Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups
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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.
1996 ◽
Vol 11
(12)
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pp. 2167-2212
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1985 ◽
Vol 38
(1)
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pp. 55-64
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2013 ◽
Vol 2013
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pp. 1-13
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2016 ◽
Vol 30
(26)
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pp. 1650186
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1984 ◽
Vol 11
(3)
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pp. 287-308
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2013 ◽
Vol 12
(08)
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pp. 1350055
2013 ◽
Vol 10
(07)
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pp. 1320011
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