Hausdorff–Young inequality

In mathematics, the Hausdorff−Young inequality bounds the L^q-norm of the Fourier coefficients of a periodic function for q ≥ 2. William Henry Young (1913) proved the inequality for some special values of q, and Hausdorff (1923) 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 Babenko (1961) and Beckner (1975) 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 ${\displaystyle f}$ on the unit circle and outputs the sequence of its Fourier coefficients

Parseval's theorem shows that T is bounded from ${\displaystyle L^{2}}$ to ${\displaystyle \ell ^{2}}$ with norm 1. On the other hand, clearly,

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