On Some Generalized Symmetric Integral Operators of Buschman-Erdelyi's Type

Virchenko, Nina

doi:10.2991/jnmp.1996.3.3-4.25

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…Among relatively recent papers dealing with Buschman-Erdélyi operators treated as integral ones, we note papers of Virchenko (see [642,643]), Kilbas, Rubin, Glushak, and their disciples. For instance, in [139,534], Buschman-Erdélyi operators in Lebesgue weight spaces and their multidimensional generalizations in the form of integrals over pyramidal regions are considered.…”

Section: First-kind Buschman-erdélyi Integral Transmutation Operators...mentioning

confidence: 99%

The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

Katrakhov¹,

Вячеславович²,

Sitnik

et al. 2018

SMFN

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The main content of this book is composed from two doctoral theses: by V. V. Katrakhov (1989) and by S. M. Sitnik (2016). In our work, for the ﬁrst time in the format of a monograph, we systematically expound the theory of transmutation operators and their applications to diﬀerential equations with singularities in coeﬃcients, in particular, with Bessel operators. Along with detailed survey and bibliography on this theory, the book contains original results of the authors. Signiﬁcant part of these results is published with detailed proofs for the ﬁrst time. In the ﬁrst chapter, we give historical background, necessary notation, deﬁnitions, and auxiliary facts. In the second chapter, we give the detailed theory of Sonin and Poisson transmutations. In the third chapter, we describe an important special class of the Buschman-Erde´lyi transmutations and their applications. In the fourth chapter, we consider new weighted boundary-value problems with Sonin and Poisson transmutations. In the ﬁfth chapter, we consider applications of the Buschman-Erde´lyi transmutations of special form to new boundary-value problems for elliptic equations with signiﬁcant singularities of solutions. In the sixth chapter, we describe a universal compositional method for construction of transmutations and its applications. In the concluding seventh chapter, we consider applications of the theory of transmutations to diﬀerential equations with variable coeﬃcients: namely, to the problem of construction of a new class of transmutations with sharp estimates of kernels for perturbed diﬀerential equations with the Bessel operator, and to special cases of the well-known Landis problem on exponential estimates of the rate of growth for solutions of the stationary Schro¨dinger equation. The book is concluded with a brief biographic essay about Valeriy V. Katrakhov, as well as detailed bibliography containing 648 references.

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Section: First-kind Buschman-erdélyi Integral Transmutation Operators...mentioning

confidence: 99%