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} (1) \quad \quad \quad \quad \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 (1) as \textbf{Functional Continuous Uncertainty Principle}. It improves the Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle obtained by K. Mahesh Krishna in [arXiv:2304.03324v1 [math.FA], 5 April 2023]. It also answers a question asked by Prof. Philip B. Stark to the author. Based on Donoho-Elad Sparsity Theorem, we formulate Measure Minimization Conjecture.

翻译：设 $(\Omega, \mu)$, $(\Delta, \nu)$ 为测度空间。令 $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ 和 $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ 为Banach空间 $\mathcal{X}$ 上的连续p-Schauder框架。则对于每个 $x \in \mathcal{X}\setminus\{0\}$，我们证明 \begin{align} (1) \quad \quad \quad \quad \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} 其中 \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*} 且 $q$ 为 $p$ 的共轭指标。我们将不等式(1)称为\textbf{函数连续不确定性原理}。该结果改进了K. Mahesh Krishna在[arXiv:2304.03324v1 [math.FA], 2023年4月5日]中得到的函数Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani不确定性原理，并回答了Philip B. Stark教授向作者提出的一个问题。基于Donoho-Elad稀疏性定理，我们提出了测度最小化猜想。