NONCOMMUTATIVE SHARP HAUSDORFF–YOUNG INEQUALITY

The classical Hausdorff-Young inequality provides an optimal relationship between the L^p norm of a function and the L^q norm of its Fourier transform. A long-standing problem in noncommutative analysis has been to determine whether an analogous sharp inequality holds for the Weyl transform and on quantum Euclidean spaces, and to identify the optimal constants in these settings. This paper addresses this gap by proving the sharp Hausdorff-Young inequality for the quantum Euclidean space.

The authors utilize an operator-valued flow, which is a noncommutative generalization of the heat flow method used in classical analysis. Specifically, they define a flow based on the Gabor transform—a time-frequency representation—and demonstrate that this flow is monotonic. By analyzing the second-order Fréchet derivatives of the L^q norm in the noncommutative setting, they establish a complex dominance property that replaces the classical hypercontractivity arguments. This allows them to prove that the L^p to L^q norm of the Weyl transform is bounded by the optimal constant, which is shown to be the same as the classical constant B_p^2.

The primary contribution is the proof of the sharp Hausdorff-Young inequality on quantum Euclidean spaces. The authors show that for 1 ≤ p ≤ 2, the norm of the Weyl transform is bounded by the optimal constant C_p = B_p^2. This result directly implies the sharp Hausdorff-Young constants for the Heisenberg group H^1, confirming the relation H_p(H^1) = B_p^3. Furthermore, the authors define a noncommutative convolution and provide partial results for the sharp Young's inequality, which has significant implications for quantum information theory, particularly regarding beam-splitter channels.

This work resolves a significant open problem in noncommutative harmonic analysis. By providing a sharp inequality, the authors offer a definitive bound that is essential for understanding the behavior of quantum systems in phase space. The method of using an operator-valued flow is highly flexible and provides a robust framework that can be applied to other problems in noncommutative geometry and quantum probability.