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Algebras of pseudo-differential operators with discontinuous symbols. (English) Zbl 1132.47059

Toft, Joachim (ed.) et al., Modern trends in pseudo-differential operators. Basel: Birkhäuser (ISBN 978-3-7643-8097-7/hbk; 978-3-7643-8116-5/e-book). Operator Theory: Advances and Applications 172, 207-233 (2007).
Let \(V(\mathbb R)\) be the Banach algebra of all functions of bounded total variation on \(\mathbb R\). The author studies the boundedness and compactness of pseudodifferential operators \(a(x,D)\) with bounded measurable \(V(\mathbb R)\)-valued symbols \(a(x,\cdot )\) on the spaces \(L^p(\mathbb R)\), \(1<p<\infty\). This extends the methods and results of the author’s earlier paper [Proc.Lond.Math.Soc.(3) 92, No.3, 713–761 (2006; Zbl 1100.47059)] devoted to the case of absolutely continuous symbols of bounded total variation. The technique is based on the integral analogue of the Carleson–Hunt theorem [see C.E.Kenig and P.A.Tomas, Stud.Math.68, 79–83 (1980; Zbl 0442.42013)].
The above framework includes symbols with first kind discontinuities in the spatial and dual variables. Three different Banach algebras of pseudodifferential operators are studied. The author constructs Fredholm symbol calculi for these algebras. An application to the Haseman boundary value problem (leading to singular integral operators with a shift) is given.
For the entire collection see [Zbl 1111.47003].

MSC:

47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
47L15 Operator algebras with symbol structure
47G30 Pseudodifferential operators
47G10 Integral operators
45P05 Integral operators
47A53 (Semi-) Fredholm operators; index theories
30E25 Boundary value problems in the complex plane
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