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<p>We propose a functional framework of fractional Sobolev spaces for a class of ultraparabolic Kolmogorov-type operators satisfying the weak Hörmander condition. We characterize these spaces as real interpolation of natural-order intrinsic Sobolev spaces recently introduced in Pascucci and Pesce (J Funct Anal 286(7):Paper No. 110344, 40, 2024) and prove continuous embeddings into <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> and intrinsic Hölder spaces from Pagliarani et al. (J Math Anal Appl 435(2):1054–1087, 2016). 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.</p>

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Fractional Sobolev spaces related to an ultraparabolic operator

  • 期刊论文

  • 发表日期：

    2025年7月28日

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  • Antonello Pesce,
  • Sascha Portaro

摘要

We propose a functional framework of fractional Sobolev spaces for a class of ultraparabolic Kolmogorov-type operators satisfying the weak Hörmander condition. We characterize these spaces as real interpolation of natural-order intrinsic Sobolev spaces recently introduced in Pascucci and Pesce (J Funct Anal 286(7):Paper No. 110344, 40, 2024) and prove continuous embeddings into \(L^p\) L p and intrinsic Hölder spaces from Pagliarani et al. (J Math Anal Appl 435(2):1054–1087, 2016). 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.