Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields

For any non-Archimedean local field $\mathbb{K}$ and any integer $n \geq 1$, we show that the Taibleson operator admits a bounded $\mathrm{H}^\infty(Σ_θ)$ functional calculus on the Bochner space $\mathrm{L}^p(\mathbb{K}^n,Y)$ for any $\mathrm{UMD}$ Banach function space $Y$ and any angle $θ> 0$, where $Σ_θ={ z \in \mathbb{C}^*: |\arg z| < θ}$ 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.

💡 Research Summary

The paper investigates functional calculus and semilinear evolution equations associated with the Taibleson operator on non‑Archimedean local fields. Let 𝕂 be any non‑Archimedean local field (e.g., a p‑adic field) and n≥1. The Taibleson operator D_α, defined for α>0 by the singular integral (or equivalently as a Fourier multiplier with symbol ‖ξ‖_𝕂^α), plays the role of the fractional Laplacian (−Δ)^α in the ultrametric setting.

The authors first recall Taibleson’s classical result that any bounded radial function m(x)=\tilde m(‖x‖_𝕂) defines a bounded Fourier multiplier on scalar L^p(𝕂^n) for 1<p<∞. Building on this, they consider vector‑valued Bochner spaces L^p(𝕂^n,Y) where Y is a UMD Banach function space. The main theorem (Theorem 1.1) states that the operator D_α⊗Id_Y admits a bounded H^∞(Σ_θ) functional calculus for every angle θ>0 and, moreover, a bounded Hörmander functional calculus of order s>3/2. The angle of sectoriality ω_H∞(D_α⊗Id_Y) is shown to be zero, which is optimal.

The proof hinges on harmonic analysis on locally compact Spector‑Vilenkin groups. The authors construct a family of convolution operators associated with radial kernels and prove that this family is R‑bounded on L^p(𝕂^n,Y). R‑boundedness is the key analytic tool that allows one to transfer scalar multiplier results to the vector‑valued setting and to invoke the abstract H^∞ functional calculus theory (e.g., Kalton–Weis). By establishing suitable Gaussian‑type heat‑kernel bounds for complex times, they obtain the required Mihlin‑type estimates for the symbol ‖ξ‖_𝕂^α, which lead to the Hörmander calculus with the dimension‑free exponent 3/2. This is in stark contrast with the Euclidean case where the order depends on the spatial dimension (s>n/2).

Having secured the functional calculus, the paper turns to maximal L^q‑regularity. It is well known that a sectorial operator with a bounded H^∞(Σ_θ) calculus for some θ<π/2 on a UMD space enjoys maximal L^q‑regularity for all 1<q<∞. Consequently, D_α⊗Id_Y has maximal L^q‑regularity on L^p(𝕂^n,Y). This yields a priori estimates of the form
‖∂t y‖{L^q}+‖D_α y‖{L^q} ≤ C‖f‖{L^q}
for the inhomogeneous Cauchy problem ∂_t y + D_α y = f with zero initial data.

The authors further consider perturbations by an R‑sectorial operator B acting on Y with ω_R(B)<π/2. They prove (Corollary 1.2) that the coupled operator D_α⊗Id + Id⊗B also possesses maximal L^q‑regularity on the Bochner space. This allows one to treat evolution equations that combine an ultrametric diffusion (the Taibleson operator) with classical spatial operators (e.g., Laplacian, transport, or higher‑order differential operators) acting on additional variables. An explicit example is given where Y=L^r(ℝ^{m+1}_+) and B is a second‑order elliptic operator satisfying the Lopatinskii–Shapiro condition; the resulting parabolic system admits a unique strong solution with maximal regularity.

Finally, the functional calculus framework is applied to semilinear equations of the form ∂_t y + D_α y = F(y). Using the bounded H^∞ calculus, the authors obtain bounded imaginary powers of D_α⊗Id_Y, which in turn give analytic semigroup generation and the necessary Lipschitz estimates on the nonlinear term. By a fixed‑point argument in the maximal regularity space W^{1,q}(0,T;L^p(𝕂^n,Y))∩L^q(0,T;Dom(D_α⊗Id_Y)), they establish both local and global existence results for a broad class of semilinear ultrametric parabolic equations (Theorem 7.1).

In summary, the paper makes three major contributions: (1) it proves that the Taibleson operator on any non‑Archimedean local field enjoys a dimension‑free Hörmander functional calculus of order 3/2 on vector‑valued L^p‑spaces; (2) it derives maximal L^q‑regularity for both the pure Taibleson operator and its perturbations by R‑sectorial operators; and (3) it leverages these analytic tools to obtain well‑posedness results for linear and semilinear evolution equations in the ultrametric setting. These results significantly advance the harmonic analysis and PDE theory on totally disconnected spaces and open new avenues for applications in p‑adic models of physics, biology, and finance.