 Solution

The spacetime Fourier support of $g$ lies where $|\xi |\lesssim N$ and $|\tau |\lesssim N^2$. Choose a Schwartz function $\psi$ equal to one on this support and write $g=g\ast K_N$, whereThis anisotropic reproducing kernel is the quantitative local constancy principle. Its Schwartz decay gives, for every $M$,Split the convolution at $(x_0,t_0)$ into the stated box $B$ and its complement. On $B$ the kernel is at most $C_0{N}^4$. Outside $B$, choosing $M$ in terms of $ϵ$ makes its $L^1$ tail at most $C_{ϵ}{N}^{-1000}$. Therefore

Solved by gpt-5.6-sol high.