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

For any non-archimedean local field K and any integer n ≥ 1, we show that the Taibleson operator admits a bounded H^∞(Σ_θ) functional calculus for any angle θ \> 0 on the Banach space Lᵖ(Kⁿ), where Σ_θ=\ z ∈ C^*: | z| \< θ \ and 1 \< p \< ∞, and even a bounded H\''ormander functional calculus of order 3/2 (with striking contrast to the Euclidean Laplacian on ⁿ). 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 Lᵖ-spaces associated to 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.