Boundary behavior of generalized Poisson integrals on buildings of type $$\widetilde{A}_2$$ A ~ 2

2016 ◽  
Vol 10 (4) ◽  
pp. 681-724
Author(s):  
A. M. Mantero ◽  
A. Zappa
Keyword(s):  
Boundary Behavior ◽  
Generalized Poisson ◽  
Poisson Integrals

scholarly journals Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups

2009 ◽  
Vol 105 (1) ◽  
pp. 31 ◽  
Author(s):  
Ewa Damek ◽  
Jacek Dziubanski ◽  
Philippe Jaming ◽  
Salvador Pérez-Esteva
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Poisson Kernel ◽  
Boundary Behavior ◽  
One Dimensional ◽  
Generalized Poisson ◽  
Poisson Kernels ◽  
Derivatives Of ◽  
Do So

In this paper, we characterize the class of distributions on a homogeneous Lie group $\mathfrak N$ that can be extended via Poisson integration to a solvable one-dimensional extension $\mathfrak S$ of $\mathfrak N$. To do so, we introduce the $\mathcal S'$-convolution on $\mathfrak N$ and show that the set of distributions that are $\mathcal S'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions. Moreover, we show that the $\mathcal S'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behavior. Finally, we show that such distributions satisfy some global weak-$L^1$ estimates.

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