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[1]

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

In 1961, Babenko[2] found this norm for even integer values of q. Finally, in 1975, using Hermite functions as eigenfunctions of the Fourier transform, Beckner[3] 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

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