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Elliptic Problems and Hörmander Spaces

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Modern Analysis and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 191))

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Abstract

The paper gives a survey of the modern results on elliptic problems on the Hörmander function spaces. More precisely, elliptic problems are studied on a Hilbert scale of the isotropic Hörmander spaces parametrized by a real number and a function slowly varying at +∞ in the Karamata sense. This refined scale is finer than the Sobolev scale and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this scale. A local refined smoothness of the elliptic problem solution is studied. An abstract construction of classes of function spaces in which the elliptic problem is a Fredholm one is found. In particular, some generalizations of the Lions-Magenes theorems are given.

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Author information

Authors and Affiliations

  1. Institute of Mathematics National Academy of Sciences of Ukraine, Tereshchenkivs’ka str. 3, 01601, Kyiv, Ukraine

    Vladimir A. Mikhailets & Aleksandr A. Murach

  2. Chernigiv State Technological University, Shevchenka str. 95, 14027, Chernigiv, Ukraine

    Aleksandr A. Murach

Authors
  1. Vladimir A. Mikhailets
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  2. Aleksandr A. Murach
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Vadim M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    Israel Gohberg

  3. Insitute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivska st., 01601, Kyiv-4, Ukraine

    Anatoly Kochubei , Yurij Berezansky , Myroslav Gorbachuk  & Valentyna Gorbachuk , ,  & 

  4. Department of Mathematical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Gennadiy Popov

  5. Institute of Analysis and Scientific Computing, Technical University of Vienna, Wiedner Hauptstrasse 8-10, 1040, Vienna, Austria

    Heinz Langer

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Mikhailets, V.A., Murach, A.A. (2009). Elliptic Problems and Hörmander Spaces. In: Adamyan, V.M., et al. Modern Analysis and Applications. Operator Theory: Advances and Applications, vol 191. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9921-4_28

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