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

Part of: Parabolic equations and systems Lie groups Elliptic 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

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