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We consider degenerate KFP operators \[Lu=\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u\equiv\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+Yu\] ($(x,t)\in\mathbb{R}^{N+1}$, $1\leq m_{0}\leq N$) s.t. the model operatorhaving constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Liegroup 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 \[ S_{T}=\mathbb{R}^{N}\times\left(-\infty,T\right) , \] \[ \omega_{f,S_{T}}(r)=\sup_{\substack{(x,t),(y,t)\inS_{T}\\\Vert x-y\Vert\leq r}}|f(x,t)-f(y,t)| \] \[ \Vertf\Vert_{\mathcal{D}(S_{T})}=\int_{0}^{1}\frac{\omega_{f,S_{T}}(r)}% {r}dr+\Vertf\Vert_{L^{\infty}\left( S_{T}\right) } \] we ask $\Verta_{ij}\Vert_{\mathcal{D}(S_{T})}<\infty$. We bound$\omega_{u_{x_{i}x_{j}},S_{T}}$, $\left\Vert u_{x_{i}x_{j}}\right\Vert_{L^{\infty}(S_{T})}$ ($i,j=1,2,...,m_{0}$), $\omega_{Yu,S_{T}}$, $\VertYu\Vert_{L^{\infty}(S_{T})}$ in terms of $\omega_{\mathcal{L}u,S_{T}}$, $\VertLu\Vert_{L^{\infty}(S_{T})}$ 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 $Lu$ is bounded and Dini-continuous in space. Under the additionalassumption that $a_{ij}$ and $\mathcal{L}u$ are log-Dini continuous, meaningthe finiteness of the quantity% \[ \int_{0}^{1}\frac{\omega_{f,S_{T}}\left(r\right) }{r}\left\vert \log r\right\vert dr, \] 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.

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