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Fractional operators and Sobolev spaces on homogeneous groups

Nicola Garofalo, Annunziata Loiudice, Dimiter Vassilev

TL;DR

This work constructs and analyzes fractional, nonlocal operators on Lie groups of homogeneous type via a Dirichlet-form framework, establishing a Sobolev-type theory that generalizes the Euclidean fractional Laplacian. It proves a sharp fractional Sobolev inequality on the group, demonstrates the existence of extremals satisfying a fractional Yamabe-type equation, and shows Rellich-type compact embeddings for the Dirichlet spaces. The paper also derives natural limiting relations as $s\to0^+$ and $s\to1^-$, linking the nonlocal operator to the $L^2$ norm and to the sum of squares of homogeneous-one vector fields, respectively, and develops concentration-compactness tools adapted to this noncommutative setting. Complementary results on approximations via convolutions provide a robust analytic toolkit for regularization and stability of the fractional spaces and operators.

Abstract

We establish foundational properties of fractional operators on Lie groups of homogeneous type. We prove embedding theorems for the associated Sobolev-type spaces.

Fractional operators and Sobolev spaces on homogeneous groups

TL;DR

This work constructs and analyzes fractional, nonlocal operators on Lie groups of homogeneous type via a Dirichlet-form framework, establishing a Sobolev-type theory that generalizes the Euclidean fractional Laplacian. It proves a sharp fractional Sobolev inequality on the group, demonstrates the existence of extremals satisfying a fractional Yamabe-type equation, and shows Rellich-type compact embeddings for the Dirichlet spaces. The paper also derives natural limiting relations as and , linking the nonlocal operator to the norm and to the sum of squares of homogeneous-one vector fields, respectively, and develops concentration-compactness tools adapted to this noncommutative setting. Complementary results on approximations via convolutions provide a robust analytic toolkit for regularization and stability of the fractional spaces and operators.

Abstract

We establish foundational properties of fractional operators on Lie groups of homogeneous type. We prove embedding theorems for the associated Sobolev-type spaces.
Paper Structure (17 sections, 17 theorems, 202 equations)

This paper contains 17 sections, 17 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Main results
  4. Background
  5. The Haar measure
  6. Taylor formulas
  7. The Dirichlet form and fractional operator
  8. The Schwartz class of functions and the fractional operator
  9. The carré du champ operator
  10. Group invariances of the fractional operator
  11. The limits of the fractional operator at 0 and 1.
  12. The Dirichlet space and proof of Theorem \ref{['t:frac homog Sobolev and Rellich embed']} a)
  13. The Rellich-Kondrachov compact embeddings
  14. Extremals of the fractional Sobolev embedding
  15. The second concentration compactness lemma
  16. ...and 2 more sections

Key Result

Theorem 1.1

The quadratic form is a symmetric form on the space $C_0^\infty(\mathbb{G})$ of smooth functions with compact support. In fact, for $u,\, \phi\in C_0^\infty(\mathbb{G})$ we have where we have defined Furthermore, we have

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 30 more