Hardy and uncertainty inequalities on stratified Lie groups

By P. Ciatti, M. Cowling, F. Ricci. 2013

85 citations recorded by the source corpus.

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_\alpha: f \mapsto |.|^{-\alpha} L^{-\alpha/2} f$, where $|.|$ is a homogeneous norm, $0 < \alpha < Q/p$, 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 $|.|^\beta f$ and the $L^r$ norm of $L^{\delta/2} f$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/109bac6d66c4e918feea43923862dc4d9c767cb7

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Citations in the corpus

Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group (2015, methodology): “6 in the more general context of stratified groups has been proved recently in the nice work [13] using different methods.”
Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function (2020, methodology): “The representative results include Ostrowski-type inequality [15], Hardy-type inequality [16], Olsen-type inequality [17], Gagliardo-Nirenberg-type inequality [18], and the most well-known inequality of, namely, the Hermite-Hadamard-type inequality [19].”
On horizontal Hardy, Rellich, Caffarelli–Kohn–Nirenberg and p-sub-Laplacian inequalities on stratified groups (2016, methodology): “[9], [42], [40] and [39] for different settings): For each f ∈ C 0 (G\{x ′ = 0}), using Hölder’s inequality and (3.”
Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups (2016, methodology): “A higher order extension of (1.2) on stratified Lie groups was recently obtained by Ciatti, Cowling and Ricci [3] for arbitrary homogeneous quasi-norms but the obtained constants are not optimal.”
Hermite‐Hadamard inequalities for Riemann‐Liouville fractional integrals of a convex function with respect to a monotone function (2019, background): “From (8) and (11), we obtain the left-hand side inequality in (6).”