Accepted Paper

Inserted: 25 may 2023
Last Updated: 24 oct 2023

Journal: Journal of Evolution equations
Year: 2023

Abstract:

We consider degenerate Kolmogorov-Fokker-Planck operators \[ \mathcal{L}u=\sum_{i,j=1}^{m_0}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2} u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u, \] (with $(x,t)\in\mathbb{R}^{N+1}$ and $1\leq m_0 \leq N$) 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 non-isotropic 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, letting $S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) $, \[ \omega_{f,S_{T}}\left( r\right) =\sup_{\left( x,t\right) ,\left( y,t\right) \in S_{T},|x-y|<r}\left\vert f\left( x,t\right) -f\left( y,t\right) \right\vert \] \[ \left\Vert f\right\Vert _{\mathcal{D}\left( S_{T}\right) } =\int_{0}^{1}\frac{\omega_{f,S_{T}}\left( r\right) }{r}dr+\left\Vert f\right\Vert _{L^{\infty}\left( S_{T}\right) } \] we require the finiteness of $\left\Vert a_{ij}\right\Vert _{\mathcal{D} \left( S_{T}\right) }$. We bound $\omega_{u_{x_{i}x_{j}},S_{T}}$, $\left\Vert u_{x_{i}x_{j}}\right\Vert _{L^{\infty}\left( S_{T}\right) }$ ($i,j=1,2,...,q$), $\omega_{Yu,S_{T}}$, $\left\Vert Yu\right\Vert _{L^{\infty }\left( S_{T}\right) }$ in terms of $\omega_{\mathcal{L}u,S_{T}}$, $\Vert\mathcal{L}u\Vert_{L^{\infty}\left( S_{T}\right) }$ 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 partially Dini-continuous. Under the additional assumption that both the coefficients $a_{ij}$ and $\mathcal{L}u$ are log-Dini continuous, meaning the finiteness of the quantity \[ \int_{0}^{1}\frac{\omega_{f,S_{T}}\left( r\right) }{r}\left\vert \log r\right\vert dr, \] we prove that $u_{x_{i}x_{j}}$ and $Yu$ are Dini continuous; moreover, in this case, the derivatives $\partial_{x_{i}x_{j}}^{2}u$ are locally uniformly continuous in space and time.