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arXiv · 2501.05898

Fractional Sobolev spaces related to an ultraparabolic operator

Antonello Pesce·Sascha Portaro
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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.

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Antonello Pesce, Sascha Portaro. 2025-01-10. Fractional Sobolev spaces related to an ultraparabolic operator. https://arxiv.org/abs/2501.05898

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