Tempered distributions and Fourier transform on the Heisenberg group
Tempered distributions and Fourier transform on the Heisenberg group
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海森堡群的调和分布和傅里叶变换
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
R. Danchin
中科院分区：
文献类型：
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作者：
H. Bahouri;J. Chemin;R. Danchin
本工作的最终目标是将海森堡群$\H^d，$上的傅立叶变换推广到回火分布。在欧氏空间中，我们的策略是首先证明傅里叶变换是施瓦茨空间上的同构，然后通过对偶定义扩张。这里遇到的困难是，$\H^d$上的可积函数的傅里叶变换不再是$\H^d$上的函数：根据标准定义，它是$L^2（\R^d）上的一族有界算子。在\ccite{bcdFH空间}中，我们将可积函数的傅里叶变换定义为集合~$\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}$上的映射，并赋予适当的距离$\wh d$.这一观点证明了傅里叶变换对$\H^d$上的Schwartz空间的范围提供了一个用户友好的描述，这使得扩展到整组回火分布是直接的。作为第一个应用程序，我们给出了一个明确的公式的傅立叶变换的光滑函数的$\H^d$是独立的垂直变量。我们还提供了其他例子。
The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on $\H^d$is no longer a function on $\H^d$ : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Fourier transform of an integrable functionto be a mapping on the set~$\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}$endowed with a suitable distance $\wh d$.This viewpoint turns out to provide a user friendly description of the range of the Schwartz space on $\H^d$ by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward. As a first application, we give an explicit formula for the Fourier transform of smooth functions on $\H^d$ that are independent of the vertical variable. We also provide other examples.