Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth

Abstract

In its simpler form, the Heisenberg–Pauli–Weyl inequality says that ‖ f ‖ 2 4 ⩽ C ( ∫ R n | x | 2 | f ( x ) | 2 d x ) ( ∫ R n | ( − Δ ) 1 2 f ( x ) | 2 d x ) . 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.