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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Imaginary powers of Laplace operators


Authors: Adam Sikora and James Wright
Journal: Proc. Amer. Math. Soc. 129 (2001), 1745-1754
MSC (2000): Primary 42B15; Secondary 35P99
Published electronically: October 31, 2000
MathSciNet review: 1814106
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Abstract:

We show that if $L$ is a second-order uniformly elliptic operator in divergence form on $\mathbf{R}^d$, then $C_1(1+\vert\alpha\vert)^{d/2} \le \Vert L^{i\alpha}\Vert _{L^1 \to L^{1,\infty}} \le C_2 (1+\vert\alpha\vert)^{d/2}$. We also prove that the upper bounds remain true for any operator with the finite speed propagation property.


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Additional Information

Adam Sikora
Affiliation: Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia (or University of Wrocław, KBN 2 P03A 058 14, Poland)
Email: sikora@maths.anu.edu.au

James Wright
Affiliation: School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia
Address at time of publication: Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, United Kingdom
Email: jimw@maths.unsw.edu.au, wright@maths.ed.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05754-3
PII: S 0002-9939(00)05754-3
Keywords: Spectral multiplier, imaginary powers
Received by editor(s): June 22, 1999
Received by editor(s) in revised form: September 27, 1999
Published electronically: October 31, 2000
Additional Notes: The research for this paper was supported by the Australian National University, the University of New South Wales, the University of Wroclaw, the Australian Research Council and the Polish Research Council KBN. We thank these institutions for their contributions.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society