Imaginary powers of Laplace operators
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- by Adam Sikora and James Wright PDF
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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 $C_1(1+|\alpha |)^{d/2} \le \|L^{i\alpha }\|_{L^1 \to L^{1,\infty }} \le C_2 (1+|\alpha |)^{d/2}$. We also prove that the upper bounds remain true for any operator with the finite speed propagation property.References
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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)
- MR Author ID: 292432
- 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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1745-1754
- MSC (2000): Primary 42B15; Secondary 35P99
- DOI: https://doi.org/10.1090/S0002-9939-00-05754-3
- MathSciNet review: 1814106