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Pseudodifferential Operators Associated with a Semigroup of Operators

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Abstract

Related to a semigroup of operators on a metric measure space, we define and study pseudodifferential operators (including the setting of Riemannian manifolds, fractals, graphs etc.). Boundedness on L p for pseudodifferential operators of order 0 is proved. We mainly focus on symbols belonging to the class \(S^{0}_{1,\delta}\) for δ∈[0,1). For the limit class \(S^{0}_{1,1}\), we describe some results by restricting our attention to the case of a sub-Laplacian operator on a Riemannian manifold.

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Notes

  1. We have to check that for fixed x, the map σ( ⋅ ,L)(f)(x) belongs to \(\bigcap_{j} {\mathcal{D}}(\Delta^{j})\). Indeed, this is the case if we consider “elementary” symbols of the form (3.8) which is sufficient due to Lemma 3.4 below.

  2. It is probably possible to extend the next results in a more general framework with different operators H, L with some commutativity assumptions. Here, we prefer to focus on this simpler situation for convenience.

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Authors and Affiliations

  1. Laboratoire Jean Leray, CNRS, Université de Nantes, 2, rue de la Houssinière, 44322, Nantes cedex 3, France

    Frédéric Bernicot

  2. Mathematical Sciences Institute, Australian National University, John Dedman Building, Canberra, ACT, 0200, Australia

    Dorothee Frey

Authors
  1. Frédéric Bernicot
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  2. Dorothee Frey
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Correspondence to Frédéric Bernicot.

Additional information

Communicated by Stéphane Jaffard.

The first author is supported by the ANR under the project AFoMEN No. 2011-JS01-001-01.

The second author is supported by the Australian Research Council Discovery grants DP110102488 and DP120103692.

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Bernicot, F., Frey, D. Pseudodifferential Operators Associated with a Semigroup of Operators. J Fourier Anal Appl 20, 91–118 (2014). https://doi.org/10.1007/s00041-013-9309-y

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  • DOI: https://doi.org/10.1007/s00041-013-9309-y

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