Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 23G / a / Solution

Taking Fourier transforms givesThusThe multiplier and all its polynomially weighted derivatives have at most polynomial growth, so this defines a tempered distribution. In fact it maps $H^s$ continuously into $H^{s+4}$. Since the multiplier never vanishes, any homogeneous tempered solution has zero Fourier transform; uniqueness follows.