Given a 2-step stratified group which does not satisfy a slight strengthening of the Moore–Wolf condition, a sub-Laplacian \({\mathcal {L}}\) and a family \({\mathcal {T}}\) of elements of the derived algebra, we study the convolution kernels associated with the operators of the form \(m({\mathcal {L}}, -\,i {\mathcal {T}})\). Under suitable conditions, we prove that: (i) if the convolution kernel of the operator \(m({\mathcal {L}},-\,i {\mathcal {T}})\) belongs to \(L^1\), then m equals almost everywhere a continuous function vanishing at \(\infty \) (‘Riemann–Lebesgue lemma’); (ii) if the convolution kernel of the operator \(m({\mathcal {L}},-\,i{\mathcal {T}})\) is a Schwartz function, then m equals almost everywhere a Schwartz function.

中文翻译：

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两步分层组的频谱乘数，I

给定一个两步分层的组，该组不能满足摩尔-沃尔夫条件的稍微增强，一个亚拉普拉斯语\（{\ mathcal {L}} \）和一个家庭\（{\ mathcal {T}} \}对于派生代数的元素，我们研究与\（m（{\ mathcal {L}}，-\，i {\ mathcal {T}}）\）形式的算子相关的卷积核。在适当的条件下，我们证明：（i）如果算子\（m（{\ mathcal {L}}，-\，i {\ mathcal {T}}）\）的卷积核属于\（L ^ 1 \），则m几乎等于在各处都消失的连续函数\（\ infty \）（'Riemann–Lebesgue lemma'）; （ii）算子的卷积核\（m（{\ mathcal {L}}，-\，i {\ mathcal {T}}）\）是Schwartz函数，则m几乎在所有Schwartz函数中都等于。