Sharp functional calculus for the Taibleson operator on non-archimedean local fields
Résumé
For any non-archimedean local field K and any integer n≥1, we show that the Taibleson operator admits a bounded H∞(Σθ) functional calculus for any angle θ>0 on the Banach space Lp(Kn), where Σθ={z∈C∗:|argz|<θ} and 1<p<∞, and even a bounded H\"ormander functional calculus of order 32 (with striking contrast to the Euclidean Laplacian on \Rn). 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.
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