We give necessary and sufficient conditions on the symbols for which the corresponding pseudo-differential operators on the affine group are Hilbert–Schimdt operators. We also give a characterization of trace class pseudo-differential operators on the affine group. A trace formula for these trace class operators is also obtained. We have also obtained the \(L^2\) boundedness of the Weyl transforms on the affine group.

中文翻译：

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仿射群上的Hilbert–Schimdt和Trace类伪微分算符和Weyl变换

我们对仿射群上对应的伪微分算子是Hilbert–Schimdt算子的符号给出必要和充分的条件。我们还给出了仿射群上跟踪类伪微分算子的表征。还获得了这些跟踪类运算符的跟踪公式。我们还获得了仿射群上Weyl变换的\（L ^ 2 \）有界。