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On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. (English) Zbl 0845.35059
Summary: We consider the class of ultraparabolic differential operators of the following type $Lu= \sum^{p_0}_{i,j= 1} a_{i, j}(x, t) \partial_{x_i x_j} u+ \sum^N_{i, j= 1} b_{i, j} x_i \partial_{x_j} u- \partial_t u,$ where $$B= (b_{i, j})$$ is a constant matrix and $$0< p_0< N$$. We give a definition of Hölder continuity related to suitable groups of translations and dilations. Then, assuming such a regularity on the coefficients $$a_{i,j}(x, t)$$, we construct the fundamental solution $$\Gamma$$ of $$L$$ by the Levi’s parametrix method. Moreover, we prove an accurate local estimate of $$\Gamma$$ and an invariant Harnack inequality for non-negative solutions of the divergence form equation $Lu= \text{div}(A(x, t) Du)+ \langle x, B Du\rangle- \partial_t u.$

##### MSC:
 35K70 Ultraparabolic equations, pseudoparabolic equations, etc.