Pitt Inequalities and Logarithmic-type Uncertainty Principles for Metaplectic Operators
Abstract
In this work, we provide a complete characterisation of Pitt's inequality for metaplectic operators. The symplectic geometry underlying a metaplectic operator distinguishes effective directions, along which its action is Fourier-type, from singular directions, along which concentration is preserved. For this reason, we adopt two complementary perspectives, thereby obtaining both an isotropic Pitt's inequality, emulating the classical theorem of Pitt, and an anisotropic inequality that adapts to the geometric features of the metaplectic group. As a by-product, we obtain Pitt's inequality for Fourier transforms along subspaces of rd and new quantitative time-dependent boundedness results for metaplectic operators on homogeneous Sobolev spaces, with applications to Schrödinger evolutions generated by quadratic Hamiltonians. Additionally, we derive logarithmic and entropic uncertainty principles for the metaplectic group. We show that the classical entropic uncertainty principle fails when both the effective and singular directions are present. In this case, a generally unbounded corrective entropy term accounts for the concentration-preserving directions and restores a meaningful lower bound. For quadratic Schrödinger evolutions, this correction follows the time-dependent geometry of the nondispersive directions and adjusts the uncertainty estimate accordingly.
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BibTeX
@misc{giacchi2026pitt,
title = {{Pitt Inequalities and Logarithmic-type Uncertainty Principles for Metaplectic Operators}},
author = {Gianluca Giacchi and Itamar Oliveira and Amy Tierney},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34946},
eprint = {2609.34946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34946},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pitt Inequalities and Logarithmic-type Uncertainty Principles for Metaplectic Operators
AU - Gianluca Giacchi
AU - Itamar Oliveira
AU - Amy Tierney
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34946
UR - https://arxiv.org/abs/2609.34946
AB - In this work, we provide a complete characterisation of Pitt's inequality for metaplectic operators. The symplectic geometry underlying a metaplectic operator distinguishes effective directions, along which its action is Fourier-type, from singular directions, along which concentration is preserved. For this reason, we adopt two complementary perspectives, thereby obtaining both an isotropic Pitt's inequality, emulating the classical theorem of Pitt, and an anisotropic inequality that adapts to the geometric features of the metaplectic group. As a by-product, we obtain Pitt's inequality for Fourier transforms along subspaces of rd and new quantitative time-dependent boundedness results for metaplectic operators on homogeneous Sobolev spaces, with applications to Schrödinger evolutions generated by quadratic Hamiltonians. Additionally, we derive logarithmic and entropic uncertainty principles for the metaplectic group. We show that the classical entropic uncertainty principle fails when both the effective and singular directions are present. In this case, a generally unbounded corrective entropy term accounts for the concentration-preserving directions and restores a meaningful lower bound. For quadratic Schrödinger evolutions, this correction follows the time-dependent geometry of the nondispersive directions and adjusts the uncertainty estimate accordingly.
ER -