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

We consider degenerate Kolmogorov–Fokker–Planck operators \(\begin{aligned} \mathcal {L}u&=\sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}} ^{2}u+\sum _{k,j=1}^{N}b_{jk}x_{k}\partial _{x_{j}}u-\partial _{t}u\\&\equiv \sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}}^{2}u+Yu \end{aligned}\) $$\begin{array}{cc}\hfill \mathcal{L}u& =\sum _{i,j=1}^{m_0}{a}_{\mathit{ij}}(x,t){\partial }_{x_i{x}_j}^{2}u+\sum _{k,j=1}^N{b}_{\mathit{jk}}{x}_k{\partial }_{x_j}u-{\partial }_{t}u\hfill \\ \hfill & \equiv \sum _{i,j=1}^{m_0}{a}_{\mathit{ij}}(x,t){\partial }_{x_i{x}_j}^{2}u+Yu\hfill \end{array}$$ (with \((x,t)\in \mathbb {R}^{N+1}\) $(x,t)\in {\mathbb{R}}^{N+1}$ and \(1\le m_{0}\le N\) $1\le m_0\le N$ ) such that the corresponding model operator having constant \(a_{ij}\) $a_{\mathit{ij}}$ is hypoelliptic, translation invariant w.r.t. a Lie group operation in \(\mathbb {R} ^{N+1}\) ${\mathbb{R}}^{N+1}$ and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix \((a_{ij})_{i,j=1}^{m_{0}}\) ${(a_{\mathit{ij}})}_{i,j=1}^{m_0}$ is symmetric and uniformly positive on \(\mathbb {R}^{m_{0}}\) ${\mathbb{R}}^{m_0}$ . The coefficients \(a_{ij}\) $a_{\mathit{ij}}$ are bounded and Dini continuous in space, and only bounded measurable in time. This means that, setting \(\begin{aligned} \mathrm {(i)}&\,\,S_{T}=\mathbb {R}^{N}\times \left( -\infty ,T\right) ,\\ \mathrm {(ii)}&\,\,\omega _{f,S_{T}}(r) = \sup _{\begin{array}{c} (x,t),(y,t)\in S_{T}\\ \Vert x-y\Vert \le r \end{array}}\vert f(x,t) -f(y,t)\vert \\ \mathrm {(iii)}&\,\,\Vert f\Vert _{\mathcal {D}( S_{T}) } =\int _{0}^{1} \frac{\omega _{f,S_{T}}(r) }{r}dr+\Vert f\Vert _{L^{\infty }\left( S_{T}\right) } \end{aligned}\) $$\begin{array}{cc}\hfill (\mathrm{i})& S_T={\mathbb{R}}^N\times \left(-,\infty ,,,T\right),\hfill \\ \hfill (\mathrm{ii})& {\omega }_{f,S_T}(r)=\underset{\begin{array}{c}(x,t),(y,t)\in S_T\\ \Vert x-y\Vert \le r\end{array}}{sup}|f(x,t)-f(y,t)|\hfill \\ \hfill (\mathrm{iii})& {\Vert f\Vert }_{\mathcal{D}(S_T)}={\int }_0^1\frac{{\omega }_{f,S_T}(r)}{r}dr+{\Vert f\Vert }_{L^{\infty }\left(S_T\right)}\hfill \end{array}$$ we require the finiteness of \(\Vert a_{ij}\Vert _{\mathcal {D}(S_{T})}\) $\Vert a_{\mathit{ij}}{\Vert }_{\mathcal{D}(S_T)}$ . We bound \(\omega _{u_{x_{i}x_{j}},S_{T}}\) ${\omega }_{u_{x_i{x}_j},S_T}$ , \(\Vert u_{x_{i}x_{j}}\Vert _{L^{\infty }( S_{T}) }\) $\Vert u_{x_i{x}_j}{\Vert }_{L^{\infty }(S_T)}$ ( \(i,j=1,2,...,m_{0}\) $i,j=1,2,...,m_0$ ), \(\omega _{Yu,S_{T}}\) ${\omega }_{Yu,S_T}$ , \(\Vert Yu\Vert _{L^{\infty }( S_{T}) }\) ${\Vert Yu\Vert }_{L^{\infty }(S_T)}$ in terms of \(\omega _{\mathcal {L}u,S_{T}}\) ${\omega }_{\mathcal{L}u,S_T}$ , \(\Vert \mathcal {L}u\Vert _{L^{\infty }( S_{T}) }\) ${\Vert \mathcal{L}u\Vert }_{L^{\infty }(S_T)}$ and \(\Vert u\Vert _{L^{\infty }\left( S_{T}\right) }\) ${\Vert u\Vert }_{L^{\infty }\left(S_T\right)}$ , getting a control on the uniform continuity in space of \(u_{x_{i}x_{j}},Yu\) $u_{x_i{x}_j},Yu$ if \(\mathcal {L}u\) $\mathcal{L}u$ is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients \(a_{ij}\) $a_{\mathit{ij}}$ and \(\mathcal {L}u\) $\mathcal{L}u$ are log-Dini continuous, meaning the finiteness of the quantity \(\begin{aligned} \int _{0}^{1}\frac{\omega _{f,S_{T}}\left( r\right) }{r}\left| \log r\right| dr, \end{aligned}\) $$\begin{array}{c}\hfill {\int }_0^1\frac{{\omega }_{f,S_T}\left(r\right)}{r}\left|log,r\right|dr,\end{array}$$ we prove that \(u_{x_{i}x_{j}}\) $u_{x_i{x}_j}$ and Yu are Dini continuous; moreover, in this case, the derivatives \(u_{x_{i}x_{j}}\) $u_{x_i{x}_j}$ are locally uniformly continuous in space and time.