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Sharp constants in higher-order heat kernel bounds

Published online by Cambridge University Press:  17 April 2009

Nick Dungey
Nick Dungey
Affiliation:
Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia e-mail: [email protected] Australia
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Abstract

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We consider a space X of polynomial type and a self-adjoint operator on L2(X) which is assumed to have a heat kernel satisfying second-order Gaussian bounds. We prove that any power of the operator has a heat kernel satisfying Gaussian bounds with a precise constant in the Gaussian. This constant was previously identified by Barbatis and Davies in the case of powers of the Laplace operator on RN. In this case we prove slightly sharper bounds and show that the above-mentioned constant is optimal.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 2 , April 2000 , pp. 189 - 200
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Barbatis, G. and Davies, E.B., ‘Sharp bounds on heat kernel of higher order uniformly elliptic operators’, J. Operator Theory 36 (1996), 179–198.Google Scholar
[2]Davies, E.B., One-parameter semigroups, London Math. Soc. Monographs 15 (Academic Press, London, New York, 1980).Google Scholar
[3]Davies, E.B., Heat kernels and spectral theory, Cambridge Tracts in Mathematics 92 (Cambridge University Press, Cambridge, 1989).CrossRefGoogle Scholar
[4]Duong, X.T. and Robinson, D.W., ‘Semigroup kernels, Poisson bounds, and holomorphic functional calculus’, J. Funct. Anal 142 (1996), 89–129.CrossRefGoogle Scholar
[5]Robinson, D.W., ‘Commutators and generators II’, Math. Scand 64 (1989), 87–108.CrossRefGoogle Scholar
[6]Robinson, D.W., Elliptic operators and Lie groups, Oxford Mathematical Monographs (Oxford University Press, Oxford, 1991).CrossRefGoogle Scholar
[7]Saloff-Coste, L., ‘A note on Poincaré, Sobolev, and Harnack inequalities’, Internat. Math. Res. Notices 2 (1992), 27–38.CrossRefGoogle Scholar
[8]Varopoulos, N.T., Saloff-Coste, L. and Coulhon, T., Analysis and geometry on groups, Cambridge Tracts in Mathematics 100 (Cambridge University Press, Cambridge, 1992).Google Scholar
[9]Yosida, K., Functional analysis (sixth edition), Grundlehren der mathematischen Wissenschaften 123 (Springer-Verlag, Berlin, Heidelberg, New York, 1980).Google Scholar