Global Sobolev theory for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and $VMO$ in space

TL;DR

The paper develops a global Sobolev theory for Kolmogorov-Fokker-Planck operators $\\mathcal{L}u=\sum_{i,j=1}^{q} a_{ij}(x,t)u_{x_i x_j}+\sum_{k,j=1}^{N} b_{jk}x_k u_{x_j}-\\partial_t u$ with coefficients $a_{ij}$ that are $VMO$ in space and merely bounded in time. The authors introduce a model operator $\\overline{\\mathcal{L}}$ with time-dependent coefficients, construct its fundamental solution $\\Gamma$, and obtain representation formulas and $L^p$-boundedness for the associated singular integrals, enabling global $W^{2,p}$ control of second spatial derivatives and the drift term. Extending these ideas to $(x,t)$-dependent coefficients under the $VMO_x$ assumption, they prove mean-oscillation bounds and $L^p$ estimates for $u_{x_i x_j}$ and $Yu$, culminating in refined global Sobolev estimates on $S_T$ and solvability of $\\mathcal{L}u-\\lambda u=f$ for large $\\lambda$ along with well-posedness of the Cauchy problem in $W_X^{2,p}$. The results substantially extend global regularity theory for degenerate KFP-type operators with rough temporal coefficients, providing rigorous tools for analysis of corresponding stochastic and kinetic PDEs. The methods blend representation formulas from the model operator, singular integral theory on spaces of homogeneous type, and Krylov-type maximal-function techniques adapted to the hypoelliptic, non-Euclidean geometry induced by the drift structure.

Abstract

We consider Kolmogorov-Fokker-Planck operators of the form $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)u_{x_{i}x_{j}}+\sum_{k,j=1}^{N} b_{jk}x_{k}u_{x_{j}}-\partial_{t}u, $$ with $\left( x,t\right) \in\mathbb{R}^{N+1},N\geq q\geq1$. We assume that $a_{ij}\in L^{\infty}\left( \mathbb{R}^{N+1}\right) $, the matrix $\left\{ a_{ij}\right\} $ is symmetric and uniformly positive on $\mathbb{R}^{q}$, and the drift \[ Y=\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}-\partial_{t} \] has a structure which makes the model operator with constant $a_{ij}$ hypoelliptic, translation invariant w.r.t. a suitable Lie group operation, and $2$-homogeneus w.r.t. a suitable family of dilations. We also assume that the coefficients $a_{ij}$ are $VMO$ w.r.t. the space variable, and only bounded measurable in $t$. We prove, for every $p\in\left( 1,\infty\right) $, global Sobolev estimates of the kind: \begin{align*} \Vert u\Vert _{W_{X}^{2,p}(S_{T})} \equiv & \sum_{i,j=1}^{q}\Vert u_{x_{i}x_{j}}\Vert_{L^{p}(S_{T})} +\Vert Yu\Vert _{L^{p}(S_{T})} +\sum_{i=1}^{q}\Vert u_{x_{i}}\Vert _{L^{p}(S_{T})} +\Vert u\Vert _{L^{p}(S_{T})} \\ & \leq c\big\{ \Vert \mathcal{L}u\Vert _{L^{p}(S_{T})}+\Vert u\Vert_{L^{p}(S_{T})}\big\} \end{align*} with $S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) $ for any $T\in(-\infty,+\infty]$. Also, the well-posedness in $W_{X}^{2,p}(Ω_{T})$, with $Ω_{T}=\mathbb{R}^{N}\times(0,T) $ and $T\in\mathbb{R}$, of the Cauchy problem% $$ \begin{cases} \mathcal{L}u=f & \text{in $Ω_{T}$} \\ u(\cdot,0) =g & \text{in $\mathbb{R}^{N}$} \end{cases} $$ is proved, for $f\in L^{p}(Ω_{T}), g\in W_{X}^{2,p}(\mathbb{R}^{N})$.

Theorems & Definitions (58)

• Remark 1.1
• Definition 1.2
• Definition 1.3: Sobolev spaces
• Theorem 1.4: Global Sobolev estimates
• Definition 1.5
• Definition 1.6
• Theorem 1.7: Well-posedness of the Cauchy problem
• Definition 1.8
• Theorem 1.9
• Definition 1.10