Babenko–Beckner inequality

In mathematics, the Babenko–Beckner inequality (after K. Ivan Babenko and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of Lp spaces. The (q, p)-norm of the n-dimensional Fourier transform is defined to be[1]

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

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 q ≥ 2 {\displaystyle q\geq 2} q\geq 2 is

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

Thus we have the Babenko–Beckner inequality that

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

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

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

then we have

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

or more simply

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

Main ideas of proof

Throughout this sketch of a proof, let

\( 1<p\leq 2,\quad {\frac 1p}+{\frac 1q}=1,\quad {\text{and}}\quad \omega ={\sqrt {1-p}}=i{\sqrt {p-1}}. \)

(Except for q, we will more or less follow the notation of Beckner.)
The two-point lemma

Let \( d\nu (x) \) be the discrete measure with weight 1 1/2 at the points \( x=\pm 1. \) Then the operator

\( {\displaystyle C:a+bx\rightarrow a+\omega bx}

maps \( L^{p}(d\nu ) \) to \( L^{q}(d\nu ) \) with norm 1; that is,

\( \left[\int |a+\omega bx|^{q}d\nu (x)\right]^{{1/q}}\leq \left[\int |a+bx|^{p}d\nu (x)\right]^{{1/p}}, \)

or more explicitly,

\( \left[{\frac {|a+\omega b|^{q}+|a-\omega b|^{q}}2}\right]^{{1/q}}\leq \left[{\frac {|a+b|^{p}+|a-b|^{p}}2}\right]^{{1/p}} \)

for any complex a, b. (See Beckner's paper for the proof of his "two-point lemma".)
A sequence of Bernoulli trials

The measure \( d\nu \) that was introduced above is actually a fair Bernoulli trial with mean 0 and variance 1. Consider the sum of a sequence of n such Bernoulli trials, independent and normalized so that the standard deviation remains 1. We obtain the measure \( d\nu _{n}(x) \) which is the n-fold convolution of \( d\nu ({\sqrt n}x) \) with itself. The next step is to extend the operator C defined on the two-point space above to an operator defined on the (n + 1)-point space of \( d\nu _{n}(x) \) with respect to the elementary symmetric polynomials.

Convergence to standard normal distribution

The sequence\( d\nu _{n}(x) \) converges weakly to the standard normal probability distribution \( d\mu (x)={\frac {1}{{\sqrt {2\pi }}}}e^{{-x^{2}/2}}\,dx \) with respect to functions of polynomial growth. In the limit, the extension of the operator C above in terms of the elementary symmetric polynomials with respect to the measure \( d\nu _{n}(x) \) is expressed as an operator T in terms of the Hermite polynomials with respect to the standard normal distribution. These Hermite functions are the eigenfunctions of the Fourier transform, and the (q, p)-norm of the Fourier transform is obtained as a result after some renormalization.