Fractional Sobolev spaces related to an ultraparabolic operator

TL;DR

This paper develops a fractional Sobolev space framework tailored for ultraparabolic operators satisfying the weak H"ormander condition, extending classical embeddings and interpolation techniques to this non-Euclidean setting.

Contribution

It introduces a new fractional Sobolev space framework for ultraparabolic operators, characterizes these spaces via real interpolation, and establishes continuous embeddings into $L^p$ and intrinsic H"older spaces.

Findings

Characterization of fractional Sobolev spaces as real interpolation spaces.

Continuous embeddings into $L^p$ and intrinsic H"older spaces.

Extension of classical Euclidean embeddings to the ultraparabolic setting.

Abstract

We propose a functional framework of fractional Sobolev spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander 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 H\"older 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.