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On the boundedness of singular integral operators with complicated singularities. (English. Russian original) Zbl 0535.47029

Sov. Math., Dokl. 27, 40-42 (1983); translation from Dokl. Akad. Nauk SSSR 268, 44-46 (1983).
Let \[ I(f)(y)=\int_{S}\frac{f(x)h(x)\omega(x)}{| \chi^ 1(x)- \chi^ 1(y)|^{k_ 1}...| \chi^ N(x)-\chi^ N(y)|^{k_ N}} \] where S is a smooth submanifold of a domain \(\Omega \subset R^ n\) of codimension d and with the volume form \(\omega\), \(\chi^{\alpha}:\Omega r\to R^{\alpha}\) are smooth nonsingular mappings, \(n_{\alpha}\leq n\), \(k_{\alpha}\geq 0 (\alpha =1,...,N)\) and h is a smooth function on S with compact support.
Let D be an appropriate domain such that \(S\subset \partial D\) and let \(I(f)(y^ 0)=\lim_{y\to y^ 0,y\in D}I(f)(y).\) Suppose that this limit exists for all \(y^ 0\in S\), independently of the choice of \(y\in L^{\infty}(S,\omega)\). There are given conditions for the operator I to transform \(L^{\infty}(S,\omega)\) into itself and to be bounded in this space.

MSC:

47Gxx Integral, integro-differential, and pseudodifferential operators
47B38 Linear operators on function spaces (general)
45H05 Integral equations with miscellaneous special kernels
45E05 Integral equations with kernels of Cauchy type
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