Fractional Sobolev spaces related to an ultraparabolic operator

2025/01/10 by Pesce, Antonello, Portaro, Sascha
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2501.05898

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.

Related

• Sobolev embeddings for kinetic Fokker-Planck equations
• Fractional Sobolev spaces via interpolation, and applications to mixed local-nonlocal operators
• Approximation and Interpolation in Kolmogorov-type groups
• Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights
• Hölder continuity of functions in the fractional Sobolev spaces: 1-dimensional case