Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Imaginary powers of Laplace operators

Author(s): Adam Sikora; James Wright
Journal: Proc. Amer. Math. Soc. 129 (2001), 1745-1754.
MSC (2000): Primary 42B15; Secondary 35P99
Posted: October 31, 2000
MathSciNet review: 1814106
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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.

References:

1.
G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc. (3) 120 (1994), 973-979. MR 95j:22016
2.
M. Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. (1) 328 (1991), 73-81. MR 92k:42017
3.
M. Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, 77, Amer. Math. Soc. Providence (1990). MR 92f:42021
4.
M. Christ, E.M. Stein, A remark on singular Calderón - Zygmund theory, Proc. Amer. Math. Soc. (3) 99 (1987), 71-75. MR 88c:42030
5.
M. Cowling, S. Meda, Harmonic analysis and ultracontractivity, Trans. Amer. Math. Soc. (2) 340 (1993), 733-752. MR 94b:47050
6.
M. Cowling, I. Doust, A. McIntosh, A. Yagi, Banach space operators with bounded $H^{\infty}$ functional calculus, J. Austral. Math. Soc. (Series A) 60 (1996), 51-89. MR 97d:47023
7.
M. Cowling, A. Sikora, Spectral multipliers for a sub-Laplacian on $\mathrm{ SU}(2)$, Math. Z., to appear.
8.
E.B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press (1989). MR 90e:35123
9.
G. Dore, A. Venni, On the closedness of the sum of two closed operators, Math. Z. (2) 196 (1987), 189-201. MR 88m:47072
10.
A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher transcendental functions. Vol. I. Mc Graw-Hill (1953). MR 15:419i
11.
C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. MR 41:2468
12.
W. Hebisch, Multiplier theorem on generalized Heisenberg groups, Colloq. Math. 65 (1993), 231-239. MR 94m:43013
13.
L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93-139. MR 22:12389
14.
L. Hörmander, The analysis of linear partial differential operators III, Springer-Verlag (1985). MR 87d:35002a; corrected reprint MR 95h:35255
15.
S. Meda, A general multiplier theorem, Proc. Amer. Math. Soc. (3) 110 (1990), 639-647. MR 91f:42010
16.
D. Müller, E.M. Stein, On spectral multipliers for Heisenberg and related groups, J.Math. Pures Appl. 73 (1994), 413-440. MR 96m:43004
17.
J. Prüss, H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. (3) 203 (1990), 429-452. MR 91b:47030
18.
A. Sikora, Sharp pointwise estimates on heat kernels, Quart. J. Math. Oxford (2) 47 (1996), 371-382. MR 97m:58189
19.
A. Sikora, On-diagonal estimates on Schrödinger semigroup kernels and reduced heat kernels, Comm. Math. Phys. 188 (1997), 233-249. MR 98k:58213
20.
E.M. Stein, Topics in harmonic analysis related to the Littlewood-Paley Theory, Princeton University Press (1970). MR 40:6176
21.
E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press (1970). MR 44:7280
22.
E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press (1993). MR 95c:42002
23.
M. Taylor, Partial differential equations, II, Springer-Verlag, New York (1996). MR 98b:35003

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42B15, 35P99

Retrieve articles in all Journals with MSC (2000): 42B15, 35P99

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 Wroclaw, 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: 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
Posted: 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 of article: Copyright 2000, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia