Kinetic Sobolev Spaces

TL;DR

This paper introduces kinetic Sobolev spaces tailored to the Kolmogorov equation, establishing their properties, embeddings, and regularity transfer, with applications to well-posedness of the associated Cauchy problem.

Contribution

It defines new kinetic Sobolev spaces for the Kolmogorov equation, proves their key properties, and provides a novel uniqueness criterion for well-posedness.

Findings

Established sharp embeddings and regularity transfer results.

Proved L^p boundedness of singular integral operators related to the Kolmogorov operator.

Provided a new criterion for uniqueness and well-posedness of the Cauchy problem.

Abstract

We define and study homogeneous kinetic Sobolev spaces adapted to the Kolmogorov equation. We consider both local and non-local diffusion. The spaces are built from the Lebesgue spaces L p for all integrability exponents p $\in$ (1, $\infty$) with regularity assumptions in the transport and diffusive directions according to the scaling of the Kolmogorov equation. The regularity scale accommodates weak and strong solutions. We prove that the proposed spaces satisfy sharp embeddings quantifying the transfer-ofregularity {\`a} la Bouchut-H{\"o}rmander, continuity-in-time in the spirit of Lions and the gainof-integrability of Sobolev and Hardy-Littlewood-Sobolev type. A core tool are mapping properties of the Kolmogorov operator, given by the fundamental solution, established between anisotropic homogeneous Sobolev spaces. To achieve this, we prove L^p boundedness of related singular integral operators, for which we deduce novel kernel estimates by a Littlewood-Paley decomposition and geometric considerations. Moreover, we provide a new uniqueness criterion which allows us to show well-posedness of the Cauchy problem.