Babenko–Beckner inequality

In mathematics, the Babenko–Beckner inequality (after Konstantin I. Babenko and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of L^p spaces. The (q, p)-norm of the n-dimensional Fourier transform is defined to be

${\displaystyle \|{\mathcal {F}}\|_{q,p}=\sup _{f\in L^{p}(\mathbb {R} ^{n})}{\frac {\|{\mathcal {F}}f\|_{q}}{\|f\|_{p}}},{\text{ where }}1<p\leq 2,{\text{ and }}{\frac {1}{p}}+{\frac {1}{q}}=1.}$

In 1961, Babenko found this norm for even integer values of q. Finally, in 1975, using Hermite functions as eigenfunctions of the Fourier transform, Beckner proved that the value of this norm for all ${\displaystyle q\geq 2}$ is

${\displaystyle \|{\mathcal {F}}\|_{q,p}=\left(p^{1/p}/q^{1/q}\right)^{n/2}.}$

Thus we have the Babenko–Beckner inequality that

${\displaystyle \|{\mathcal {F}}f\|_{q}\leq \left(p^{1/p}/q^{1/q}\right)^{n/2}\|f\|_{p}.}$

To write this out explicitly, (in the case of one dimension,) if the Fourier transform is normalized so that

${\displaystyle g(y)\approx \int _{\mathbb {R} }e^{-2\pi ixy}f(x)\,dx{\text{ and }}f(x)\approx \int _{\mathbb {R} }e^{2\pi ixy}g(y)\,dy,}$

then we have

${\displaystyle \left(\int _{\mathbb {R} }|g(y)|^{q}\,dy\right)^{1/q}\leq \left(p^{1/p}/q^{1/q}\right)^{1/2}\left(\int _{\mathbb {R} }|f(x)|^{p}\,dx\right)^{1/p}}$

or more simply

${\displaystyle \left({\sqrt {q}}\int _{\mathbb {R} }|g(y)|^{q}\,dy\right)^{1/q}\leq \left({\sqrt {p}}\int _{\mathbb {R} }|f(x)|^{p}\,dx\right)^{1/p}.}$