Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth

In its simpler form, the Heisenberg–Pauli–Weyl inequality says that‖f‖24⩽C(∫Rn|x|2|f(x)|2dx)(∫Rn|(−Δ)12f(x)|2dx). In this paper, we extend this inequality to positive self-adjoint operators L on measure spaces with a “gauge function” such that (a) measures of balls are controlled by powers of the radius (possibly different powers for large and small balls); (b) the semigroup generated by L satisfies ultracontractive estimates with polynomial bounds of the same type. We give examples of applications of this result to sub-Laplacians on groups of polynomial volume growth and to certain higher-order left-invariant hypoelliptic operators on nilpotent groups. We finally show that these estimates also imply generalized forms of local uncertainty inequalities.