$L^r$-Multipliers on compact $p$-adic Lie groups

1. Problem Statement

The paper addresses the fundamental problem of characterizing Fourier multipliers on compact $p$-adic Lie groups acting on weighted $L^r$ spaces ($L^r_\alpha$). Specifically, it seeks to establish sufficient conditions on the symbol $\sigma$ of a left-invariant linear operator $T_\sigma$ to ensure its boundedness on $L^r_\alpha(\mathbb{G})$ for $1 < r < \infty$ and $\alpha > 0$.

While multiplier theorems (such as Mikhlin and Hörmander theorems) are well-established for Euclidean spaces and real compact Lie groups, the $p$-adic setting presents unique challenges:

• Totally Disconnected Structure: The underlying space is a profinite group, lacking the smooth manifold structure of real Lie groups.
• Non-Commutativity: The focus is on non-abelian groups (e.g., the Heisenberg group), where the unitary dual $\hat{\mathbb{G}}$ consists of matrix representations rather than characters, complicating the definition of "derivatives" of symbols.
• Lack of Canonical Laplacian: Defining appropriate analogues of differential operators (like the Laplacian) on general $p$-adic manifolds is an open problem. The paper focuses on Vladimirov-Taibleson (VT) operators as the primary class of pseudo-differential operators.

2. Methodology

The author employs a synthesis of techniques from representation theory, harmonic analysis on totally disconnected groups, and interpolation theory.

A. Representation-Theoretic Framework

• Group Fourier Transform: Operators are analyzed via their symbols $\sigma(\pi)$, where $\pi$ ranges over the unitary dual $\hat{\mathbb{G}}$. For compact $p$-adic Lie groups, the dual is parametrized by specific subsets of trees (ultrametric spaces), often identified with quotients of $\mathbb{Q}_p^d$.
• Difference Operators: Since standard derivatives do not exist, the paper utilizes Ruzhansky-Turunen (RT) difference operators and Saloff-Coste difference operators. These are defined via the tensor product of representations:
  $$\Delta_\eta \sigma(\xi) = \sigma(\eta \otimes \xi) - \sigma(I_{d_\eta} \otimes \xi)$$
  This allows for the measurement of the "smoothness" or regularity of the symbol $\sigma$ on the discrete dual space.

B. Norm Equivalences and Lower Bounds

A critical technical step involves establishing an equivalence between the weighted $L^2$ norm and a norm defined by difference operators on the Fourier side.

• Lower Bound Property: The paper introduces a condition (Definition 3.11) requiring that for any non-identity representation $\pi$ and $x \neq e$, the matrix $\pi(x)$ has a sufficient number of eigenvalues distinct from 1. This ensures that $\|\pi(x) - I\|_{HS}$ is bounded below by the group metric, which is essential for proving norm equivalences.
• Equivalence: It is proven that for groups satisfying this property (e.g., Heisenberg, Engel groups):
  $$\|f\|_{L^2_\alpha}^2 \asymp \sum_{\xi, \eta} d_\xi d_\eta \|\eta\|_p^{-(\alpha+d)} \|\Delta_\eta \hat{f}(\xi)\|_{HS}^2$$

C. Interpolation and Calderón-Zygmund Decomposition

To extend results from $L^2$ to $L^r$ ($r \neq 2$), the author adapts the Calderón-Zygmund decomposition to the $p$-adic setting.

• Condition $H(t)$: A key condition is imposed on the symbol's kernel (inverse Fourier transform) to establish weak-type $(1,1)$ boundedness. This condition controls the oscillation of the kernel outside the support of the function.
• Interpolation: By combining the $L^2$ boundedness (derived from difference operator estimates) with the weak-type $(1,1)$ bound (derived from Condition $H(t)$), the Marcinkiewicz interpolation theorem is used to prove strong $L^r$ boundedness for $1 < r < \infty$.

