Generalized uncertainty inequalities

Abstract

The Heisenberg–Pauli–Weyl (HPW) uncertainty inequality on ${\mathbb{R}^n}$ says that

$$\| f \|_2 \leq C_{\alpha,\beta}\| |x|^\alpha f\|_2^\frac{\beta}{\alpha+\beta}\| (-\Delta)^{\beta/2}f\|_2^\frac{\alpha}{\alpha+\beta}.$$

Let H be a Hilbert space; we obtain inequalities of the form

$$\| f \|_H \leq C_{\alpha,\beta}\| T^\alpha f\|_H^\frac{\beta}{\alpha+\beta}\| L^\beta{f\|_H^\frac{\alpha}{\alpha+\beta}}$$

for a pair of positive self-adjoint operators T, L on H satisfying a “balance condition” involving certain operator norms of their spectral projectors. This extends a result of Ciatti et al. (Adv Math 215(2):616–625, 2007) since our hypotheses allow growth rates other than polynomial, e.g., exponential ones. As examples of applications, we obtain HPW-type inequalities on Riemannian manifolds, Riemannian symmetric spaces of non-compact type, homogeneous graphs and unimodular Lie groups.