Bernstein-Sato-Polynome und Faltungsgruppen zu Differentialoperatoren. (Bernstein-Sato polynomials and convolution groups for differential operators)
The work relies on I. N. Bernstein's method of analytic continuation of the distribution-valued function \(P^{\lambda}\) (P a polynomial) with respect to the complex exponent \(\lambda\). In the case of a homogeneous, elliptic partial differential operator P(i\(\partial /2\pi)\), the convolvability and the validity of the relation \(T_{\lambda}*T_{\mu}=T_{\lambda +\mu}\) in the ``convolution group'' \(\{T_{\lambda}\}\), which corresponds to \(P^{\lambda}\) through Fourier transform, is characterized by a condition on the indices \(\lambda\), \(\mu\). In this way, the known convolution properties of the elliptic Riesz kernels, which represent the convolution group of the Laplacian operator, are generalized. 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 \(x^ m_ 1+...+x^ m_ n\). The significance of this process for the computation of fundamental solutions of the powers of the corresponding differential operator P(i\(\partial /2\pi)\) is illustrated in working out the example \(x^ 4_ 1+x^ 4_ 2\).
scientific article; zbMATH DE number 4136433