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The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation

Francesca Corni, Fausto Ferrari

TL;DR

The paper develops an analytic-continuation framework to define fractional powers of the sub-Laplacian $\mathcal{L}$ on Carnot groups and relates these operators to generalized Riesz potentials via the heat kernel. It first establishes analyticity of the map $\alpha\mapsto \psi(x,\alpha)$ on $(0,Q)$ and then extends to negative and broader ranges in the Heisenberg group, enabling the construction of $\mathcal{L}^s$ for $-\frac{Q}{2}<s<1$ (and beyond in the Heisenberg setting). Through inductive continuation across strips $(-4,Q)$, $(-6,Q)$, and $(-8,Q)$, the authors derive explicit representations for $\psi(x,\alpha)$ and compute heat-kernel moments, yielding concrete geometric consequences. They also prove semigroup properties and demonstrate consistency with known Euclidean and Heisenberg cases, providing tools for nonlocal potential theory on Carnot groups. Overall, the work bridges fractional sub-Laplacians with generalized Riesz potentials and paves the way for potential-theoretic analysis on stratified groups.

Abstract

In this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the $k$-th order momenta with respect to the heat kernel.

The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation

TL;DR

The paper develops an analytic-continuation framework to define fractional powers of the sub-Laplacian on Carnot groups and relates these operators to generalized Riesz potentials via the heat kernel. It first establishes analyticity of the map on and then extends to negative and broader ranges in the Heisenberg group, enabling the construction of for (and beyond in the Heisenberg setting). Through inductive continuation across strips , , and , the authors derive explicit representations for and compute heat-kernel moments, yielding concrete geometric consequences. They also prove semigroup properties and demonstrate consistency with known Euclidean and Heisenberg cases, providing tools for nonlocal potential theory on Carnot groups. Overall, the work bridges fractional sub-Laplacians with generalized Riesz potentials and paves the way for potential-theoretic analysis on stratified groups.

Abstract

In this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the -th order momenta with respect to the heat kernel.
Paper Structure (40 sections, 33 theorems, 366 equations)

This paper contains 40 sections, 33 theorems, 366 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stratified groups, Pansu differentiabily, coarea formula
  4. Pansu differentiability
  5. Coarea formula in stratified groups
  6. Taylor polynomial on stratified groups
  7. Fractional sub-Laplacian on stratified groups
  8. Generalized Riesz potentials
  9. Regularity
  10. Main properties of the map $\sigma(\alpha)$
  11. Analytic continuation in $\mathbb{G}$
  12. The map $\alpha \to \psi(x, \alpha)$ is analytic on $(0,Q)$
  13. Further properties of the map $\alpha \to \psi(x,\alpha)$
  14. The map $\alpha \to \psi(x, \alpha)$ is analytically continued on (-2,0]
  15. Extension of the definition of $\mathcal{L}^s$ to $-\frac{Q}{2}<s<1$
  16. ...and 25 more sections

Key Result

Theorem 2.4

Let $A \subset \mathbb{G}$ be a measurable set and consider a Lipschitz function $u: A \to \mathbb{R}$. For any measurable function $h:A \to [0, \infty)$ the equality holds. Notice that $\theta^g_{Q-1}( \nabla_H u(x))$ and $\mathcal{S}^{Q-1}$ are both considered with respect to $d$.

Theorems & Definitions (68)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 58 more