This article is cited in 17 papers

The cauchy problem and other related problems for convolution equations

L. R. Volevich, S. G. Gindikin


Abstract: Spaces of generalized functions with exponential asymptotic behaviour are considered. Convolutors in these spaces are completely described. It is shown that a convolution equation is uniquely soluble if and only if there exists a fundamental solution that is a convolutor. The explicit description of convolutors renders this condition effective. In particular, Petrovskii's correctness condition is obtained in the case of differential equations. A calculus of pseudodifferential operators with inhomogeneous symbols of constant strength is constructed; the solubility of the Cauchy problem can be proved by means of this calculus for a certain class of differential equations with variable coefficients.

UDC: 517.9

MSC: 45E10, 42A85, 42A38, 35Sxx

Received: 02.01.1972

 English version: Russian Mathematical Surveys, 1972, 27:4, 71–160

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