Unexpected Uncertainty Principle for Disc Banach Spaces

Let $(\{f_n{\}}_{n=1}^{\mathrm{\infty }},\{{\tau }_n{\}}_{n=1}^{\mathrm{\infty }})$ and $(\{g_n{\}}_{n=1}^{\mathrm{\infty }},\{{\omega }_n{\}}_{n=1}^{\mathrm{\infty }})$ be unbounded continuous p-Schauder frames ($0<p<1$) for a disc Banach space $\mathcal{X}$. Then for every $x\in (\mathcal{D}({\theta }_f)\cap \mathcal{D}({\theta }_g))\setminus \{0\}$, we show that \begin{align}\label{UB} (1) \quad \quad \quad \quad |θ_f x|0|θ_g x|0 \geq \frac{1}{\left(\displaystyle\sup{n,m \in $\mathbb{N}$ }|f_n(ω_m)|\right)^p\left(\displaystyle\sup{n, m \in $\mathbb{N}$}|g_m(τ_n)|\right)^p}, \end{align} where \begin{align*} & θ_f: $\mathcal{D}$(θ_f) \ni x \mapsto θ_fx := {f_n(x)}{n=1}^\infty\in \ell^p($\mathbb{N}$), \quad θ_g: $\mathcal{D}$(θ_g) \ni x \mapsto θ_gx := {g_n(x)}{n=1}^\infty\in \ell^p($\mathbb{N}$). \end{align*} Inequality (1) is unexpectedly different from both bounded uncertainty principle arXiv:2308.00312v1 and unbounded uncertainty principle arXiv:2312.00366v1 for Banach spaces.