A SHORT SURVEY OF RECENT RESULTS ON BUSCHMAN-ERDELYI TRANSMUTATIONS

This short survey paper contains brief historical information, main known facts and original author's results on the theory of the Buschman-Erdelyi transmutations and some of their applications. The operators of Buschman-Erdelyi type were fi rst studied by E. T. Copson, R. G. Buschman and A. Erdelyi as integral operators. In 1990' s the author was fi rst to prove the transmutational nature of these operators and published papers with detailed study of their properties. This class include as special cases such famous objects as the Sonine-Poisson-Delsarte transmutations and the fractional Riemann-Lioville integrals. In this paper, the Buschman-Erdelyi transmutations are logically classi fi ed as operators of the fi rst kind with special case of zero order smoothness operators, second kind and third kind with special case of unitary Sonine-Katrakhov and Poisson-Katrakhov transmutations. We study such properties as transmutational conditions, factorizations, norm estimates, connections with classical integral transforms. Applications are considered to singular partial di ff erential equations, embedding theorems with sharp constants in Kipriyanov spaces, Euler-Poisson-Darboux equation including Copson lemma, generalized translations, Dunkl operators, Radon transform, generalized harmonics theory, Hardy operators, V. Katrakhov's results on pseudodi ff erential operators and boundary-value problems of new kind for equations with solutions of arbitrary growth at the isolated singularity for elliptic partial di ff erential equations.