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

We consider degenerate KFP operators $$Lu=\sum _{i,j=1}^{m_0}{a}_{ij}(x,t){\mathrm{\partial }}_{x_i{x}_j}^{2}u+\sum _{k,j=1}^N{b}_{jk}{x}_k{\mathrm{\partial }}_{x_j}u-{\mathrm{\partial }}_{t}u\equiv \sum _{i,j=1}^{m_0}{a}_{ij}(x,t){\mathrm{\partial }}_{x_i{x}_j}^{2}u+Yu$$ ($(x,t)\in {\mathbb{R}}^{N+1}$, $1\le m_0\le N$) s.t. the model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Lie group in ${\mathbb{R}}^{N+1}$ and $2$-homogeneous w.r.t. a family of dilations; $(a_{ij}{)}_{i,j=1}^{m_0}$ is symmetric and uniformly positive on ${\mathbb{R}}^{m_0}$; $a_{ij}$ are bounded and Dini continuous in space, bounded measurable in time, i.e.: setting $$S_T={\mathbb{R}}^N\times (-\mathrm{\infty },T),$$ $${\omega }_{f,S_T}(r)=\underset{\begin{array}{c}(x,t),(y,t)\in S_T\\ \mathrm{\parallel }x-y\mathrm{\parallel }\le r\end{array}}{\sup }\mathrm{\mid }f(x,t)-f(y,t)\mathrm{\mid }$$ \Vert f\Vert_{\mathcal{D}(S_{T})}=\int_{0}^{1}\frac{ω_{f,S_{T}}(r)}% {r}dr+\Vert f\Vert_{L^{\infty}\left( S_{T}\right) } we ask $\mathrm{\parallel }{a}_{ij}{\mathrm{\parallel }}_{\mathcal{D}(S_T)}<\mathrm{\infty }$. We bound ${\omega }_{u_{x_i{x}_j},S_T}$, ${\parallel u_{x_i{x}_j}\parallel }_{L^{\mathrm{\infty }}(S_T)}$ ($i,j=1,2,\mathrm{.}\mathrm{.}\mathrm{.},m_0$), ${\omega }_{Yu,S_T}$, $\mathrm{\parallel }Yu{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(S_T)}$ in terms of ${\omega }_{\mathcal{L}u,S_T}$, $\mathrm{\parallel }Lu{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(S_T)}$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}\left(S_T\right)}$, getting a control on the uniform continuity in space of $u_{x_i{x}_j},Yu$ if $Lu$ is bounded and Dini-continuous in space. Under the additional assumption that $a_{ij}$ and $\mathcal{L}u$ are log-Dini continuous, meaning the finiteness of the quantity% $${\int }_0^1\frac{{\omega }_{f,S_T}\left(r\right)}{r}\mid \log r\mid dr,$$ 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.