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Fourier multipliers, symbols, and nuclearity on compact manifolds

Authors
  • Delgado, Julio1
  • Ruzhansky, Michael1
  • 1 Imperial College London, Department of Mathematics, 180 Queen’s Gate, London, SW7 2AZ, United Kingdom , London (United Kingdom)
Type
Published Article
Journal
Journal d'Analyse Mathématique
Publisher
The Hebrew University Magnes Press
Publication Date
Jun 01, 2018
Volume
135
Issue
2
Pages
757–800
Identifiers
DOI: 10.1007/s11854-018-0052-9
Source
Springer Nature
License
Green

Abstract

The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite-dimensional subspaces. As a consequence, given a compact manifold M endowed with a positive measure, we introduce a notion of the operator’s full symbol adapted to the Fourier analysis relative to a fixed elliptic operator E. We give a description of Fourier multipliers, or of operators invariant relative to E. We apply these concepts to study Schatten classes of operators on L2(M) and to obtain a formula for the trace of trace class operators. We also apply it to provide conditions for operators between Lp-spaces to be r-nuclear in the sense of Grothendieck.

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