Functional calculus and semilinear evolution equations 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 Bochner space $\mathrm{L}^p(\mathbb{K}^n,Y)$ for any $\mathrm{UMD}$ Banach function space $Y$ and any angle $\theta > 0$, 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 $\frac{3}{2}$. In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups and establish the $R$-boundedness of a family of convolution operators. Our results contribute to the theory of functional calculi for operators acting on vector-valued $\mathrm{L}^p$-spaces over totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator.