Second-order subelliptic operators on Lie groups I: complex uniformly continuous principal coefficients

We consider second-order subelliptic operators with complex coefficients over a connected Lie group G. If the principal coefficients are right uniformly continuous then we prove that the operators generate strongly continuous holomorphic semigroups with kernels K satisfying Gaussian bounds. Moreover, the kernels are Holder continuous and for each [nu] [is an element of][left angle bracket]0, 1[right angle bracket] and [kappa] > 0 one has estimates |K_{z}(k^{-1}g;l^{-1}h) - K_{z}(g;h)| \leqslant a|z|^{-D'/2}\mathrm{e}^{\omega|z|}\Big(\frac{|k|'+|l|'}{|z|^{1/2}+|gh^{-1}|'} \Big)^{\nu} \mathrm{e}^{-b(|gh^{-1}|')^{2}|z|^{-1}} for g, h, k, l [is an element of] G and all z in a subsector of the sector of holomorphy with |k|'+|l|'\leqslant \kappa|z|^{1/2}+2^{-1}|gh^{-1}|' where |\cdot|' denotes the canonical subelliptic modulus and D " the local dimension. These results are established by a blend of elliptic and parabolic techniques in which De Giorgi estimates and Morrey-Campanato spaces play an important role.