In mathematics, the Hausdorff−Young inequality bounds the Lq-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
Parseval's theorem shows that T is bounded from
so T is bounded from
In a short formula, this says that
This is the well known Hausdorff–Young inequality. For p > 2 the natural extrapolation of this inequality fails, and the fact that a function belongs to
Optimal estimates
The constant involved in the Hausdorff–Young inequality can be made optimal by using careful estimates from the theory of harmonic analysis. If
where