Sharp functional calculus for the Taibleson operator on non-archimedean local fields

Abstract: For any non-Archimedean local field $\mathbb{K}$ and any integer $n \geq 1$, we show that the Taibleson operator admits a bounded $\mathrm{H}^\infty(\Sigma_\theta)$ functional calculus on the Banach space $\mathrm{L}^p(\mathbb{K}^n)$ for any angle $\theta > 0$, which is optimal, where $\Sigma_\theta={ z \in \mathbb{C}^*: |\arg z| < \theta }$ and $1 < p < \infty$. Moreover, we prove that it even admits a bounded H\"ormander functional calculus of order $\frac{3}{2}$, revealing a new phenomenon absent from the existing literature, as this order is independent of the dimension, unlike previous results in various settings. In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups establishing the $R$-boundedness of a family of convolution operators. Our results enhance the understanding of functional calculi of operators acting on $\mathrm{L}^p$-spaces associated with totally disconnected spaces and have implications for the maximal regularity of the fundamental evolution equations associated to the Taibleson operator, relevant in various physical models.