Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 22F / iv

The implication is true in every dimension despite the failure of part (iii). The hypothesis gives $\mathcal{F}f\in L^2$, since its weight is at least one. Let $g$ be its inverse Fourier-Plancherel transform; then $g\in H^s$ by construction.

Use the Gaussian approximate identity $G_{\epsilon }(x)={\epsilon }^{-n}{e}^{-\pi |x{|}^2/{\epsilon }^2}$. Its Fourier transform is $e^{-\pi {\epsilon }^2|\xi {|}^2}$. Fourier inversion of this integrable multiplier times the bounded transform of $f$, justified by Fubini, gives the same function both as $G_{\epsilon }\ast f$ and as $G_{\epsilon }\ast g$. The first converges to $f$ in $L^1$, while the second converges to $g$ in $L^2$ by Plancherel and dominated convergence. Along a common sequence, select subsequences converging almost everywhere for both norms. Their limits agree, giving $f=g$ almost everywhere. ConsequentlyThis proves the requested conclusion without relying on the incorrect general-dimensional kernel assertion.