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

Abstract

AbstractWe consider degenerate Kolmogorov–Fokker–Planck operators $$\begin{aligned} \mathcal {L}u&=\sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}} ^{2}u+\sum _{k,j=1}^{N}b_{jk}x_{k}\partial _{x_{j}}u-\partial _{t}u\\&\equiv \sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}}^{2}u+Yu \end{aligned}$$ L u = ∑ i , j = 1 m 0 a ij ( x , t ) ∂ x i x j 2 u + ∑ k , j = 1 N b jk x k ∂ x j u - ∂ t u ≡ ∑ i , j = 1 m 0 a ij ( x , t ) ∂ x i x j 2 u + Y u (with $$(x,t)\in \mathbb {R}^{N+1}$$ ( x , t ) ∈ R N + 1 and $$1\le m_{0}\le N$$ 1 ≤ m 0 ≤ N ) such that the corresponding model operator having constant $$a_{ij}$$ a ij is hypoelliptic, translation invariant w.r.t. a Lie group operation in $$\mathbb {R} ^{N+1}$$ R N + 1 and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix $$(a_{ij})_{i,j=1}^{m_{0}}$$ ( a ij ) i , j = 1 m 0 is symmetric and uniformly positive on $$\mathbb {R}^{m_{0}}$$ R m 0 . The coefficients $$a_{ij}$$ a ij are bounded and Dini continuous in space, and only bounded measurable in time. This means that, setting $$\begin{aligned} \mathrm {(i)}&\,\,S_{T}=\mathbb {R}^{N}\times \left( -\infty ,T\right) ,\\ \mathrm {(ii)}&\,\,\omega _{f,S_{T}}(r) = \sup _{\begin{array}{c} (x,t),(y,t)\in S_{T}\\ \Vert x-y\Vert \le r \end{array}}\vert f(x,t) -f(y,t)\vert \\ \mathrm {(iii)}&\,\,\Vert f\Vert _{\mathcal {D}( S_{T}) } =\int _{0}^{1} \frac{\omega _{f,S_{T}}(r) }{r}dr+\Vert f\Vert _{L^{\infty }\left( S_{T}\right) } \end{aligned}$$ ( i ) S T = R N × - ∞ , T , ( ii ) ω f , S T ( r ) = sup ( x , t ) , ( y , t ) ∈ S T ‖ x - y ‖ ≤ r | f ( x , t ) - f ( y , t ) | ( iii ) ‖ f ‖ D ( S T ) = ∫ 0 1 ω f , S T ( r ) r d r + ‖ f ‖ L ∞ S T we require the finiteness of $$\Vert a_{ij}\Vert _{\mathcal {D}(S_{T})}$$ ‖ a ij ‖ D ( S T ) . We bound $$\omega _{u_{x_{i}x_{j}},S_{T}}$$ ω u x i x j , S T , $$\Vert u_{x_{i}x_{j}}\Vert _{L^{\infty }( S_{T}) }$$ ‖ u x i x j ‖ L ∞ ( S T ) ($$i,j=1,2,...,m_{0}$$ i , j = 1 , 2 , . . . , m 0 ), $$\omega _{Yu,S_{T}}$$ ω Y u , S T , $$\Vert Yu\Vert _{L^{\infty }( S_{T}) }$$ ‖ Y u ‖ L ∞ ( S T ) in terms of $$\omega _{\mathcal {L}u,S_{T}}$$ ω L u , S T , $$\Vert \mathcal {L}u\Vert _{L^{\infty }( S_{T}) }$$ ‖ L u ‖ L ∞ ( S T ) and $$\Vert u\Vert _{L^{\infty }\left( S_{T}\right) }$$ ‖ u ‖ L ∞ S T , getting a control on the uniform continuity in space of $$u_{x_{i}x_{j}},Yu$$ u x i x j , Y u if $$\mathcal {L}u$$ L u is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients $$a_{ij}$$ a ij and $$\mathcal {L}u$$ L u are log-Dini continuous, meaning the finiteness of the quantity $$\begin{aligned} \int _{0}^{1}\frac{\omega _{f,S_{T}}\left( r\right) }{r}\left| \log r\right| dr, \end{aligned}$$ ∫ 0 1 ω f , S T r r log r d r , we prove that $$u_{x_{i}x_{j}}$$ u x i x j and Yu are Dini continuous; moreover, in this case, the derivatives $$u_{x_{i}x_{j}}$$ u x i x j are locally uniformly continuous in space and time.