Let \({\mathcal D}^\prime_{L^1}\) denote the space of integrable distributions and \(B\) the space of smooth functions with bounded derivatives of arbitrary order and let us consider the dual pair \(\left({\mathcal D}^\prime_{L^1},B\right).\) Given two tempered distributions \(T\) and \(S\), the \({\mathcal S}^\prime\)-convolution of \(T\) and \(S\) exists if \(T\left(\check{S}\ast \varphi\right)\in{\mathcal D}^\prime_{L^1}\) for every \(\varphi \in {\mathcal S}\). In this case, \(T\ast S\) denotes the tempered distribution given by the map \(\varphi \mapsto \left(T\left(\check{S}\ast \varphi\right),1\right).\) The product domain version of the Poisson kernel in the Cartesian product of \(n\) copies of the upper half plane is \[ P_{(y)}(x) = \prod_{j=1}^n\left(\frac{1}{\pi}\frac{y_j}{x_j^2+y_j^2}\right), \] where \(y_1 > 0, \dots, y_n > 0.\)
J. Alvarez, M. Guzmán-Partida and U. Skórnik [Stud. Math. 156, No. 2, 143–163 (2003; Zbl 1023.46041)] characterized those tempered distributions \(T\) with the property that \(T\ast P_{(y)}\) exists whenever \(y_1 > 0, \dots, y_n > 0.\)
In the present paper, it is proved that, for such a tempered distribution \(T\), the \({\mathcal S}^\prime\)-convolution \(T\ast P_{(y)}\) converges to \(T\) in an appropriate sense as \(y_1,\dots,y_n\to 0^+.\) The paper also includes a characterization of those \(n\)-harmonic functions on \({\mathbb R}_+^2 \times \ldots \times {\mathbb R}_+^2\) that can be written as finite sums of products of one-dimensional Poisson integrals of appropriate distributions, as well as some \({\mathcal S}^\prime\)-convolvability results with general Poisson like kernels.