Kinetic Sobolev Spaces

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 {à} la Bouchut-H{ö}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.