out in a constructive manner in the special case of the polynomial $P$ being of the form $x_1^m+\cdots +{x^{\prime }}_n^m$. The importance of this process for the computation of fundamental solutions of the powers of the corresponding differential operator $P(\mathrm{i}\partial /2\pi )$ is illustrated in working out the example ${\partial }_1^4+{\partial }_2^4$.