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 $f$ on the unit circle and outputs the sequence of its Fourier coefficients

$\widehat{f}(n)=\frac{1}{2\pi}\int_0^{2\pi}e^{-inx}f(x)\,dx,\quad n=0,\pm1,\pm2,\dots.$

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

$|(Tf)(n)|=|\widehat{f}(n)|=\left|\frac{1}{2\pi}\int_0^{2\pi}e^{-int}f(t)\,dt\right|\leq \frac{1}{2\pi} \int_0^{2\pi}|f(t)|\,dt$

so T is bounded from $L^1$ to $\ell^\infty$ with norm 1. Therefore we may invoke the Riesz–Thorin theorem to get, for any 1 < p < 2 that T, as an operator from $L^p$ to $\ell^q$, is bounded with norm 1, where

$\frac{1}{p}+\frac{1}{q}=1.$

In a short formula, this says that

$\left(\sum_{n=-\infty}^\infty |\widehat{f}(n)|^q\right)^{1/q}\leq \left( \frac{1}{2\pi}\int_0^{2\pi}|f(t)|^p\,dt\right)^{1/p}.$

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 $L^p$, does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in $\ell^2$.

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 $f\in L^p$ for $1<p\leq 2$, the optimal bound is

$\|\hat{f}\|_{L^q}\leq p^{1/2p}q^{-1/2q}\|f\|_{L^p}$

where $q=p/(p-1)$ is the Hölder conjugate of $p$ (Cifuentes 2010)

References

• Babenko, K. Ivan (1961), "An inequality in the theory of Fourier integrals", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 25: 531–542, ISSN 0373-2436, MR 0138939 English transl., Amer. Math. Soc. Transl. (2) 44, pp. 115–128
• Beckner, William (1975), "Inequalities in Fourier analysis", Annals of Mathematics. Second Series 102 (1): 159–182, doi:10.2307/1970980, ISSN 0003-486X, JSTOR 1970980, MR 0385456
• Cifuentes, Patricio (2010), Harmonic Analysis and Partial Differential Equations, Contemporary Mathematics 505, American Mathematical Society, p. 94, ISBN 9780821858318.
• Hausdorff, Felix (1923), "Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen", Mathematische Zeitschrift 16: 163–169, doi:10.1007/BF01175679
• Young, W. H. (1913), "On the Determination of the Summability of a Function by Means of its Fourier Constants", Proc. London Math. Soc. 12: 71–88, doi:10.1112/plms/s2-12.1.7