Poisson kernels on semi-direct products of abelian groups

Rok, strany: 2016, 1375 - 1386

Let $G$ be a semi direct product $G=\mathbb{R}^d\rtimes \mathbb{R}^k$. On $G$ we consider a class of second order left-invariant differential operators of the form $\mathcal L_α= ∑_j=1^d e^2λ_j(a)∂_xj² + ∑_j=1^k(∂_aj²-2α_j∂_aj)$, where $a\in\mathbb{R}^k$ and $λ₁,…,λ_d \in (\mathbb{R}^k)^*$. It is known that bounded $\mathcal L_α$-harmonic functions on $G$ are precisely the ``Poisson integrals'' of $L^∞(\R^d)$ against the Poisson kernel $ν_α$ which is a smooth function on $\mathbb{R}^d$. We prove an upper bound of $ν_α$ and its derivatives.

ISO 690:

Penney, R., Urban, R. 2016. Poisson kernels on semi-direct products of abelian groups. In Mathematica Slovaca, vol. 66, no.6, pp. 1375-1386. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0230

APA:

Penney, R., Urban, R. (2016). Poisson kernels on semi-direct products of abelian groups. Mathematica Slovaca, 66(6), 1375-1386. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0230