Geodesic
Equations with homogeneous kernels and Mellin transformation of generalized functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 27 (1976) no. 2, pp. 149-162 Cet article a éte moissonné depuis la source Math-Net.Ru

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If an integrodifferential operator $A$ with homogeneous kernel on a half-axis is to be continuous in the space of tempered distributions, it is necessary and sufficient that its kernel satisfy a smoothness condition (Theorem 4, Definition 6). Under this condition, the eigenvalue $A^{-1}(\xi)$ corresponding to the eigenhtaction $x_{+}^{-i\xi}$ has growth not higher than a power as $|\xi|\to\infty$, $|\operatorname{Im}\xi|\leqslant C<\infty$. The operator $A$ is normally solvable if (and only if, under certain restrictions) $A^{-1}(\xi)$ also has growth not higher than a power for the same $\xi$. Expressions (2.12) are obtained for the general solution of the equation $Au=f$ in the form of convergent, i.e., regularized, integrals. The formalism of the Mellin transformation of generalized functions is developed for this purpose. 
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     author = {A. I. Komech},
     title = {Equations with homogeneous kernels and {Mellin} transformation of generalized functions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {149--162},
     year = {1976},
     volume = {27},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1976_27_2_a2/}
}
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A. I. Komech. Equations with homogeneous kernels and Mellin transformation of generalized functions. Teoretičeskaâ i matematičeskaâ fizika, Tome 27 (1976) no. 2, pp. 149-162. http://geodesic.mathdoc.fr/item/TMF_1976_27_2_a2/

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