Second-order subelliptic operators on lie groups I: Complex uniformly continuous principal coefficients

We consider second-order subelliptic operators with complex coefficients ov er a connected Lie group G. If the principal coefficients are right uniform ly 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 [0, 1] and kappa > 0 one has estimates \K-z(k(-1)g;l(-1)h) - K-z(g;h)\ less than or equal to a\z(-D'/2)e(omega\z\) (\k\' + \l\'/\z\(1/2) + \gh(-1)\')(nu)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\' less than or equal to kappa\z\(1/2) + 2(-1)\gh (-1)\' where \.\' denotes the canonical subelliptic modulus and D' the loca l dimension. These results are established by a blend of elliptic and parabolic techniqu es in which De Giorgi estimates and Morrey-Campanato spaces play an importa nt role.