# A sharp nonlinear Hausdorff-Young inequality for small potentials

## TL;DR

This paper proves a sharper nonlinear Hausdorff-Young inequality for small functions, improving bounds and extending previous linear results through perturbative methods and refined inequalities.

## Contribution

It establishes a better upper bound for the nonlinear Hausdorff-Young quotient for small potentials, enhancing understanding of nonlinear Fourier analysis.

## Key findings

- Nonlinear Hausdorff-Young quotient has improved bounds for small functions.
- The proof combines perturbative techniques with a sharpened linear inequality.
- Results extend linear inequalities to a nonlinear setting for small potentials.

## Abstract

The nonlinear Hausdorff-Young inequality follows from the work of Christ and Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen independently of the exponent. We show that the nonlinear Hausdorff-Young quotient admits an even better upper bound than the linear one, provided that the function is sufficiently small in the $L^1$ norm. The proof combines perturbative techniques with the sharpened version of the linear Hausdorff-Young inequality due to Christ.