FUNCTIONAL CONTINUOUS UNCERTAINTY PRINCIPLE

Let $(\Omega, \mu)$,  $(\Delta, \nu)$ be   measure spaces. Let  $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$  and   $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$   be    continuous p-Schauder frames  for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$,  we show that 
\begin{align}\label{CUE}
    \mu(\operatorname{supp}(\theta_f x))^\frac{1}{p}    \nu(\operatorname{supp}(\theta_g x))^\frac{1}{q} \geq     \frac{1}{\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|}, \quad      \nu(\operatorname{supp}(\theta_g x))^\frac{1}{p}    \mu(\operatorname{supp}(\theta_f x))^\frac{1}{q}\geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|}.
\end{align}
where 
\begin{align*}
    &\theta_f: \mathcal{X} \ni x \mapsto \theta_fx \in \mathcal{L}^p(\Omega, \mu); \quad   \theta_fx: \Omega \ni \alpha \mapsto  (\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\
    &\theta_g: \mathcal{X} \ni x \mapsto \theta_gx \in \mathcal{L}^p(\Delta, \nu); \quad   \theta_gx: \Delta \ni \beta \mapsto  (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K}
\end{align*}
and $q$ is the conjugate index of $p$. We call Inequality (\ref{CUE}) as \textbf{Functional Continuous  Uncertainty Principle}. It improves the Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle obtained by M. Krishna in   \textit{[arXiv:2304.03324v1, 2023]}.   It also answers a question asked by Prof. Philip B. Stark  to the author.