Fourier multipliers, symbols, and nuclearity on compact manifolds

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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Correspondence to Michael Ruzhansky.

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Supported by Marie Curie IIF 301599 and by the Leverhulme Grant RPG-2014-02.

Supported by EPSRC grant EP/K039407/1.

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Delgado, J., Ruzhansky, M. Fourier multipliers, symbols, and nuclearity on compact manifolds. JAMA 135, 757–800 (2018). https://doi.org/10.1007/s11854-018-0052-9

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