In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. William Henry Young () proved the inequality for some special values of q, and Hausdorff () proved it in general. More generally the inequality also applies to the Fourier transform of a function on a locally compact group, such as Rn, and in this case and gave a sharper form of it called the Babenko–Beckner inequality. We consider the Fourier operator, namely let T be the operator that takes a function Parseval's theorem shows that T is bounded from to to to .