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 $g=g*K_N$, where
$$
K_N(x,t)=N^4\check\psi(Nx,N^2t).
$$
This anisotropic reproducing kernel is the quantitative <local constancy principle>. Its Schwartz decay gives, for every $M$,
$$
|K_N(x,t)|\leq C_MN^4(1+N|x|+N^2|t|)^{-M}.
$$
Split the convolution at $(x_0,t_0)$ into the stated box $B$ and its complement. On $B$ the kernel is at most $C_0N^4$. Outside $B$, choosing $M$ in terms of $\epsilon$ makes its $L^1$ tail at most $C_\epsilon N^{-1000}$. Therefore
$$
|g(x_0,t_0)|
\leq C_0N^4\int_B|g(x,t)|\,dxdt
+C_\epsilon N^{-1000}\|g\|_\infty.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 163 / 2 / b / Solution

A function whose Fourier support has widths $N$ in spatial-frequency directions and $N^2$ in a time-frequency direction varies only on the dual scales $N^{-1}$ and $N^{-2}$. Convolution with an adapted Schwartz kernel makes this quantitative.


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 Incoming links: Local constancy principle