I am interested in studying the Poisson kernel on the flat $n$-torus, $T^n$, and am trying to understand how to reconcile two separate definitions, both given in Grafakos's book, Classial Fourier Analysis. The two definitions are as follows:
\begin{align} P_z(x) &=\frac{\Gamma\left(\frac{n+1}{2}\right)}{\pi^{\frac{n+1}{2}}}\sum_{m\in\mathbb{Z}^n}\frac{z^{-n}}{\left(1 + |\frac{x+m}{z}|^2\right)^{\frac{n+1}{2}}}\\ &= \sum_{m\in\mathbb{Z}^n} \exp(-2\pi|m|z)\exp(2\pi i m\cdot x), \hspace{1em} (1) \end{align}
which comes from periodizing the Poisson kernel on upper-half space (Section 4.3, Multipliers, Transference and A.E. Convergence; proof of theorem 4.3.4), and
\begin{align} P_{r_1, \dots, r_n}(x) &= \prod_{k=1}^{n}\left(\frac{1 - r_{k}^2}{1 - 2r_{k}\cos(x_k) + r_{k}^2}\right)\\ &= \sum_{m\in \mathbb{Z}^n} r_{1}^{|m_1|}r_{2}^{|m_2|}\cdots r_{n}^{|m_n|} \exp(2\pi i m\cdot x), \hspace{1em} (2) \end{align}
which is given by a product of $n$ 1D Poisson kernels on the unit disc (Section 3.1, Fourier Coefficients; exercise 3.1.7).
If we let $r_{1} = r_{2} = \cdots = r_{n} = r$ in case (2) then we may write:
\begin{align} P_{r}(x) &= \prod_{k=1}^{n}\left(\frac{1 - r^2}{1 - 2r\cos(x_k) + r^2}\right)\\ &= \sum_{m\in \mathbb{Z}^n} r^{|m_1|+|m_2| + \cdots + |m_n|}\exp(2\pi i m\cdot x). \hspace{1em} (3) \end{align}
Letting $r = \exp(-2\pi z),$ (3) becomes
\begin{align} P_{r}(x) &= \prod_{k=1}^{n}\left(\frac{1 - r^2}{1 - 2r\cos(x_k) + r^2}\right)\\ &= \sum_{m\in \mathbb{Z}^n} \exp(-2\pi(|m_1|+|m_2| + \cdots + |m_n|)z)\exp(2\pi i m\cdot x). \hspace{1em} (4) \end{align}
Now, (1) and (4) are very similar, the difference being the metric on the lattice points $m$. That is, if we impose the taxicab metric on $\mathbb{Z}^n$ so that $|m| = |m_1| + |m_2| + \cdots |m_n|$ then these two definitions become equivalent.
I am hoping for insight into some fundamental (topological? geometric?) distinction between these two definitions.
I believe I have at least a partial answer to this question. In Stein and Weiss's book Introduction to Fourier Analysis on Euclidean Spaces in Chapter III.3 Tubes over Cones, they construct the Cauchy kernel using the cone $$ \Gamma = \{(y_1, y_2, \dots, y_n) \in \mathbb{R}^n \mid y_k > 0 \; \text{ for } k = 1, \dots, n\} $$ and the tube $T_{\Gamma}$ where $z = (z_1, \dots, z_n) \in T_{\Gamma}$ if for each $k$ we have $z_k = x_k + i y_k$ and $(y_1, \dots, y_k) \in \Gamma$. The Cauchy kernel associated to $T_{\Gamma}$ is then given by \begin{align*} K(z) &= \int_{0}^{+\infty}\int_{0}^{+\infty} \cdots \int_{0}^{+\infty}e^{2\pi i(z_1 t_1 + z_2t_2 + \cdots + z_n t_n)}\; dt_1\;dt_2\;\cdots\;dt_n\\ &= \prod_{k=1}^{n} \frac{-1}{2\pi i z_{k}}. \end{align*}
The Poisson kernel can be computed by $$ P(x, y) = \frac{|K(z)|^2}{K(2iy)} = \prod_{k=1}^{n} \frac{1}{\pi}\frac{y_{k}}{x_k^2 + y_k^2} $$ where $y_k > 0$ for each $k = 1, \dots, n$. The multidimensional Fourier transform of this function in the $x_k$ coordinates is: \begin{align*} \hat{P}_{y}(\xi) &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\cdots \int_{-\infty}^{\infty} \prod_{k=1}^{n} \left(\frac{1}{\pi}\frac{y_{k}}{x_k^2 + y_k^2}e^{-2\pi i x_k \xi_k}\right)\;dx_1 \; dx_2 \cdots dx_{n}\\ &= \prod_{k=1}^{n} \int_{-\infty}^{\infty}\left(\frac{1}{\pi}\frac{y_{k}}{x_k^2 + y_k^2}e^{-2\pi i x_k \xi_1}\right)\; dx_k\\ &= e^{-2\pi(|\xi_1|y_1 + |\xi_2|y_2 + \cdots |\xi_n| y_n)}. \end{align*} So, again using Poisson summation, we have (at least formally) the identity \begin{align*} \sum_{m \in \mathbb{Z}^n} \prod_{k=1}^{n} \frac{1}{\pi}\frac{y_{k}}{(x_k - m_k)^2 + y_k^2} &= \sum_{m \in \mathbb{Z}^n} e^{-2\pi(|m_1|y_1 + |m_2|y_2 + \cdots |m_n| y_n)} e^{2\pi i m \cdot x}\\ &= \sum_{m \in \mathbb{Z}^n} r_1^{|m_1}| r_2^{|m_2|} \cdots r_n^{|m_n|} e^{2\pi m \cdot x}\\ &= \prod_{k=1}^{n} \frac{1 - r_k^2}{1 - 2r_k \cos(x_k) + r_k^2}, \end{align*} where $r_k = e^{-2\pi y_k}$ for $k = 1, \dots, n$. If we like, we can consider $y_1 = y_2 = \cdots y_k$ to unify the different radii. In any case, if we consider the definition of the $n$-Torus as a subset of $\mathbb{C}^n$ given by: $$ T^n = \left\{(e^{2\pi i x_1}, e^{2\pi i x_2}, \dots, e^{2\pi i x_n}) \mid x_k \in [0,1] \text{ for } k = 1, \dots, n\right\}, $$ we can consider the related subset $$ \left\{(e^{2\pi i (x_1 + iy_1)}, e^{2\pi i (x_2 + iy_2)}, \dots, e^{2\pi i (x_n + iy_n}) \mid x_k \in [0,1], y_k > 0 \text{ for } k = 1, \dots, n\right\} = \left\{(r_1 e^{2\pi i x_1}, r_2 e^{2\pi i x_2}, \dots, r_n e^{2\pi i x_n}) \mid x_k, r_k \in [0,1] \text{ for } k = 1, \dots, n\right\} $$ on which to consider this Poisson kernel. I am still unsure of what to make of this geometrically, but it at least further justifies the validity of the product of the 1D Poisson kernels on the unit disk as a valid Poisson kernel in some setting, and provides some more distinction between the two definitions.