We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G. In particular, we show that the operators T_α : f → |·|^{−α} L^{−α/2}f, where |·| is a homogeneous norm, 0 < α < Q/p, and L is the sub-Laplacian, are bounded on the Lebesgue space L^p(G). As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli– Weyl inequality, relating the L^p norm of a function f to the L^q norm of |·|^β f and the L^r norm of L^{δ/2}f.

Hardy and uncertainty inequalities on stratified Lie groups

RICCI, Fulvio
2015

Abstract

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G. In particular, we show that the operators T_α : f → |·|^{−α} L^{−α/2}f, where |·| is a homogeneous norm, 0 < α < Q/p, and L is the sub-Laplacian, are bounded on the Lebesgue space L^p(G). As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli– Weyl inequality, relating the L^p norm of a function f to the L^q norm of |·|^β f and the L^r norm of L^{δ/2}f.
Settore MAT/05 - Analisi Matematica
Hardy's inequality; Heisenberg's inequality; Stratified group; Uncertainty principle;
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/63200
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