Mean value formula for harmonic functions on product of spheres

Question

Is there a mean value formula for harmonic functions over product of spheres?
I mean, let us consider $\mathbb{R}^4$ with coordinates $\alpha$, $\beta$, $\gamma$ and $\delta$ and the laplacian $\Delta:=\dfrac{\partial^2}{\partial\alpha^2}+\dfrac{\partial^2}{\partial\beta^2}+\dfrac{\partial^2}{\partial\gamma^2}+\dfrac{\partial^2}{\partial\delta^2}$.
Let $f:\mathbb{R}^4\times\mathbb{R}^4\rightarrow\mathbb{R}$ be a separately harmonic function, which means that $f=f(\alpha_1,\beta_1,\gamma_1,\delta_1, \alpha_2,\beta_2,\gamma_2,\delta_2)$ satisfies $\Delta_1f=\Delta_2f=0$ and consider two 3 dimensional spheres $\partial B_{r_1}(x_1)$, $\partial B_{r_2}(x_2)$, where $x_i=(\alpha_i,\beta_i,\gamma_i,\delta_i)$.
Is there a mean value formula of the form \begin{equation} f(x_1,x_2)=\dfrac{1}{Area\left(\partial B_{r_1}(x_1)\times\partial B_{r_2}(x_2)\right)}\int_{\partial B_{r_1}(x_1)\times\partial B_{r_2}(x_2)}f(y_1,y_2)d\sigma(y_1)\times d\sigma(y_2)? \end{equation} Thank you for any help.

Answer 1

Yes, that formula holds. By assumption for every fixed $x_2$ the function $x\mapsto f(x,x_2)$ is harmonic, same for $x\mapsto f(x_1,x)$ for every $x_1$. Therefore \begin{align*} f(x_1,x_2) &= \frac{1}{|\partial B_{r_1}(x_1)|} \int_{\partial B_{r_1}(x_1)} f(y_1,x_2)\, \sigma(dy_1)\\ &= \frac{1}{|\partial B_{r_1}(x_1)|} \int_{\partial B_{r_1}(x_1)} \frac{1}{|\partial B_{r_2}(x_2)|} \int_{\partial B_{r_2}(x_2)}f(y_1,y_2)\,\sigma(dy_2)\, \sigma(dy_1)\\ &=\frac{1}{|\partial B_{r_1}(x_1)||\partial B_{r_2}(x_2)|} \int_{\partial B_{r_1}(x_1)} \int_{\partial B_{r_2}(x_2)}f(y_1,y_2)\,\sigma(dy_2)\, \sigma(dy_1). \end{align*}