A sharpened Hausdorff-Young inequality

Abstract

The Hausdorff-Young inequality for Euclidean space, in its sharp form due to Beckner, gives an upper bound for the Fourier transform in terms of Lebesgue space norms, with an optimal constant. The extremizers have been identified by Lieb to be the Gaussians. We establish an improved upper bound, for functions that nearly extremize the inequality, with a negative second term roughly proportional to the square of the distance to the set of extremizers. One formulation of this term comes with its own sharp constant. The main step is to show that any extremizing sequence is precompact, modulo the action of the group of natural symmetries of the inequality. This step relies on inverse theorems of additive combinatorial nature.

References in corpus (5)

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• Near-extremizers of Young's Inequality for R^d
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Cited by in corpus (9)

• Remainder Terms for Several Inequalities on Some Groups of Heisenberg-type
• Quantitative inequalities for the expected lifetime of Brownian motion
• On an extremization problem concerning Fourier coefficients
• A sharp nonlinear Hausdorff-Young inequality for small potentials
• Stability of trace theorems on the sphere
• On nearly radial product functions
• Asymptotically sharp discrete nonlinear Hausdorff-Young inequalities for the SU(1,1)-valued Fourier products
• On some determinant and matrix inequalities with a geometrical flavour
• A constrained optimization problem for the Fourier transform: Existence