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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.
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