Introduction and definitions
When considering the norms of a function $f(x)$ and its Fourier transform, $\tilde{f}$, \begin{equation} \tilde{f}(y)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R}e^{-iyx}f(x) \,, \end{equation} defined according to \begin{equation} \|f\|_p = \left(\int_\mathbb{R} |f(x)|^p dx\right)^{\frac{1}{p}}\,, \end{equation} the following inequality holds \begin{equation} \|\tilde f\|_q \leq k(p,q) \|f\|_p \,, \end{equation} with \begin{equation} \frac{1}{p}+\frac{1}{q}=1 \,, \quad q\geq 2 \,. \end{equation}
The optimal constant for $k(p,q)$ was calculated by Beckner [1] (as well as others) and is equal to: \begin{equation} k(p,q) = \left(\frac{2\pi}{q}\right)^{\frac{1}{2q}} \left(\frac{2\pi}{p}\right)^{\frac{-1}{2p}}\,. \end{equation}
The question
Now, instead of considering the Fourier transform, consider the fractional Fourier transform for some angle $2\pi/n$, where $n\in\mathbb{N}\,, n\geq 3$. Then denoting by \begin{equation} \hat{f}^{(j)}=\mathcal{F}\ldots \mathcal{F} [f] \,, 1\leq j \leq n-1 \end{equation} the result of applying the fractional Fourier transform to $f$ $j$ times, one has the modified Beckner inequality \begin{equation} \|\hat f ^{(j)} \|_q \leq c(p,q) \|\hat f ^{(j-1)}\|_p \,, \end{equation} again with \begin{equation} \frac{1}{p}+\frac{1}{q}=1 \,, \quad q\geq 2 \,. \end{equation}
Similarly to the case of the Fourier transform, the optimal constant can be calculated and is found to be equal to: \begin{equation} c(p,q) = d(p) \, k(p,q) \,, \end{equation} and where $d(p)$ is a factor that essentially depends on the angle $2\pi/n$. (see e.g. [2] or [3], behind paywall)
I am wondering if there exists an inequality for the norms of all the $\hat{f}^{(j)}$'s (or at least more than two) at the same time, which does not trivially follows by combining the Beckner-type ones. For example, with $n=3$, is there some inequality of the form \begin{equation} a(p,q,r)\| \hat{f}^{(2)}\|_p + b(p,q,r)\|\hat{f}^{(1)}\|_q + c(p,q,r)\|f\|_r \geq 0 \,, \end{equation} for certain $a,b,c$ and potentially some constraints on the $p,q,r$?
Any relevant results and references would be highly appreciated.
References
[1] W. Beckner, Annals of Mathematics, 102, 159-182 (1975)
[2] E.H. Lieb, Inventions Math., 102,179-208 (1990)
[3] X.Guanlei, W.Xiaotong, X.Xiaogang, IET Signal Process 3, 392–402 (2009)