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Second-order strongly elliptic operators on Lie groups with Hölder continuous coefficients

Part of: Lie groups Elliptic equations and systems Parabolic equations and systems

Published online by Cambridge University Press:  09 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 TechnologyP.O. Box 513 5600 MB EindhovenThe Netherlands e-mail: terelst@win.tue.nl
Derek W. Robinson
Affiliation:
Centre for Mathematics and its Applications School of Mathematical Sciences Australian National UniversityCanberra, ACT 0200Australia e-mail: derek.robinson@anu.edu.au
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Abstract

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Let G be a connected Lie group with Lie algebra g and a1, …, ad an algebraic basis of g. Further let Ai denote the generators of left translations, acting on the Lp-spaces Lp(G; dg) formed with left Haar measure dg, in the directions ai. We consider second-order operators in divergence form corresponding to a quadratic form with complex coefficients, bounded Hölder continuous principal coefficients cij and lower order coefficients ci, c′ii, c0 ∈ L∞ such that the matrix C= (cij) of principal coefficients satisfies the subellipticity condition uniformly over G.

We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel and smoothness of the domain of powers of H on the Lρ-spaces. Moreover, we present Gaussian type bounds for the kernel and its derivatives.

Similar theorems are proved for strongly elliptic operators in non-divergence form for which the principal coefficients are at least once differentiable.

MSC classification

Secondary: 35J15: Second-order elliptic equations 35K05: Heat equation 22E30: Analysis on real and complex Lie groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 3 , December 1997 , pp. 297 - 363
Copyright
Copyright © Australian Mathematical Society 1997

References

[ADM]Albrecht, D., Duong, X. and McIntosh, A., ‘Operator theory and harmonic analysis’, in: Instructional workshop on analysis and geometry, part III, vol. 34 of Proceedings of the Centre for Mathematics and its Applications, Australian National University, Canberra 1996, 77–136.Google Scholar
[Aus]Auscher, P., ‘Regularity theorems and heat kernel for elliptic operators’, J. London Math. Soc. 54 (1996), 284–296.Google Scholar
[ACT]Auscher, P., Coulhon, T. and Tchamitchian, P., ‘Absence de principe du maximum pour certaines équations paraboliques complexes’, Colloq. Math. 71 (1996), 87–95.Google Scholar
[AMT]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, 1994).Google Scholar
[Bur]Burns, R. J., Sobolev spaces on Lie groups (PhD thesis, The Australian National University, Canberra, 1991).Google Scholar
[BER]Burns, R. J., ter Elst, A. F. M. and Robinson, D. W., ‘L p-regularity of subelliptic operators on Lie groups’, J. Operator Theory 31 (1994), 165–187.Google Scholar
[BuB]Butzer, P. L. and Berens, H., Semi-groups of operators and approximation, Grundlehren Math. Wiss. 145 (Springer, Berlin, 1967).Google Scholar
[CoW]Coifman, R. R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogénes, Lecture Notes in Math. 242 (Springer, Berlin, 1971).CrossRefGoogle Scholar
[CDMY]Cowling, M., Doust, I., McIntosh, A. and Yagi, A., ‘Banach space operators with a bounded H∞ functional calculus’, J. Austral. Math. Soc. (Series A) 60 (1996), 51–89.CrossRefGoogle Scholar
[DuR]Duong, X. T. and Robinson, D. W., ‘Semigroup kernels, Poisson bounds and holomorphic functional calculus’, J. Funct. Anal. 142 (1996), 89–129.CrossRefGoogle Scholar
[EIR1]ter Elst, A. F. M. and Robinson, D. W., ‘Subelliptic operators on Lie groups: regularity’, J. Austral. Math. Soc. (Series A) 57 (1994), 179–229.Google Scholar
[EIR2]ter Elst, A. F. M. and Robinson, D. W., ‘Subcoercivity and subelliptic operators on Lie groups I: Free nilpotent groups’, Potential Anal. 3 (1994), 283–337.Google Scholar
[EIR3]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
[EIR4]ter Elst, A. F. M. and Robinson, D. W., ‘Reduced heat kernels on nilpotent Lie groups’, Comm. Math. Phys. 173 (1995), 475–511.Google Scholar
[EIR5]ter Elst, A. F. M. and Robinson, D. W., ‘On Kato's square root problem’, Hokkaido Math. J. 26 (1997), 365–376.Google Scholar
[EIR6]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, 1996).Google Scholar
[EIR7]ter Elst, A. F. M. and Robinson, D. W., ‘High order divergence-form elliptic operators on Lie groups’, Bull. Austral. Math. Soc. 55 (1997), 335–348.Google Scholar
[EIR8]ter Elst, A. F. M. and Robinson, D. W., ‘Second-order subelliptic operators on Lie groups II: real measurable principal coefficients’, Research Report MRR 037–96 (The Australian National University, Canberra, Australia, 1996).Google Scholar
[Gia1]Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. of Math. Stud. 105 (Princeton University Press, Princeton, 1983).Google Scholar
[Gia2]Giaquinta, M., Introduction to regularity theory for nonlinear elliptic systems, Lectures in Math. ETH Zürich (Birkhäuser, Basel, 1993).Google Scholar
[Kat1]Kato, T., ‘Fractional powers of dissipative operators’, J. Math. Soc. Japan 13 (1961), 246–274.Google Scholar
[Kat2]Kato, T., Perturbation theory for linear operators, 2nd edition, Grundlehren Math. Wiss. 132 (Springer, Berlin, 1984).Google Scholar
[Kor]Kordyukov, Y. A., ‘L p-Theory of elliptic differential operators on manifolds of bounded geometry’, Acta Appl. Math. 23 (1991), 223–260.Google Scholar
[Lio]Lions, J. L., ‘Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs’, J. Math. Soc. Japan 14 (1962), 233–241.Google Scholar
[LiM]Lions, J. L. and Magenes, E., Non-homogeneous boundary value problems and applications, Volume I, Grundlehren Math. Wiss. 181 (Springer, Berlin, 1972).Google Scholar
[McI]McIntosh, A., ‘Square roots of elliptic operators’, J. Funct. Anal. 61 (1985), 307–327.Google Scholar
[Rob]Robinson, D. W., Elliptic operators and Lie groups, Oxford Math. Monographs (Oxford University Press, Oxford, 1991).CrossRefGoogle Scholar
[Ste]Stein, E. M., Singular integrals and differential properties of functions, Princeton Math. Series 30 (Princeton University Press, Princeton, 1970).Google Scholar
[Tay]Taylor, M. E., Pseudodifferential operators, Princeton Math. Series 34 (Princeton University Press, Princeton, 1981).Google Scholar
[Tri]Triebel, H., Interpolation theory, function spaces, differential operators (North-Holland, Amsterdam, 1978).Google Scholar
[VSC]Varopoulos, N. T., Saloff-Coste, L. and Coulhon, T., Analysis and geometry on groups, Cambridge Tracts in Math. 100 (Cambridge University Press, Cambridge, 1992).Google Scholar