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

Authors:

Institution:

(1) Department of Mathematics and Computing Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands;(2) Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

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 ngr

0, 1

and

> 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 isin

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.

Keywords:

subelliptic operators Gaussian bounds kernel bounds De Giorgi estimates

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