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
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.
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ter Elst, A.F.M., Robinson, D.W. Second-Order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Principal Coefficients. Acta Applicandae Mathematicae 59, 299–331 (1999). https://doi.org/10.1023/A:1006373625999
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DOI: https://doi.org/10.1023/A:1006373625999