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where $x \in \Bbb{F}_q$ and $v \in \Bbb{F}_q\big( \sqrt{\delta}\big)$ are parameters. I would argue that $\mathrm{G}_x$ should be regarded as $\Bbb{F}_q\big( \sqrt{\delta} \big)$ analogue of the real-valued Gaussian $z \mapsto \exp ( -x |z|^2)$ on the complex plane for reasons connected to the cuspidal representation theory of $\mathrm{SL}_2\big( \Bbb{F}_q \big)$ which are briefly outlined in my response to the Abdesselam's post. Both $\mathrm{G}_x$ and $\mathrm{H}_v$ are near eigenfunctions of the $\Bbb{F}_q\big( \sqrt{\delta} \big)$-Fourier transform. Let's check this to be sure:

where $x \in \Bbb{F}_q$ and $v \in \Bbb{F}_q\big( \sqrt{\delta}\big)$ are parameters. I would argue that $\mathrm{G}_x$ should be regarded as $\Bbb{F}_q\big( \sqrt{\delta} \big)$ analogue of the real-valued Gaussian $z \mapsto \exp ( -x |z|^2)$ on the complex plane for reasons connected to the cuspidal representation theory of $\mathrm{SL}_2\big( \Bbb{F}_q \big)$ which are briefly outlined in my response to Abdesselam's post. Both $\mathrm{G}_x$ and $\mathrm{H}_v$ are near eigenfunctions of the $\Bbb{F}_q\big( \sqrt{\delta} \big)$-Fourier transform. Let's check this to be sure:

where $\widehat{\mathrm{H}}_v(0)$ is a (normalized) Gauss sum whose complex modulus is $1$. Clearly $\mathrm{H}_v$ will be a eigenfunction if and only if $v^4 = -{1 \over 4}$ in $\Bbb{F}_q \big( \sqrt{\delta} \big)$.

where $\widehat{\mathrm{H}}_v(0)$ is a (normalized) Gauss sum with unit complex modulus. Clearly $\mathrm{H}_v$ will be a eigenfunction if and only if $v^4 = -{1 \over 4}$ in $\Bbb{F}_q \big( \sqrt{\delta} \big)$.

\begin{equation} \begin{array}{ll} \displaystyle \mathrm{G}_x(z) &\displaystyle := \ \psi \big( x\, \mathrm{N}(z) \big) \\ \displaystyle \mathrm{H}_y(z) &\displaystyle := \ \psi \big( y\, \mathrm{Tr} \, (z^2) \big) \end{array} \end{equation}

where $x, y \in \Bbb{F}_q$ are parameters. I would argue that $\mathrm{G}_x$ should be regarded as $\Bbb{F}_q\big( \sqrt{\delta} \big)$ analogue of the real-valued Gaussian $z \mapsto \exp ( -x |z|^2)$ on the complex plane for reasons connected to the cuspidal representation theory of $\mathrm{SL}_2\big( \Bbb{F}_q \big)$ which are briefly outlined in my response to the Abdesselam's post. Both $\mathrm{G}_x$ and $\mathrm{H}_y$ are near eigenfunctions of the $\Bbb{F}_q\big( \sqrt{\delta} \big)$-Fourier transform. Let's check this to be sure:

\begin{equation} \widehat{\mathrm{H}}_y \ = \ \widehat{\mathrm{H}}_y(0) \, \mathrm{H}_{- (4y)^{-1}} \end{equation}

where $\widehat{\mathrm{H}}_y(0)$ is a (normalized) Gauss sum whose complex modulus is $1$. Clearly $\mathrm{H}_v$ will be a eigenfunction if and only if $y^2 = -{1 \over 4}$ in $\Bbb{F}_q$.

Note that $\| \mathrm{G}_x \|_s = \| \mathrm{H}_y \|_s = q^{2 \over s}$.

valid within the range $1 < s <2$? Is it sharp and which functions saturate the inequality if it is? Clearly these would include functions of the form $f(z) = c \, \mathrm{G}_x(z-w)$ or $f(z) = c \, \mathrm{H}_y(z-w)$ for some choice of parameters $x,y \in \Bbb{F}_q$, some shift $w \in \Bbb{F}_q\big( \sqrt{\delta} \big)$, and some overall scalaring factor $c \in \Bbb{C}$. But are there others?

\begin{equation} \begin{array}{ll} \displaystyle \mathrm{G}_x(z) &\displaystyle := \ \psi \big( x\, \mathrm{N}(z) \big) \\ \displaystyle \mathrm{H}_v(z) &\displaystyle := \ \psi \big( \mathrm{Tr} \, (vz)^2 \big) \end{array} \end{equation}

where $x \in \Bbb{F}_q$ and $v \in \Bbb{F}_q\big( \sqrt{\delta}\big)$ are parameters. I would argue that $\mathrm{G}_x$ should be regarded as $\Bbb{F}_q\big( \sqrt{\delta} \big)$ analogue of the real-valued Gaussian $z \mapsto \exp ( -x |z|^2)$ on the complex plane for reasons connected to the cuspidal representation theory of $\mathrm{SL}_2\big( \Bbb{F}_q \big)$ which are briefly outlined in my response to the Abdesselam's post. Both $\mathrm{G}_x$ and $\mathrm{H}_v$ are near eigenfunctions of the $\Bbb{F}_q\big( \sqrt{\delta} \big)$-Fourier transform. Let's check this to be sure:

\begin{equation} \widehat{\mathrm{H}}_v \ = \ \widehat{\mathrm{H}}_v(0) \, \overline{\mathrm{H}}_{(2v)^{-1}} \end{equation}

where $\widehat{\mathrm{H}}_v(0)$ is a (normalized) Gauss sum whose complex modulus is $1$. Clearly $\mathrm{H}_v$ will be a eigenfunction if and only if $v^4 = -{1 \over 4}$ in $\Bbb{F}_q \big( \sqrt{\delta} \big)$.

Note that $\| \mathrm{G}_x \|_s = \| \mathrm{H}_v \|_s = q^{2 \over s}$.

valid within the range $1 < s <2$? Is it sharp and which functions saturate the inequality if it is? Clearly these would include functions of the form $f(z) = c \, \mathrm{G}_x(z-w)$ or $f(z) = c \, \mathrm{H}_v(z-w)$ for some choice of parameters $x \in \Bbb{F}_q$ or $v \in \Bbb{F}_q\big( \sqrt{\delta}\big)$, some shift $w \in \Bbb{F}_q\big( \sqrt{\delta} \big)$, and some overall scaling factor $c \in \Bbb{C}$. But are there others?