For any non-Archimedean local field <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation> and any integer <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \geqslant 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that the Taibleson operator admits a bounded <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{H}^\infty (\Sigma _\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mtext>H</mtext> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Σ</mi> <mi>θ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> functional calculus on the Bochner space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{L}^p(\mathbb {K}^n,Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mtext>L</mtext> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">K</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{UMD}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>UMD</mtext> </math></EquationSource> </InlineEquation> Banach function space Y and any angle <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\theta > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Sigma _\theta =\{ z \in \mathbb {C}^*: |\arg z| < \theta \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Σ</mi> <mi>θ</mi> </msub> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> </mrow> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mo>∗</mo> </msup> <mrow> <mo>:</mo> <mo stretchy="false">|</mo> <mo>arg</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo><</mo> <mi>θ</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$1 <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we prove that it even admits a bounded Hörmander functional calculus of order <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\frac{3}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation>. In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups and establish the R-boundedness of a family of convolution operators. Our results contribute to the theory of functional calculi for operators acting on vector-valued <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\textrm{L}^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>L</mtext> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-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.

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

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发表日期：

2026年4月28日

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Cédric Arhancet,
Christoph Kriegler

摘要

For any non-Archimedean local field $\mathbb {K}$ $K$ and any integer $n \geqslant 1$ $n ⩾ 1$ , we show that the Taibleson operator admits a bounded $\textrm{H}^\infty (\Sigma _\theta )$ $H^{\infty} (Σ_{θ})$ functional calculus on the Bochner space $\textrm{L}^p(\mathbb {K}^n,Y)$ $L^{p} (K^{n}, Y)$ for any $\textrm{UMD}$ $UMD$ Banach function space Y and any angle $\theta > 0$ $θ > 0$ , where $\Sigma _\theta =\{ z \in \mathbb {C}^*: |\arg z| < \theta \}$ $Σ_{θ} = {z \in C^{*} : | arg z | < θ}$ and $1< p < \infty $ $1 < p < \infty$ . Moreover, we prove that it even admits a bounded Hörmander functional calculus of order $\frac{3}{2}$ $\frac{3}{2}$ . 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 $\textrm{L}^p$ $L^{p}$ -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.