Abstract
By definition a homogeneous Lie group is a Lie group equipped with a family of dilations compatible with the group law. The abelian group \( \left( {{\mathbb{R}}^n , + } \right) \) is the very first example of homogeneous Lie group. Homogeneous Lie groups have proved to be a natural setting to generalise many questions of Euclidean harmonic analysis. Indeed, having both the group and dilation structures allows one to introduce many notions coming from the Euclidean harmonic analysis. There are several important differences between the Euclidean setting and the one of homogeneous Lie groups. For instance the operators appearing in the latter setting are usually more singular than their Euclidean counterparts. However it is possible to adapt the technique in harmonic analysis to still treat many questions in this more abstract setting
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Fischer, V., Ruzhansky, M. (2016). Homogeneous Lie groups. In: Quantization on Nilpotent Lie Groups. Progress in Mathematics, vol 314. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29558-9_3
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DOI: https://doi.org/10.1007/978-3-319-29558-9_3
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Publisher Name: Birkhäuser, Cham
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