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Pointwise estimates for the Poisson kernel on \(NA\) groups by the Ancona method. (English) Zbl 0876.22008

Let \(S = NA\) be the semidirect product of a homogeneous group \(N\) and \(A = {\mathbb R}^+\), where \(A\) acts by dilations on \(N\). On the Lie algebra level the action is given by \([H,Y_j] = d_j Y_j\), where \(j \in \{ 1,\ldots,n \} \), for a basis \(H\) of the Lie algebra \({\mathfrak a}\) of \(A\), a basis \(Y_1,\ldots,Y_n\) for the Lie algebra \({\mathfrak n}\) of \(N\) and \(1 = d_1 \leq d_2 \leq \ldots \leq d_n\). Let \(L = X_1^2 + \ldots + X_n^2 + X_0\) be the left-invariant second order operator where \(X_0,X_1,\ldots,X_m\) are in the Lie algebra \({\mathfrak s}\) of \(S\) and \(X_1,\ldots,X_m\) generate \({\mathfrak s}\). Then by possibly taking a different decomposition \(S = NA\), the operator \(L\) has the form \[ Lf(xa) = \Big((a \partial_a)^2 + \alpha a \partial_a + \sum_{i,j=1}^n \alpha_{ij} a^{d_i + d_j} Y_i Y_j + \sum_{i=1}^n \alpha_i a^{d_i} Y_i \Big) f(xa) \] for all \(x \in N\) and \(a \in A\). Here \(\alpha\) and the \(\alpha_j\) and \(\alpha_{ij}\) are real numbers and the matrix \([\alpha_{ij}]\) is strictly positive. Suppose \(\alpha < 0\). Let \(\nu\) be the Poisson kernel with respect to \(L\), i.e., \(\nu\) is a smooth positive bounded integrable function on \(N\) such that for every bounded \(L\)-harmonic function \(F\) on \(S\) there exists an \(f \in L^\infty(N)\) such that \(F(s) = \int_N f(sx) \nu(x) dx\) for all \(s \in S\). One of the main results of this paper is that there exists a \(c > 0\) such that \(c^{-1} (1 + |x|)^{-\alpha - Q} \leq \nu(x) \leq c (1 + |x|)^{-\alpha - Q}\) uniformly for all \(x \in N\), where \(x \mapsto |x|\) is a homogeneous norm on \(N\) and \(Q = d_1 + \ldots + d_n\). The proof uses the Martin boundary for the pair \((S,L)\), as developed in the paper of A. Ancona [Ann. Math., II. Ser. 125, 495-536 (1987; Zbl 0652.31008)] and it is shown that the Martin boundary is equal to the one point compactification of \(N\).

MSC:

22E25 Nilpotent and solvable Lie groups
22E30 Analysis on real and complex Lie groups

Citations:

Zbl 0652.31008
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References:

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