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

Authors: ter Elst A.F.M.¹; Robinson D.W.²

Source: Acta Applicandae Mathematicae, Volume 59, Number 3, December 1999 , pp. 299-331(33)

Publisher: Springer

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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 isin lang 0, 1 rang 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 isin 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.

Keywords: subelliptic operators; Gaussian bounds; kernel bounds; De Giorgi estimates

Language: English

Document Type: Regular paper

Affiliations: 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

Publication date: 1999-12-01