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Trace ideals for pseudo-differential operators and their commutators with symbols in α-modulation spaces

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Abstract

That symbols in the modulation space M 1,1 generate pseudo-differential operators of the trace class was first stated by Feichtinger and proved by Gröchenig in [13]. In this paper, we show that the same is true if we replace M 1,1 by the more general α-modulation spaces, which include modulation spaces (α = 0) and Besov spaces (α = 1) as special cases. The result with α = 0 corresponds to that of Gröchenig, and the one with α = 1 is a new result which states the trace property of the operators with symbols in the Besov space. As an application, we discuss the trace property of the commutator [α (X, D), a], where; a(χ) is a Lipschitz function and σ(χ, ξ) belongs to an α-modulation space.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku, Tokyo, 162-8601, Japan

    Masaharu Kobayashi

  2. Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-8602, Japan

    Mitsuru Sugimoto

  3. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Naohito Tomita

Authors
  1. Masaharu Kobayashi
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  2. Mitsuru Sugimoto
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  3. Naohito Tomita
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Correspondence to Masaharu Kobayashi.

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Kobayashi, M., Sugimoto, M. & Tomita, N. Trace ideals for pseudo-differential operators and their commutators with symbols in α-modulation spaces. J Anal Math 107, 141–160 (2009). https://doi.org/10.1007/s11854-009-0006-3

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  • DOI: https://doi.org/10.1007/s11854-009-0006-3

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