Geodesic
The cauchy problem and other related problems for convolution equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 27 (1972) no. 4, pp. 71-160

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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. 
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     author = {L. R. Volevich and S. G. Gindikin},
     title = {The cauchy problem and other related problems for convolution equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {71--160},
     publisher = {mathdoc},
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     number = {4},
     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_1972_27_4_a2/}
}
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L. R. Volevich; S. G. Gindikin. The cauchy problem and other related problems for convolution equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 27 (1972) no. 4, pp. 71-160. http://geodesic.mathdoc.fr/item/RM_1972_27_4_a2/