Uncertainty principles measure different notions of localisation

Three inequivalent notions of localisation are in play on this page and their hypotheses and conclusions are not interchangeable: the variance pair $(V_x,V_{\xi })$ of Spatial and frequency centres and variances of an $L^2$ function with finite second moments with the product bound The n-dimensional Heisenberg uncertainty inequality; support measure with the product bound The support-measure uncertainty inequality $\mathrm{\mid }E\mathrm{\mid }\mathrm{\mid }F\mathrm{\mid }\ge 1$ and the compact-support dichotomy; and Gaussian decay with the Hardy threshold Hardy's Gaussian uncertainty principle in ${\mathbb{R}}^n$, whose critical rigidity is isolated in Entire rigidity under Gaussian growth and real-axis decay. Gaussian decay implies finite second moments in both domains, but need not give compact support. Finite variance is not compact support — the Gaussian of Finite variance is not compact support ↗ has finite variances in both domains and full support — while Gaussian decay and its critical rigidity remain a stronger, distinct formulation. The finite product bound Finite support-product uncertainty for the unitary DFT is not the Heisenberg product: its right side is $N$, not $(4\pi {)}^{-1}$, and its equality set is different. No implication among these statements is asserted beyond the ones proved on this page.