A spectral multiplier theorem for non-self-adjoint operators
HTML articles powered by AMS MathViewer
- by El Maati Ouhabaz PDF
- Trans. Amer. Math. Soc. 361 (2009), 6567-6582 Request permission
Abstract:
We prove a spectral multiplier theorem for non-self-adjoint operators. More precisely, we consider non-self-adjoint operators $A: D(A) \subset L^2 \to L^2$ having numerical range in a sector $\Sigma (w)$ of angle $w,$ and whose heat kernel satisfies a Gaussian upper bound. We prove that for every bounded holomorphic function $f$ on $\Sigma (w),$ $f(A)$ acts on $L^p$ with $L^p-$norm estimated by the behavior of a finite number of derivatives of $f$ on the boundary of $\Sigma (w).$References
- G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc. 120 (1994), no. 3, 973–979. MR 1172944, DOI 10.1090/S0002-9939-1994-1172944-4
- Georgios Alexopoulos and Noël Lohoué, Riesz means on Lie groups and Riemannian manifolds of nonnegative curvature, Bull. Soc. Math. France 122 (1994), no. 2, 209–223 (English, with English and French summaries). MR 1273901
- Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, Basel, 2001. MR 1886588, DOI 10.1007/978-3-0348-5075-9
- Khristo Boyadzhiev and Ralph deLaubenfels, Boundary values of holomorphic semigroups, Proc. Amer. Math. Soc. 118 (1993), no. 1, 113–118. MR 1128725, DOI 10.1090/S0002-9939-1993-1128725-X
- Gilles Carron, Thierry Coulhon, and El-Maati Ouhabaz, Gaussian estimates and $L^p$-boundedness of Riesz means, J. Evol. Equ. 2 (2002), no. 3, 299–317. MR 1930609, DOI 10.1007/s00028-002-8090-1
- Thierry Coulhon and Xuan Thinh Duong, Riesz transforms for $1\leq p\leq 2$, Trans. Amer. Math. Soc. 351 (1999), no. 3, 1151–1169. MR 1458299, DOI 10.1090/S0002-9947-99-02090-5
- Michael Cowling, Ian Doust, Alan McIntosh, and Atsushi Yagi, Banach space operators with a bounded $H^\infty$ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), no. 1, 51–89. MR 1364554
- Michel Crouzeix and Bernard Delyon, Some estimates for analytic functions of strip or sectorial operators, Arch. Math. (Basel) 81 (2003), no. 5, 559–566. MR 2029717, DOI 10.1007/s00013-003-0569-7
- Bernard Delyon and François Delyon, Generalization of von Neumann’s spectral sets and integral representation of operators, Bull. Soc. Math. France 127 (1999), no. 1, 25–41 (English, with English and French summaries). MR 1700467
- Xuan Thinh Duong, From the $L^1$ norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups, Pacific J. Math. 173 (1996), no. 2, 413–424. MR 1394398
- Xuan T. Duong and Derek W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), no. 1, 89–128. MR 1419418, DOI 10.1006/jfan.1996.0145
- Xuan Thinh Duong and Alan MacIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), no. 2, 233–265. MR 1715407, DOI 10.4171/RMI/255
- Xuan Thinh Duong, El Maati Ouhabaz, and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), no. 2, 443–485. MR 1943098, DOI 10.1016/S0022-1236(02)00009-5
- O. El Mennaoui, Trace des Semi-groupes Holomorphes Singuliers à l’Origine et Comportement Asymptotique, Ph.D. Thesis, Université de franche-Comté, Besançon 1992.
- J. E. Galé and P.J. Miana, $H^\infty -$functional calculus and Mikhlin-type multiplier conditions, preprint, Universidad de Zaragoza 2006.
- Markus Haase, Spectral properties of operator logarithms, Math. Z. 245 (2003), no. 4, 761–779. MR 2020710, DOI 10.1007/s00209-003-0569-0
- W. Hebisch, Functional calculus for slowly decaying kernels, preprint 1995.
- Matthias Hieber, Integrated semigroups and differential operators on $L^p$ spaces, Math. Ann. 291 (1991), no. 1, 1–16. MR 1125004, DOI 10.1007/BF01445187
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
- T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag 1980.
- Ermanno Lanconelli, Valutazioni in $L_{p}\,({\bf {\rm }R}^{n})$ della soluzione del problema di Cauchy per l’equazione di Schrödinger, Boll. Un. Mat. Ital. (4) 1 (1968), 591–607 (Italian). MR 0234133
- Alan McIntosh, Operators which have an $H_\infty$ functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210–231. MR 912940
- El Maati Ouhabaz, Analysis of heat equations on domains, London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005. MR 2124040
- Sigrid Sjöstrand, On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 24 (1970), 331–348. MR 270219
Additional Information
- El Maati Ouhabaz
- Affiliation: Institut de Mathématiques de Bordeaux, CNRS UMR 5251, Equipe d’Analyse et Géométrie, Université Bordeaux 1, 351, Cours de la Libération, 33405 Talence, France
- Email: Elmaati.Ouhabaz@math.u-bordeaux1.fr
- Received by editor(s): June 28, 2007
- Received by editor(s) in revised form: January 30, 2008
- Published electronically: July 17, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 6567-6582
- MSC (2000): Primary 42B15; Secondary 47F05
- DOI: https://doi.org/10.1090/S0002-9947-09-04754-0
- MathSciNet review: 2538605