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Pseudodifferential Operators with Compound Slowly Oscillating Symbols

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Book cover The Extended Field of Operator Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 171))

Abstract

Let V (ℝ) denote the Banach algebra of absolutely continuous functions of bounded total variation on ℝ. We study an algebra \( \mathfrak{B} \) of pseudodifferential operators of zero order with compound slowly oscillating V (ℝ)-valued symbols (x, y) ↦ a(x, y, ·) of limited smoothness with respect to x, y ∈ ℝ. Sufficient conditions for the boundedness and compactness of pseudodifferential operators with compound symbols on Lebesgue spaces L p(ℝ) are obtained. A symbol calculus for the algebra \( \mathfrak{B} \) is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. A Fredholm criterion and an index formula for pseudodifferential operators A ∈ \( \mathfrak{B} \) are obtained. These results are carried over to Mellin pseudodifferential operators with compound slowly oscillating V (ℝ)-valued symbols. Finally, we construct a Fredholm theory of generalized singular integral operators on weighted Lebesgue spaces L p with slowly oscillating Muckenhoupt weights over slowly oscillating Carleson curves.

Partially supported by the CONACYT (México) Project No. 47220.

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Authors and Affiliations

  1. Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62209, Cuernavaca Morelos, México

    Yuri I. Karlovich

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  1. Yuri I. Karlovich
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Editors and Affiliations

  1. School of Mathematics and Statistics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, UK

    Michael A. Dritschel

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Karlovich, Y.I. (2006). Pseudodifferential Operators with Compound Slowly Oscillating Symbols. In: Dritschel, M.A. (eds) The Extended Field of Operator Theory. Operator Theory: Advances and Applications, vol 171. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7980-3_10

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