齐次李群可解扩张的广义泊松核可卷积分布
Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups
摘要：
本文刻画了一类齐次李群$\fN$上的分布，它可以通过Poisson积分扩展为$\fN$的可解一维扩展$\fS$.为此，我们介绍了$ß'$ -卷积$\fN$和显示的一组分布，$ß'$ -卷积的泊松核正是一组适当加权的衍生物的$L^1$ -功能。此外，我们证明了这样的分布与泊松核的$ß'$卷积是调和的，并具有预期的边界行为。最后，我们证明了这样的分布满足一些整体弱$L^1$估计。
英文摘要：
In this paper, we characterize the class of distributions on an homogeneous Lie group $\fN$ that can be extended via Poisson integration to a solvable one-dimensional extension $\fS$ of $\fN$. To do so, we introducte the $ß'$-convolution on $\fN$ and show that the set of distributions that are $ß'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions. Moreover, we show that the $ß'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behaviour. Finally, we show that such distributions satisfy some global weak-$L^1$ estimates.
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