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 Mat. Sb. (N.S.), 1986, Volume 129(171), Number 1, Pages 3–19 (Mi msb1803)

Algebras generated by multidimensional singular integral operators and by coefficients admitting discontinuities of homogeneous type

N. L. Vasilevskii

Abstract: Let $\mathscr L$ be a union of finitely many smooth orientable bounded disjoint surfaces in $\mathbf R^n$ of various dimensions (between $1$ and $n-1$), and let $PC(\dot{\mathbf R}^n,\mathscr L)$ be the algebra of functions continuous on $\dot{\mathbf R}^n\setminus\operatorname{Int}\mathscr L$ ($\dot{\mathbf R}^n=\mathbf R^n\cup\{\infty\}$) and having discontinuities of homogeneous type on surfaces in $\mathscr L$. This article includes a description of the algebra of symbols for the algebra $\mathscr R$ generated by all the operators of the form $A=a(x)M$ acting in $L_2(\mathbf R^n)$, where $a(x)\in PC(\dot{\mathbf R}^n,\mathscr L)$ and $M=F^{-1}m(\xi)F$, with $F$ and $F^{-1}$ the direct and inverse Fourier transformations, respectively, and with $m(\xi)$ a homogeneous function on $\mathbf R^n$ of degree zero whose restriction to the unit sphere in $\mathbf R^n$ is continuous. A criterion for operators in $\mathscr R$ to be Noetherian operators is given.
Bibliography: 25 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 57:1, 1–19

Bibliographic databases:

UDC: 517.983+517.986.3
MSC: Primary 45E99, 47G05, 47D25; Secondary 35S99, 46L05, 47A53

Citation: N. L. Vasilevskii, “Algebras generated by multidimensional singular integral operators and by coefficients admitting discontinuities of homogeneous type”, Mat. Sb. (N.S.), 129(171):1 (1986), 3–19; Math. USSR-Sb., 57:1 (1987), 1–19

Citation in format AMSBIB
\Bibitem{Vas86} \by N.~L.~Vasilevskii \paper Algebras generated by multidimensional singular integral operators and by coefficients admitting discontinuities of homogeneous type \jour Mat. Sb. (N.S.) \yr 1986 \vol 129(171) \issue 1 \pages 3--19 \mathnet{http://mi.mathnet.ru/msb1803} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=830092} \zmath{https://zbmath.org/?q=an:0623.47064|0608.47053} \transl \jour Math. USSR-Sb. \yr 1987 \vol 57 \issue 1 \pages 1--19 \crossref{https://doi.org/10.1070/SM1987v057n01ABEH003052} 

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This publication is cited in the following articles:
1. Vasilev V., “Many-Dimensional Riemann Problem and Related Singular Internal Equations”, Differ. Equ., 31:3 (1995), 492–494
2. Karapetyants A. Rabinovich V. Vasilevski N., “On Algebras of Two Dimensional Singular Integral Operators with Homogeneous Discontinuities in Symbols”, Integr. Equ. Oper. Theory, 40:3 (2001), 278–308
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