论文标题
Schauder估计Kolmogorov-Fokker-Planck运营商，其系数可以在时间上测量，而Hölder在太空中连续连续
Schauder estimates for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and Hölder continuous in space
论文作者
论文摘要
我们考虑退化Kolmogorov-Fokker-Planck运营商$$ \ Mathcal {l} u = \ sum_ {i，j = 1}^{q} a_ {ij}（x，x，x，t）\ partial_ {x_ {x_ {i} x_ {j}}}}}^{2} {2} u +\ sum_ {k，j = 1}^{n} b_ {jk} x_ {k} \ partial_ {x_ {x_ {j}} u- \ partial_ {t} u，\ qquad （x，t）\ in \ mathbb {r}^{n+1}，n \ geq q \ geq1 $$，使得具有常数$ a_ {ij {ij} $的相应模型运算符是低纤维化的，转换不变的W.R.T. $ \ mathbb {r}^{n+1} $和$ 2 $ -HOMOYOUS W.R.T.中的Lie Group操作非偶然扩张的家族。系数$ a_ {ij} $是有界的，并且在太空中连续hölder连续（W.R.T.在$ \ MATHCAL {l} $引起的某些距离中，$ \ Mathbb {r}^{n} $），并且仅在时间上可测量的界限；矩阵$ \ {a_ {ij} \} _ {i，j = 1}^{q} $在$ \ mathbb {r}^{q} $上是对称且均匀的阳性。我们证明了“部分Schauder先验估计” $$ \ sum_ {i，j = 1}^{q} {q} \ vert \ partial_ {x_ {x_ {i} x_ {j {j}}}^{2}^{2} {2} u \ vert_ yu \ vert_ {c_ {x}^α（s_ {t}）}} \ leq c \ left \ left \ {\ vert \ mathcal {l} u \ vert _ {c_ {c_ {x}} u \ vert_ {c^{0}（s_ {t}）}} \ right \} $} $} $ u $，其中$$ \ vert f \ vert f \ vert_ {c_ {x}^al（s_ {t}} t} \ sup_ {x_ {1}，x_ {2} \ in \ mathbb {r}^{n}，x_ {1} \ neq x_ {2}} \ frac {\ frac {\ left \ left \ left \ left \ welet f \ weft（ \ right \ vert} {\ left \ vert x_ {1} -x_ {2} \ right \ vert ^α}。 $$我们还证明了衍生物$ \ partial_ {x_ {x_ {i} x_ {j}}}^{2} u $在空间和时间上是本地连续的，而$ \ partial_ {x_ {x_ {x_ {i}} u $和$ u $在空间和空间中持续持续持续。
We consider degenerate Kolmogorov-Fokker-Planck operators $$ \mathcal{L}u=\sum_{i,j=1}^{q}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,\qquad (x,t)\in\mathbb{R}^{N+1},N\geq q\geq1 $$ such that the corresponding model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Lie group operation in $\mathbb{R}^{N+1}$ and $2$-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients $a_{ij}$ are bounded and Hölder continuous in space (w.r.t. some distance induced by $\mathcal{L}$ in $\mathbb{R}^{N}$) and only bounded measurable in time; the matrix $\{ a_{ij}\}_{i,j=1}^{q}$ is symmetric and uniformly positive on $\mathbb{R}^{q}$. We prove "partial Schauder a priori estimates" the kind $$ \sum_{i,j=1}^{q}\Vert\partial_{x_{i}x_{j}}^{2}u\Vert_{C_{x}^α(S_{T})}+\Vert Yu\Vert_{C_{x}^α(S_{T})}\leq c\left\{ \Vert\mathcal{L}u\Vert _{C_{x}^α(S_{T})}+\Vert u\Vert_{C^{0}(S_{T})}\right\} $$ for suitable functions $u$, where $$ \Vert f\Vert_{C_{x}^α(S_{T})}=\sup_{t\leq T}\sup_{x_{1},x_{2}\in\mathbb{R}^{N},x_{1}\neq x_{2}}\frac{\left\vert f\left( x_{1},t\right) -f\left( x_{2},t\right) \right\vert }{\left\Vert x_{1}-x_{2}\right\Vert ^α}. $$ We also prove that the derivatives $\partial_{x_{i}x_{j}}^{2}u$ are locally Hölder continuous in space and time while $\partial_{x_{i}}u$ and $u$ are globally Hölder continuous in space and time.
