mp.mathematical physics – Propagation of Klein-Gordon solutions in extra dimensions

In his paper “Von Neumann Algebras of Local Observables for Free Scalar Field” Araki used the solutions of the equation

$$frac{partial ^{2}h}{partial x^2}-frac{partial ^{2}h}{partial t^2}-m^2h=0$$

where he put the suport of the initial data $h(t, 0)$ and $frac{partial h}{partial x}(t, 0)$ at $x=0$.

Then He used this solution (noted as $h_{0}$) to pose another Klein-Gordon Problem in $mathbb{R}^{4}$:
$$(Box+m^2)G=0 $$
with initial conditions

$G(0,vec{x})=h_{0}(0,x_{1})h_{1}(vec{x})$ and $frac{partial G}{partial x^0}(0,vec{x})=frac{partial h_{0}}{partial t}(0,x_{1})h_{1}(vec{x})$

where $h_{1}$ is a function wich is equal to 1 in a ball (of radius $t_{2}$) which contains the suports of the initial conditions for $h_{0}$ and then goes to zero smoothly.

Araki asserts that $G(x^0,x^1,0,0)=h_{0}(x^0,x^1)$ for $|x^0|+|x^1|<t_{2}$, where $t_2$ satisfies $h_{1}(vec{x})=1$ for $|vec{x} |<t_{2}$.

My question is why the extra dimensions doesn’t affect the equality, at least for near points. I know the initial contitions for $G$ has translation simmetry in $x^3$ and $x^4$ and I think this would be important.

Some parts of Araki’s paper

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And the problematic part for my

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Many thanks in advance.