Second-Order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Principal Coefficients

Abstract

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 Hölder continuous and for each ν ∈〈0, 1〉 and κ > 0 one has estimates

$$\left| {K_z \left( {k^{ - 1} g;l^{ - 1} h} \right) - K_z \left( {g;h} \right)} \right| \leqslant a\left| z \right|^{ - D'/2_e {\omega }\left| z \right|} \left( {\frac{{\left| k \right|^\prime + \left| l \right|^\prime }}{{\left| z \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \left| {gh^{ - 1} } \right|^\prime }}} \right)^v {e - b}\left( {\left| {gh^{ - 1} } \right|^\prime } \right)^2 \left| z \right|^{ - 1} $$

for g, h, k, l ∈ G and all z in a subsector of the sector of holomorphy with $\left| k \right|^\prime + \left| l \right|^\prime \leqslant \kappa \left| z \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 2^{ - 1} \left| {gh^{ - 1} } \right|^\prime$ where $\left| {\; \cdot \;} \right|^\prime$ 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.