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Fractional Sobolev spaces related to an ultraparabolic operator

Antonello Pesce, Sascha Portaro

TL;DR

This work develops a fractional Sobolev/Besov framework adapted to ultraparabolic Kolmogorov-type operators under a weak Hörmander condition. By introducing intrinsic Besov spaces Λ^{n+s}_{p,q,B} and proving their real interpolation characterization Λ^{n+s}_{p,q,B} = (L^p, W^{n+1,p}_B)_{(n+s)/(n+1), q}, the authors connect fractional regularity to intrinsic geometry. They establish sharp embeddings into Lorentz and intrinsic Hölder spaces, governed by the homogeneous dimension of the underlying Kolmogorov group, and derive these results using an intrinsic Taylor remainder estimate. A key tool is a Taylor inequality for the intrinsic polynomial T_n and an approximation scheme that facilitates the interpolation analysis. Overall, the paper provides a robust, geometry-aware regularity framework for degenerate, hypoelliptic evolution operators arising in kinetic and path-dependent models, with potential applications to Fokker-Planck-type problems and related Kolmogorov equations.

Abstract

We propose a functional framework of fractional Sobolev spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition. We characterize these spaces as real interpolation of natural order intrinic Sobolev spaces recently introduced in [27], and prove continuous embeddings into $L^p$ and intrinsic Hölder spaces from [24]. These embeddings naturally extend the standard Euclidean ones, coherently with the homogeneous structure of the associated Kolmogorov group. Our approach to interpolation is based on approximation of intrinsically regular functions, the latter heavily relying on integral estimates of the intrinsic Taylor remainder. The embeddings exploit the aforementioned interpolation property and the corresponding embeddings of natural order intrinsic spaces.

Fractional Sobolev spaces related to an ultraparabolic operator

TL;DR

This work develops a fractional Sobolev/Besov framework adapted to ultraparabolic Kolmogorov-type operators under a weak Hörmander condition. By introducing intrinsic Besov spaces Λ^{n+s}_{p,q,B} and proving their real interpolation characterization Λ^{n+s}_{p,q,B} = (L^p, W^{n+1,p}_B)_{(n+s)/(n+1), q}, the authors connect fractional regularity to intrinsic geometry. They establish sharp embeddings into Lorentz and intrinsic Hölder spaces, governed by the homogeneous dimension of the underlying Kolmogorov group, and derive these results using an intrinsic Taylor remainder estimate. A key tool is a Taylor inequality for the intrinsic polynomial T_n and an approximation scheme that facilitates the interpolation analysis. Overall, the paper provides a robust, geometry-aware regularity framework for degenerate, hypoelliptic evolution operators arising in kinetic and path-dependent models, with potential applications to Fokker-Planck-type problems and related Kolmogorov equations.

Abstract

We propose a functional framework of fractional Sobolev spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition. We characterize these spaces as real interpolation of natural order intrinic Sobolev spaces recently introduced in [27], and prove continuous embeddings into and intrinsic Hölder spaces from [24]. These embeddings naturally extend the standard Euclidean ones, coherently with the homogeneous structure of the associated Kolmogorov group. Our approach to interpolation is based on approximation of intrinsically regular functions, the latter heavily relying on integral estimates of the intrinsic Taylor remainder. The embeddings exploit the aforementioned interpolation property and the corresponding embeddings of natural order intrinsic spaces.
Paper Structure (12 sections, 13 theorems, 105 equations)

This paper contains 12 sections, 13 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Intrinsic vs anisotropic Hölder and Sobolev spaces
  3. Main results
  4. Interpolation characterization
  5. Embeddings into Lorentz and intrinsic Hölder spaces
  6. Intrinsic weak derivatives and Taylor expansion
  7. Preliminaries
  8. Taylor inequality
  9. Approximation and proof of the main interpolation result
  10. Proof of Theorem \ref{['th:interpolation']}
  11. Embeddings of $\Lambda^{n+s}_{p,q,B}$ spaces
  12. Proof of Theorem \ref{['th:embeddings']}

Key Result

Theorem 1.6

For any $n\in \mathbb N_0$, $s\in (0,1)$ and $p,q\geqslant 1$, we have

Theorems & Definitions (19)

  • Definition 1.2
  • Definition 1.3: Intrinsic Hölder spaces
  • Definition 1.4: Intrinsic Besov spaces
  • Definition 1.5: Intrinsic Sobolev spaces
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4
  • ...and 9 more