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

Abstract

We consider degenerate KFP operators \[ Lu=\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u\equiv\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+Yu \] ($(x,t)\in {\mathbb{R}}^{N+1}$, $1\le m_0\le N$) 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 \[ S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) , \] \[ ω_{f,S_{T}}(r)=\sup_{\substack{(x,t),(y,t)\in S_{T}\\\Vert x-y\Vert\leq r}}|f(x,t)-f(y,t)| \] \[ \Vert f\Vert_{\mathcal{D}(S_{T})}=\int_{0}^{1}\frac{ω_{f,S_{T}}(r)}% {r}dr+\Vert f\Vert_{L^{\infty}\left( S_{T}\right) } \] we ask $\Vert a_{ij}{\Vert }_{\mathcal{D}(S_T)}<\infty$. 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 ${\omega }_{\mathcal{L}u,S_T}$, $\Vert Lu{\Vert }_{L^{\infty }(S_T)}$ 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 additional assumption that $a_{ij}$ and $\mathcal{L}u$ are log-Dini continuous, meaning the finiteness of the quantity% \[ \int_{0}^{1}\frac{ω_{f,S_{T}}\left( r\right) }{r}\left\vert \log r\right\vert dr, \] 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.