A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations (Q1399402)

The authors prove a global upper bound for the fundamental solution \(\Gamma\) of the second order partial differential equation in divergence form \[ \sum_{i,j=1}^m \partial_{x_i} (a_{ij}(x,t)\partial_{x_j} u(x,t)) + \sum_{i,j=1}^N b_{ij} x_i \partial_{x_j} u(x,t) - \partial_t u(x,t) =0 \] where \((x,t)= (x_1,x_2,\dots ,x_N,t)\in \mathbb{R}^{N+1}\) and \(1 \leq m\leq N .\) The existence of a fundamental solution of the ultraparabolic equation is ensured by some structural conditions. Let us observe that the Gaussian estimates from above of the function \(\Gamma\) is independent of the modulus of continuity of the coefficients.