The author generalizes the Heisenberg-Pauli-Weyl uncertainty principle inequality
\[ \|f\|_2\leq C_{\alpha,\beta} \big\| |x|^\alpha f\big\|_2^{\frac{\beta}{\alpha+\beta}} \big\|(-\Delta)^{\beta/2} f\big\|_2^{\frac{\alpha}{\alpha+\beta}} \]
on \(L^2(\mathbb{R}^n)\) with \(\alpha,\beta>0\) to an inequality of the form
\[ \|f\|_{H}\leq C_{\alpha,\beta} \| T\alpha f\|_H^{\frac{\beta}{\alpha+\beta}} \|L^{\beta} f\|_H^{\frac{\alpha}{\alpha+\beta}} \]
on a Hilbert space \(H\) where \(T\) and \(L\) satisfy certain growth properties expressed in spectral terms. One defines \((H,V^0)\) to be a regularized conjugate dual couple for \((H,V)\) where \(H\) is Hilbert and \(V\) a Banach space when \(V^0\) is the closure in \(V\) of \(H\cap V\). Define \(E_\lambda = E([0,\lambda))\) where \(E\) is the spectral measure of the possibly unbounded self-adjoint operator \(L\) (and \(F_\lambda\) corresponding to \(T\)). The growth conditions on \(E\) and \(F\) are formulated as follows. Fix \(\eta, \delta>0\). Let \(\Phi\) be a nonnegative measurable function defined on \([0, \eta b^\delta)\) and satisfying a moderate growth condition of the form
\[ \int_0^r s^{-2\gamma}\, \Phi (s)\, \frac{ds}{s} \leq M r^{-2\gamma} \,\Phi (r) \]
for any \(r\in (\eta a^\delta,\, \eta b^\delta)\). Suppose that \(\|F_t\|_{V^\circ\to V}\leq \Phi (r)\) for any \(r\in [0,\eta b^\delta)\) and \(\|E_{1/t}\|_{V\to V^\circ}\leq K^2/\Phi (\eta t^\delta)\) for any \(t\in [0,b)\). The first main result states that, then, for any \(t\in [a,b]\) and any \(f\in H\) one has
\[ \|E_{1/t} f\|_H \leq C t^{-\gamma\delta}\|T^\gamma f\|_{H}, \]
for a suitably determined constant \(C\).
The second main result states that, with the definitions above and the extended conclusion
\[ \|E_{1/t} f\|_H \leq C t^{-\gamma\delta}\|T^\gamma f\|_{H} \quad (\text{all }t>0) \]
one has, for all \(\alpha\geq \gamma\) and \(\beta>0\) and a suitable constant \(D_{\alpha,\beta}\),
\[ \|f\|_{H}\leq D_{\alpha,\beta} \| T\alpha f\|_H^{\frac{\beta}{\alpha+\beta}} \|L^{\beta\delta} f\|_H^{\frac{\alpha}{\alpha+\beta}} . \]
Specific \(L^2\)-uncertainty principles are proved as consequences for cases of Riemannian manifolds with a spectral gap, Riemannian symmetric spaces of non-compact type, and compact Riemannian manifolds. Still more precise bounds are given in the discrete cases of an \(n\)-dimensional square lattice and a homogeneous tree of degree \(n\), as well as for specific unimodular Lie groups. For example, in the case of a compact Riemannian manifold \(M\) with \(-L=\Delta\), the Laplace-Beltrami operator, and \(\rho=d(x_0,\cdot)\) the distance to a distinguished point, one has
\[ \|f\|_{L^2(M)}\leq C_{\alpha,\beta} \|\rho^\alpha f\|_{L^2}^{\frac{\beta}{\alpha+\beta}}\|(-\Delta)^{\beta/2} f\|_{L^2}^{\frac{\alpha}{\alpha+\beta}}. \]