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Modulation spaces of symbols for representations of nilpotent Lie groups. (English) Zbl 1257.47056
Summary: We investigate continuity properties of operators obtained as values of the Weyl correspondence constructed by N. V. Pedersen [Invent. Math. 118, No. 1, 1–36 (1994; Zbl 0848.22016)] for arbitrary irreducible representations of nilpotent Lie groups. To this end, we introduce modulation spaces for such representations and establish some of their basic properties. The situation of square-integrable representations is particularly important and, in the special case of the Schrödinger representation of the Heisenberg group, we recover the classical modulation spaces used in the time-frequency analysis.

MSC:
47G30 Pseudodifferential operators
22E25 Nilpotent and solvable Lie groups
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
35S05 Pseudodifferential operators as generalizations of partial differential operators
43A80 Analysis on other specific Lie groups
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