Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth

By P. Ciatti, F. Ricci, M. Sundari. 2007

64 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4808e768002f1677a363e388104bf70f2776bbb2

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In Applied Mathematics

1. Saifallah Ghobber
2. Philippe Jaming
3. Alexandru Kristály
4. Eugenia Malinnikova

In Computational Theory and Mathematics

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2. Philippe Jaming

In Mathematical Physics

1. Saifallah Ghobber
2. Philippe Jaming
3. Alexandru Kristály

Citations in the corpus

• Uncertainty principles for integral operators (2012, background): “Then (3.4) is Donoho–Stark’s theorem [19, 33], Corollary 3.7 is Amrein–Berthier’s theorem [1], while the local and the global uncertainty principles for the Fourier transform go back respectively to [41, 42] and [13].”
• Sharp uncertainty principles on Riemannian manifolds: the influence of curvature (2013, methodology): “Without the sake of completeness, Heisenberg-Pauli-Weyl principle has been studied in various contexts, see Ciatti, Ricci and Sundari [13] (for positive self-adjoint operators on measure spaces), Fefferman [19], Folland and Sitaram [21], and Nahmod [33] (locating eigenvalues for selfadjoint differen”