Noncommutative sharp Hausdorff-Young inequality
Abstract
We prove the sharp Hausdorff--Young inequality on the quantum Euclidean space. Our result implies the sharp Hausdorff--Young constants for the Weyl transform, as well as that for Heisenberg groups. The key ingredient is a novel flow related to the mixed-norm of noncommutative Gabor transform. This, meanwhile, implies a new proof of the classical sharp Hausdorff--Young inequality. We then apply the sharp Hausdorff--Young inequality to establish the sharp Young inequality for noncommutative convolution in the range 1≤ p,q≤2≤ r≤∞, with 1/p+1/q=1+1/r. After appropriate rescaling and trace normalization, this convolution coincides with beam-splitter convolution for bosonic systems, yielding the corresponding Young inequalities with optimal constants in the same range.
Related
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BibTeX
@misc{qiu2026noncommutative,
title = {{Noncommutative sharp Hausdorff-Young inequality}},
author = {Hongsen Qiu and Xiao Xiong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10424},
eprint = {2609.10424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10424},
}
RIS
TY - UNPB
TI - Noncommutative sharp Hausdorff-Young inequality
AU - Hongsen Qiu
AU - Xiao Xiong
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.10424
UR - https://arxiv.org/abs/2609.10424
AB - We prove the sharp Hausdorff--Young inequality on the quantum Euclidean space. Our result implies the sharp Hausdorff--Young constants for the Weyl transform, as well as that for Heisenberg groups. The key ingredient is a novel flow related to the mixed-norm of noncommutative Gabor transform. This, meanwhile, implies a new proof of the classical sharp Hausdorff--Young inequality. We then apply the sharp Hausdorff--Young inequality to establish the sharp Young inequality for noncommutative convolution in the range 1≤ p,q≤2≤ r≤∞, with 1/p+1/q=1+1/r. After appropriate rescaling and trace normalization, this convolution coincides with beam-splitter convolution for bosonic systems, yielding the corresponding Young inequalities with optimal constants in the same range.
ER -