Bernstein-Sato-Polynome und Faltungsgruppen zu Differentialoperatoren
Peter E. Wagner · Zeitschrift für Analysis und ihre Anwendungen · 1989
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 .