My question is a follow-up to Abdelmalek Abdesselam’s recent post

What makes Gaussian distributions special? Local field version?

asking about various characterizations of (real-valued) Gaussian distributions which remain valid for other analogues of Gaussian distributions/functions (e.g. in the p-adic context). One interesting

characterization of this kind arises with Babenko-Beckner’s fine-tuning of the Hausdorff inequality (see https://en.wikipedia.org/wiki/Babenkoâ€“Beckner_inequality). For real numbers $s, t$ with ${1 over s} + {1 over t} = 1$ and $1 < s leq 2$ it is known that

the Fourier transform $f mapsto hat{f}$ maps $L^s(Bbb{R}^n)$

to $L^t(Bbb{R}^n)$ and satisfies the inequality

begin{equation}

| hat{f} , |_t leq Big( s^{1 over s} , t^{-{1 over t}} Big)^{n over 2} , | f |_s quad

left( {scriptstyle begin{array}{l} text{Babenko}

\ text{Beckner} \ text{inequality} end{array}} right)

end{equation}

When $s = t = 2$ this inequality becomes an equality which is

valid for all $f in L^2(Bbb{R}^n)$. For $s < 2$ equality

is achieved if and only if $f$ is a Gaussian function.

My question concerns an analogue of this inequality

for finite fields: Let $q$ be a power of a prime $p$

and let $Bbb{F}_q$ be the finite field with $q$ elements.

Choose a non-square $delta in Bbb{F}_q$ and form the

quadratic extension $Bbb{F}_qbig( sqrt{delta} big)$.

We view elements of $Bbb{F}_qbig( sqrt{delta} big)$

as linear combinations of the form $z = x + sqrt{delta} y$

with $x, y in Bbb{F}_q$ subject to the usual rules of

addition and multiplication. Conjugation and norm are

expressed, respectively, as $bar{z} = x – sqrt{delta} y$

and $mathrm{N}(z)= x^2 – delta y^2$. Furthermore

define $mathrm{Tr}(z) := z + bar{z}$. Choose any non-trivial

additive character $psi: Bbb{F}_q longrightarrow Bbb{C}^*$

and define the $Bbb{F}_qbig( sqrt{delta} big)$-Fourier transform

$widehat{f}$ of a complex-valued function $f: Bbb{F}_qbig( sqrt{delta} big) longrightarrow Bbb{C}$ by

begin{equation}

widehat{f}(z) :=

{1 over q} , sum_{w in Bbb{F}_qbig( sqrt{delta} big)} ,

f(w) , psi Big(-mathrm{Tr}(zw) Big)

end{equation}

If we endow the function space $Bbb{C}big( Bbb{F}_qbig( sqrt{delta} big) big)$ with the hermitian inner product

begin{equation}

langle f , g rangle := {1 over q} , sum_{w in Bbb{F}_qbig( sqrt{delta} big)} ,

f(w) , overline{g(w)} end{equation}

then Plancherel holds, i.e. $| widehat{f} , |_2 = | f |_2$ and the Babenko-Beckner inequality should take the form

begin{equation} (dagger)

quad | widehat{f} , |_t leq |f , |_s

end{equation}

for any pair of real numbers $s,t$ with ${1 over s} + {1 over t} = 1$ and $1 < s leq 2$. This is a finite field rendering of a more general version of the Babenko-Beckner inequality that holds for

finite abelian groups (see for example https://www.e-periodica.ch/cntmng?pid=ens-001:2000:46::190). As a side note, I would very keen to learn what shape this equality takes in the non-abelian setting.

For $s<2$ inequality is not strict. Indeed $(dagger)$ becomes

an equality for what I’ll call the

$Bbb{F}_qbig( sqrt{delta} big)$-Gaussian functions

defined by $G_x(z) := psi big( x, mathrm{N}(z) big)$

with $x in Bbb{F}_q$. This is because $G_x$is an eigenfunction of the $Bbb{F}_qbig( sqrt{delta} big)$-Fourier transform whose

eigenvalue will be a unit complex number; indeed the Fourier

transform is unitary!

**Question:** Within the range $1 < s < 2$

does inequality $(dagger)$ become an equality

if and only if $f(z) = c ,G_x(z-w)$ for some parameter $x in Bbb{F}_q$, some shift $w in Bbb{F}_qbig( sqrt{delta} big)$, and some overall scalar factor $c in Bbb{C}$?

thanks, ines.