Proof cost and complex-analysis interface for Hardy uncertainty

The complex-analysis cost of Hardy's Gaussian uncertainty principle in ${\mathbb{R}}^n$ is the one-variable rigidity in Entire rigidity under Gaussian growth and real-axis decay. For the critical function $\mathrm{\Phi }(z)=e^{\pi z^2\mathrm{/}a}F(z)$, that lemma first bounds $q_{\delta }\mathrm{\Phi }$ on two sectors of aperture $\theta <\pi \mathrm{/}2$, with $q_{\delta }=e^{i\delta z^2}$ on the first and $q_{\delta }=e^{-i\delta z^2}$ on the second. It then uses $h_{\epsilon }(z)=\exp (i\epsilon e^{i\mu }{z}_{\mathrm{p}\mathrm{r}}^{2+\epsilon })$, with the sector-dependent phase $\mu$ specified in its step 1.2, so that $\mathrm{\mid }{h}_{\epsilon }(z)\mathrm{\mid }\le e^{-\epsilon {\sigma }_{\epsilon }\mathrm{\mid }z{\mathrm{\mid }}^{2+\epsilon }}$ uniformly in angle. Boundary and infinity control give the sector bound; removing the perturbations and applying Liouville gives critical rigidity. Coordinate slices then yield the higher-dimensional theorem.

The estimate $\mathrm{\mid }F(x+iy)\mathrm{\mid }\le C_1{e}^{\pi y^2\mathrm{/}a}$ comes from the spatial Gaussian bound; the critical decay is $e^{-\pi x^2\mathrm{/}a}$. Tao's real-variable proof in the cited post proves a weaker, non-sharp threshold $ab>C_0$. This describes that particular proof, not a limitation of all real-variable methods: the cited survey, §1, printed p. 2, also records a complete sharp real-variable proof.