A short survey of recent results on Buschman-Erdelyi transmutations

TL;DR

This survey reviews recent developments in Buschman-Erdélyi transmutations, highlighting their properties, classifications, and applications in various areas of mathematical analysis and partial differential equations.

Contribution

It provides a comprehensive classification of Buschman-Erdélyi transmutations and discusses their properties and applications, including original results by the author.

Findings

Classified Buschman-Erdélyi transmutations into three kinds.

Established transmutational properties and factorizations.

Connected these operators with classical integral transforms and applications.

Abstract

This paper was published in the special issue of the Journal of Inequalities and Special Functions dedicated to Professor Ivan Dimovski's contributions to different fields of mathematics: transmutation theory, special functions, integral transforms, function theory etc. This short survey paper contains brief historical information, main known facts and original author's results on the theory of the Buschman-Erd\'{e}lyi transmutations and some of their applications. The operators of Buschman-Erd\'{e}lyi type were first studied by E.T.~Copson, R.G.~Buschman and A.~Erd\'{e}lyi as integral operators. In 1990's the author was first 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 classified as operators of the first 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 differential 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 pseudodifferential operators and boundary-value problems of new kind for equations with solutions of arbitrary growth at the isolated singularity for elliptic partial differential equations.