Equations with homogeneous kernels and Mellin transformation of generalized functions

A. I. Komech

Teoretičeskaâ i matematičeskaâ fizika, Tome 27 (1976) no. 2, pp. 149-162

Résumé

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.