We consider degenerate Kolmogorov–Fokker–Planck operators Lu=∑i,j=1m0aij(x,t)∂xixj2u+∑k,j=1Nbjkxk∂xju-∂tu≡∑i,j=1m0aij(x,t)∂xixj2u+Yu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathcal {L}u&=\\sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\\partial _{x_{i}x_{j}} ^{2}u+\\sum _{k,j=1}^{N}b_{jk}x_{k}\\partial _{x_{j}}u-\\partial _{t}u\\\\&\\equiv \\sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\\partial _{x_{i}x_{j}}^{2}u+Yu \\end{aligned}$$\\end{document}(with (x,t)∈RN+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x,t)\\in \\mathbb {R}^{N+1}$$\\end{document} and 1≤m0≤N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le m_{0}\\le N$$\\end{document}) such that the corresponding model operator having constant aij\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_{ij}$$\\end{document} is hypoelliptic, translation invariant w.r.t. a Lie group operation in RN+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R} ^{N+1}$$\\end{document} and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix (aij)i,j=1m0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(a_{ij})_{i,j=1}^{m_{0}}$$\\end{document} is symmetric and uniformly positive on Rm0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{m_{0}}$$\\end{document}. The coefficients aij\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_{ij}$$\\end{document} are bounded and Dini continuous in space, and only bounded measurable in time. This means that, setting (i)ST=RN×-∞,T,(ii)ωf,ST(r)=sup(x,t),(y,t)∈ST‖x-y‖≤r|f(x,t)-f(y,t)|(iii)‖f‖D(ST)=∫01ωf,ST(r)rdr+‖f‖L∞ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathrm {(i)}&\\,\\,S_{T}=\\mathbb {R}^{N}\ imes \\left( -\\infty ,T\\right) ,\\\\ \\mathrm {(ii)}&\\,\\,\\omega _{f,S_{T}}(r) = \\sup _{\\begin{array}{c} (x,t),(y,t)\\in S_{T}\\\\ \\Vert x-y\\Vert \\le r \\end{array}}\\vert f(x,t) -f(y,t)\\vert \\\\ \\mathrm {(iii)}&\\,\\,\\Vert f\\Vert _{\\mathcal {D}( S_{T}) } =\\int _{0}^{1} \\frac{\\omega _{f,S_{T}}(r) }{r}dr+\\Vert f\\Vert _{L^{\\infty }\\left( S_{T}\\right) } \\end{aligned}$$\\end{document}we require the finiteness of ‖aij‖D(ST)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert a_{ij}\\Vert _{\\mathcal {D}(S_{T})}$$\\end{document}. We bound ωuxixj,ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _{u_{x_{i}x_{j}},S_{T}}$$\\end{document}, ‖uxixj‖L∞(ST)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert u_{x_{i}x_{j}}\\Vert _{L^{\\infty }( S_{T}) }$$\\end{document} (i,j=1,2,...,m0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i,j=1,2,...,m_{0}$$\\end{document}), ωYu,ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _{Yu,S_{T}}$$\\end{document}, ‖Yu‖L∞(ST)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert Yu\\Vert _{L^{\\infty }( S_{T}) }$$\\end{document} in terms of ωLu,ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _{\\mathcal {L}u,S_{T}}$$\\end{document}, ‖Lu‖L∞(ST)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert \\mathcal {L}u\\Vert _{L^{\\infty }( S_{T}) }$$\\end{document} and ‖u‖L∞ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert u\\Vert _{L^{\\infty }\\left( S_{T}\\right) }$$\\end{document}, getting a control on the uniform continuity in space of uxixj,Yu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{x_{i}x_{j}},Yu$$\\end{document} if Lu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}u$$\\end{document} is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients aij\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_{ij}$$\\end{document} and Lu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}u$$\\end{document} are log-Dini continuous, meaning the finiteness of the quantity ∫01ωf,STrrlogrdr,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{0}^{1}\\frac{\\omega _{f,S_{T}}\\left( r\\right) }{r}\\left| \\log r\\right| dr, \\end{aligned}$$\\end{document}we prove that uxixj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{x_{i}x_{j}}$$\\end{document} and Yu are Dini continuous; moreover, in this case, the derivatives uxixj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{x_{i}x_{j}}$$\\end{document} are locally uniformly continuous in space and time.