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Zeitschrift für Analysis und ihre Anwendungen

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Volume 8, Issue 5, 1989, pp. 407–423
DOI: 10.4171/ZAA/362

Published online: 1989-10-31

Bernstein-Sato-Polynome und Faltungsgruppen zu Differentialoperatoren

Peter Wagner[1]

(1) Universität Innsbruck, Austria

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 $x_1^m + \cdots + x'_n^m$. 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 $\partial _1^4 + \partial _2^4$.

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Wagner Peter: Bernstein-Sato-Polynome und Faltungsgruppen zu Differentialoperatoren. Z. Anal. Anwend. 8 (1989), 407-423. doi: 10.4171/ZAA/362