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Smoothness Scales and Calderón–Zygmund Theory in the Scalar–Valued Case

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Multi-Layer Potentials and Boundary Problems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2063))

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Abstract

While one of the main goals of this monograph is the systematic development of a Calderón–Zygmund theory for multi-layer type operators associated with higher-order operators with matrix-valued coefficients, the starting point is the consideration of the scalar-valued case. As such, the aim of this introductory chapter is to present an account of those aspects of the scalar theory which are most relevant for the current work.

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Notes

  1. 1.

    A quick inspection reveals that the same result and proof are valid in the more general case of domains whose complement is regular in the sense of Whitney (in the terminology of [57, p. 52]; cf. also [101, p. 1372] where the notion of quasi-Euclideanity is employed), i.e., subsets of \({\mathbb{R}}^{n}\) with the property that any two points X, Y in the complement may be joined with a rectifiable curve \({\gamma }_{X,Y }\) disjoint from the set in question and which satisfies (2.318).

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Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, USA

    Irina Mitrea

  2. Department of Mathematics, University of Missouri, Columbia, MO, USA

    Marius Mitrea

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Mitrea, I., Mitrea, M. (2013). Smoothness Scales and Calderón–Zygmund Theory in the Scalar–Valued Case. In: Multi-Layer Potentials and Boundary Problems. Lecture Notes in Mathematics, vol 2063. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32666-0_2

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