3. Key Contributions

1. Non-Commutative Multiplier Theorems: The paper extends the classical multiplier theorems of Taibleson and Saloff-Coste (originally for abelian Vilenkin groups) to non-abelian compact $p$-adic Lie groups. It provides explicit conditions on matrix-valued symbols involving RT-difference operators.
2. Generalization of Mikhlin Conditions: It formulates a $p$-adic analogue of the Mikhlin-Hörmander condition. Specifically, if a symbol $\sigma$ satisfies decay estimates on its difference operators (e.g., $\|\Delta_\eta^{\beta} \sigma(\xi)\| \lesssim \|\xi\|^{-\beta}$), the associated operator is bounded on $L^r_\alpha$.
3. Littlewood-Paley Decomposition: A Littlewood-Paley square function $Sf$ is constructed for compact $p$-adic Lie groups. The paper proves the equivalence:
   $$\|f\|_{L^r_\alpha} \asymp \|Sf\|_{L^r_\alpha}$$
   This decomposition is crucial for analyzing the boundedness of functions of operators.
4. Application to Vladimirov-Taibleson Operators: The theory is applied to the Vladimirov-Taibleson operator ($D_\alpha$) and its sub-Laplacian variants. The paper proves that bounded functions of these operators (including negative powers/inverses) are bounded on $L^r_\alpha$ spaces.
5. Explicit Examples: The methodology is demonstrated on specific non-trivial examples:
   • The Heisenberg group $\mathbb{H}_d(\mathbb{Z}_p)$.
   • The Engel group $\mathcal{B}_4$.
   • A 5-dimensional nilpotent group $\mathbb{G}_{5,2}$.
     For these groups, the unitary dual is explicitly parametrized, and the lower bound property is verified.

4. Main Results

• Theorem 2.1 ($L^2$ Boundedness): If the symbol $\sigma$ satisfies a difference operator decay condition (analogous to a derivative bound), then $T_\sigma$ is bounded on $L^2_\alpha(\mathbb{G})$.
• Theorem 2.5 ($L^r$ Boundedness): If $\sigma$ satisfies Condition $H(t)$ and is bounded on a specific $L^2_{\alpha_0}$ space, then $T_\sigma$ is bounded on $L^r_\alpha(\mathbb{G})$ for a range of $r$ and $\alpha$.
• Theorem 2.7 (Main Multiplier Theorem): Combines the above to provide concrete sufficient conditions on the symbol (in terms of RT-difference operators) for $L^r$ boundedness on compact $p$-adic Lie groups.
• Theorem 2.8 (Littlewood-Paley): Establishes the $L^r$ equivalence of the function and its square function $Sf$ for $1 < r < \infty$.
• Theorem 7.2 (Sub-Laplacians): Extends the results to Vladimirov sub-Laplacians (operators acting only on specific directions of the Lie algebra), proving boundedness for functions of these operators.

5. Significance

• Bridging Real and $p$-adic Analysis: The work successfully adapts the sophisticated machinery of pseudo-differential operators from real Lie groups (Ruzhansky-Turunen theory) to the totally disconnected $p$-adic setting, respecting the ultrametric geometry.
• Foundational for $p$-adic Physics: By establishing the boundedness of operators related to the Vladimirov-Taibleson operator (often used as a fractional derivative or Laplacian in $p$-adic quantum mechanics and string theory), the paper provides rigorous functional analytic tools for studying $p$-adic differential equations.
• Resolution of Open Questions: It addresses the lack of a canonical "Laplacian" on general $p$-adic manifolds by providing a robust framework for translation-invariant operators on groups, which serve as local models for manifolds.
• New Tools for Representation Theory: The explicit computation of tensor product decompositions and difference operators for specific $p$-adic Lie groups (like Heisenberg) offers new insights into the structure of their unitary duals.

In summary, this paper provides a comprehensive framework for the $L^r$-theory of pseudo-differential operators on compact $p$-adic Lie groups, unifying representation theory, harmonic analysis, and interpolation theory to solve long-standing problems regarding multiplier boundedness in the non-Archimedean, non-commutative setting.