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

N. L. Vasilevskii

Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 1-19

Résumé

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.