Abstract
We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion’s directions, i.e. \(x_{1}, \ldots , x_{m_{0}}\). The result is that u is also regular with respect to the remaining directions. The approach we propose is based on the commutators identities and allows us to obtain a Sobolev exponent that does not depend on the integrability assumption of the right-hand side. Lastly, we provide a new proof for the optimal spatial regularity.
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Anceschi, F. Spatial regularity for a class of degenerate Kolmogorov equations. Ricerche mat 71, 271–281 (2022). https://doi.org/10.1007/s11587-022-00685-6
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DOI: https://doi.org/10.1007/s11587-022-00685-6