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High order divergence-form elliptic operators on Lie groups

Published online by Cambridge University Press:  17 April 2009

A. F. M. ter Elst  and
Derek W. Robinson
A. F. M. ter Elst
Affiliation:
Department of Mathematics and Computing Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
Derek W. Robinson
Affiliation:
Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia
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We give a straightforward proof that divergence-form elliptic operators of order m on a d-dimensional Lie group with m ≥ d have Hölder continuous kernels satisfying Gaussian bounds.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 2 , April 1997 , pp. 335 - 348
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Arendt, W. and Buchvalov, A.V., ‘Integral representations of resolvents and semigroups’, Forum Math. 6 (1994), 111–135.CrossRefGoogle Scholar
[2]Auscher, P., Coulhon, T. and Tchamitchian, P., ‘Absence de principe du maximum pour certaines équations paraboliques complexes’, Colloq. Math. 71 (1996), 87–95.CrossRefGoogle Scholar
[3]Auscher, P., McIntosh, A. and Tchamitchian, P., Heat kernels of second order complex elliptic operators and their applications, Research Report 94–164 (Maquarie University, Sydney, Australia, 1994.).Google Scholar
[4]Bensoussan, A., Lions, J.L. and Papanicolaou, G., Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications 5 (North-Holland, Amsterdam, New York, Oxford, Tokyo, 1978).Google Scholar
[5]Burns, R.J., ter Elst, A.F.M. and Robinson, D.W., ‘Lp-regularity of subelliptic operators on Lie groups’, J. Operator Theory 31 (1994), 165–187.Google Scholar
[6]Cowling, M., Doust, I., McIntosh, A. and Yagi, A., ‘Banach space operators with a bounded H ∞ functional calculus’, J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.CrossRefGoogle Scholar
[7]Davies, E.B., ‘Uniformly elliptic operators with measurable coefficients’, J. Funct. Anal. 132 (1995), 141–169.CrossRefGoogle Scholar
[8]Davies, E.B., ‘Limits on Lp regularity of self-adjoint elliptic operators.’, J. Differential Equations (to appear).Google Scholar
[9]ter Elst, A.F.M. and Robinson, D.W.. ‘Functional analysis of subelliptic operators on Lie groups’, J. Operator Theory 31 (1994), 277–301.Google Scholar
[10]ter Elst, A.F.M. and Robinson, D.W., ‘Elliptic operators on Lie groups’, Acta Appl. Math. 44 (1996), 133–150.Google Scholar
[11]ter Elst, A.F.M. and Robinson, D.W., Second-order subelliptic operators on Lie groups I: complex uniformly continuous principal coefficients, Research Report MRR 035−96 (The Australian National University, Canberra, Australia, 1996).Google Scholar
[12]Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105 (Princeton University Press, Princeton, 1983).Google Scholar
[13]Giorgi, E.D., ‘Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari’, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur. 3 (1957), 25–43.Google Scholar
[14]Kato, T., Perturbation theory for linear operators, Grundlehren der mathematischen Wissenschaften 132, (second edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1984).Google Scholar
[15]Morrey, C.B., Multiple integrals in the calculus of variations (Springer-Verlag, Berlin, Heidelberg, New York, 1966).Google Scholar
[16]Nash, J., ‘Continuity of solutions of parabolic and elliptic equations’, Amer. J. Math. 80 (1958), 931–954.CrossRefGoogle Scholar
[17]Robinson, D.W., Elliptic operators and Lie groups, Oxford Mathematical Monographs (Oxford University Press, Oxford, 1991).Google Scholar