Schauder estimates on bounded domains for KFP operators with coefficients measurable in time and Hölder continuous in space
Stefano Biagi, Marco Bramanti · Analysis and Geometry in Metric Spaces · 2024
Abstract We consider degenerate Kolmogorov-Fokker-Planck operators ℒ u = ∑ i , j = 1 q a i j ( x , t ) u x i x j + ∑ k , j = 1 N b j k x k u x j − u t , {\mathcal{ {\mathcal L} }}u=\mathop{\sum }\limits_{i,j=1}^{q}{a}_{ij}\left(x,t){u}_{{x}_{i}{x}_{j}}+\mathop{\sum }\limits_{k,j=1}^{N}{b}_{jk}{x}_{k}{u}_{{x}_{j}}-{u}_{t}, such that the corresponding model operator having constant a i j {a}_{ij} is hypoelliptic, translation invariant w.r.t. a Lie group operation in R N + 1 {{\mathbb{R}}}^{N+1} and 2-homogeneous w.r.t. a family of nonisotropic dilations. We assume that the a i j {a}_{ij} ’s are globally bounded and Hölder continuous in x x (w.r.t. some intrinsic distance induced by