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arXiv · 2107.04441

A note on the weak regularity theory for degenerate Kolmogorov equations

Francesca Anceschi·Annalaura Rebucci
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Abstract

The aim of this work is to prove a Harnack inequality and the H\"older continuity for weak solutions to the Kolmogorov equation $\mathscr{L} u = f$ with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a functional space $\mathcal{W}$, suitable for the study of weak solutions to $\mathscr{L}u = f$, that allows us to prove a weak Poincar\'e inequality. More precisely, our goal is to prove a weak Harnack inequality for non-negative super-solutions by considering their Log-transform and following S. N. Kruzkov (1963). Then this functional inequality is combined with a classical covering argument (Ink-Spots Theorem) that we extend for the fist time to the case of ultraparabolic equations.

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Francesca Anceschi, Annalaura Rebucci. 2021-07-09. A note on the weak regularity theory for degenerate Kolmogorov equations. https://arxiv.org/abs/2107.04441

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