*Submitted Paper*

**Inserted:** 16 jul 2024

**Last Updated:** 29 jul 2024

**Year:** 2024

**Doi:** 10.13140/RG.2.2.31072.70402/1

**Links:**
Link to the PDF on ResearchGates

**Abstract:**

We deal with the following wide class of kinetic equations, \[ \partial_t f \,+\, v\cdot\nabla_x f \, = \, \mathcal{L}_v f. \] Above, the diffusion term $\mathcal{L}_v$ is an integro-differential operator, whose nonnegative kernel is of fractional order $s\in(0,1)$ having merely measurable coefficients.

Firstly, we obtain a general $L^\infty$-interpolation inequality with a natural nonlocal tail term in velocity, in turn giving local boundedness even for weak subsolutions $f$ without any sign assumptions. This is a veritable novelty, being boundedness usually assumed a priori in such a setting.
Then, we prove that nonnegative weak solutions $f$ do satisfy
\[
\sup_{Q^-} f \ \leq \ c\inf_{Q^+} f \,+ \, c\,(1-s) |

{Tail} (f) |

_{L^{2+\varepsilon}_{t,x}}
\]
where $Q^{\pm}$ are suitable slanted cylinders provided that their nonlocal tail in velocity is $(2+\varepsilon)$-summable along {the transport variables}.
This is the very first Strong Harnack inequality for kinetic integro-differential-type equations under the tail integrability assumption, which can be deduced for instance from the usual mass density boundedness (as for the Boltzmann equation without cut-off), and in clear accordance with the very recent surprising counterexample by Kassmann and Weidner; see arxiv:2405.05223

A new standalone result, a Besicovitch-Vitali-type covering argument for very general kinetic geometries, independent on the involved equation, is also needed, stated and proved.

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