KFP operators with coefficients measurable in time and Dini continuous in space

Abstract

We consider degenerate Kolmogorov–Fokker–Planck operators $$\begin{aligned} \mathcal {L}u&=\sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}} ^{2}u+\sum _{k,j=1}^{N}b_{jk}x_{k}\partial _{x_{j}}u-\partial _{t}u\\&\equiv \sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}}^{2}u+Yu \end{aligned}$$ (with \((x,t)\in \mathbb {R}^{N+1}\) and \(1\le m_{0}\le N\) ) such that the corresponding model operator having constant \(a_{ij}\) is hypoelliptic, translation invariant w.r.t. a Lie group operation in \(\mathbb {R} ^{N+1}\) and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix \((a_{ij})_{i,j=1}^{m_{0}}\) is symmetric and uniformly positive on \(\mathbb {R}^{m_{0}}\) . The coefficients \(a_{ij}\) are bounded and Dini continuous in space, and only bounded measurable in time. This means that, setting $$\begin{aligned} \mathrm {(i)}&\,\,S_{T}=\mathbb {R}^{N}\times \left( -\infty ,T\right) ,\\ \mathrm {(ii)}&\,\,\omega _{f,S_{T}}(r) = \sup _{\begin{array}{c} (x,t),(y,t)\in S_{T}\\ \Vert x-y\Vert \le r \end{array}}\vert f(x,t) -f(y,t)\vert \\ \mathrm {(iii)}&\,\,\Vert f\Vert _{\mathcal {D}( S_{T}) } =\int _{0}^{1} \frac{\omega _{f,S_{T}}(r) }{r}dr+\Vert f\Vert _{L^{\infty }\left( S_{T}\right) } \end{aligned}$$ we require the finiteness of \(\Vert a_{ij}\Vert _{\mathcal {D}(S_{T})}\) . We bound \(\omega _{u_{x_{i}x_{j}},S_{T}}\) , \(\Vert u_{x_{i}x_{j}}\Vert _{L^{\infty }( S_{T}) }\) ( \(i,j=1,2,...,m_{0}\) ), \(\omega _{Yu,S_{T}}\) , \(\Vert Yu\Vert _{L^{\infty }( S_{T}) }\) in terms of \(\omega _{\mathcal {L}u,S_{T}}\) , \(\Vert \mathcal {L}u\Vert _{L^{\infty }( S_{T}) }\) and \(\Vert u\Vert _{L^{\infty }\left( S_{T}\right) }\) , getting a control on the uniform continuity in space of \(u_{x_{i}x_{j}},Yu\) if \(\mathcal {L}u\) is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients \(a_{ij}\) and \(\mathcal {L}u\) are log-Dini continuous, meaning the finiteness of the quantity $$\begin{aligned} \int _{0}^{1}\frac{\omega _{f,S_{T}}\left( r\right) }{r}\left| \log r\right| dr, \end{aligned}$$ we prove that \(u_{x_{i}x_{j}}\) and Yu are Dini continuous; moreover, in this case, the derivatives \(u_{x_{i}x_{j}}\) are locally uniformly continuous in space and time.