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 $n\ge 1$, we show that the Taibleson operator admits a bounded ${\mathrm{H}}^{\infty }(\hat{I}{\pounds }_{\hat{I}}¸)$ functional calculus on the Bochner space ${\mathrm{L}}^p({\mathbb{K}}^n,Y)$ for any $\mathrm{UMD}$ Banach function space $Y$ and any angle $\hat{I}¸>0$, where $\hat{I}{\pounds }_{\hat{I}}¸=\{z\in {\mathbb{C}}^{\ast }:|\arg z|<\hat{I}¸\}$ and $1<p<\infty$. Moreover, we prove that it even admits a bounded Hörmander 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.

41 pages, small improvements, final version