A question on theorem 1.1 of Fritz John ultrahyperbolic pde

Question

I have the following paper:

Fritz John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938), no. 2, pages 300-322
doi:10.1215/S0012-7094-38-00423-5

Now I want to check Fritz John's claim in the proof of Theorem 1.1, where he says that equation (7) can be easily verified for the case i=1, k=2.

I used maple to check the validity of this assertion, and I get false. Can someone help me check if this assertion made in Fritz John's indeed valid?

Thanks in advance. P.S I am adding information crucial for this question of mine, since it maybe the case that there are some who cannot view the paper.

We have $\frac{\partial^2 u(x_1,x_2,x_3,x_4)}{\partial x_1^2}+\frac{\partial^2 u(x_1,x_2,x_3,x_4)}{\partial x_2^2}=\frac{\partial^2 u(x_1,x_2,x_3,x_4)}{\partial x_3^2}+\frac{\partial^2 u(x_1,x_2,x_3,x_4)}{\partial x_4^2}$. And $v(\xi_1,\xi_2,\xi_3,\eta_1,\eta_2,\eta_3) = (\sum_i (q_i/q_3)^2)^{1/2}u(\frac{p_2+q_2}{q_3},\frac{-p_1-q_1}{q_3},\frac{p_2-q_2}{q_3},\frac{-p_1+q_1}{q_3})$, where $q_i=\xi_i-\eta_i$, and $p_i=\xi_j \eta_k - \xi_k \eta_j$, where $ijk$ is a cyclic permutation of $\{ 1,2,3\}$. Now, equation (7) is: $$(\frac{\partial^2}{\partial \xi_i \partial \eta_k} - \frac{\partial^2}{\partial \xi_k \partial \eta_i})\frac{v}{|\xi - \eta|} = 0$$ where $|\xi - \eta| := \sum_i (\xi_i-\eta_i)^2$.

Answer 1

I don't have John's paper available at the moment, but are you sure

$$ | \xi - \eta| = \sum (\xi_i - \eta_i)^2 $$

and not

$$ | \xi - \eta|^2 = \sum (\xi_i - \eta_i)^2 ? $$

For one, I don't think John would define a "norm" with the wrong scaling, and for two, because if the latter holds you have that

$$ \frac{v}{ |\xi - \eta|} = \frac{1}{|q_3|} u( \dots) $$

and noting that $p_1, p_2, q_1, q_2$ are linear in $\xi_1, \xi_2, \eta_1, \eta_2$ you can write

$$ \partial_{\eta_1} = \xi_3 \partial_1 + \partial_2 + \xi_3 \partial_3 - \partial_4 $$

with a change of variables and verify that

$$ \partial^2_{\xi_1 \eta_2} - \partial^2_{\xi_2 \eta_1} = - q_3 ( \partial^2_{11} + \partial^2_{22} - \partial^2_{33} - \partial^2_{44} ) $$

so that the claimed equation is true by virtue of $u$ solving the ultrahyperbolic PDE.