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Generalized Gaussian estimates and riesz means of Schrödinger groups

Part of: Abstract harmonic analysis General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Sönke Blunck
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Abstract

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We show that generalized Gaussian estimates for selfadjoint semigroups (e-tA)t ∈ R+ on L2 imply Lp boundedness of Riesz means and other regularizations of the Schrödinger group (eitA)t ∈ R. This generalizes results restricted to semigroups with a heat kernel, which are due to Sjöstrand, Alexopoulos and more recently Carron, Coulhon and Ouhabaz. This generalization is crucial for elliptic operators A that are of higher order or have singular lower order terms since, in general, their semigroups fail to have a heat kernel.

MSC classification

Secondary: 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 47A60: Functional calculus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 2 , April 2007 , pp. 149 - 162
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Alexopoulos, G., ‘Lp bounds for spectral multipliers from Gaussian estimates on the heat kernel’, preprint.Google Scholar
[2]Auscher, P. an Tchamitchian, Ph., ‘Square root problem for divergence operators and related topics’, Astérisque 249 (1998).Google Scholar
[3]Arendt, W., El Mennaoui, O. and Hieber, M., ‘Boundary values of holomorphic semigroups’, Proc. Amer. Math. Soc. 125 (1997), 635–647.CrossRefGoogle Scholar
[4]Blunck, S., ‘A Hörmander-type spectral multiplier theorem for operators without heat kernel’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2 (2003), 449–459.Google Scholar
[5]Blunck, S. and Kunstmann, P. C., ‘Weighted norm estimates and maximal regularity’, Adv. Differential Equations 7 (2002), 1513–1532.CrossRefGoogle Scholar
[6]Blunck, S., ‘Calderon-Zygmund theory for non-integral operators and the H infin; functional calculus’, Rev. Mat. Iberoamericana 19 (2003), 919–942.CrossRefGoogle Scholar
[7]Blunck, S., ‘Weak type (p, p) estimates for Riesz transforms’, Math. Z. 247 (2004), 137–148.CrossRefGoogle Scholar
[8]Blunck, S., ‘Generalized Gaussian estimates and the Legendre transform’, J. Operator Theory 53 (2005), 351–365.Google Scholar
[9]Boyadzhiev, K. and de Laubenfels, R., ‘Boundary values of holomorphic semigroups’, Proc. Amer. Math. Soc. 118 (1993), 113–118.CrossRefGoogle Scholar
[10]Carron, G., Coulhon, T. and Ouhabaz, E. M., ‘Gaussian estimates and Lp-boundedness of Rieszmeans’, J. Evol. Equ. 2 (2002), 299–317.CrossRefGoogle Scholar
[11]Davies, E. B., ‘Uniformly elliptic operators with measurable coefficients’, J. Funct. Anal. 132 (1995), 141–169.CrossRefGoogle Scholar
[12]Davies, E. B., ‘Limits on Lp-regularity of self-adjoint elliptic operators’, J. Differential Equations 135 (1997), 83–102.CrossRefGoogle Scholar
[13]de Laubenfels, R., ‘Integrated semigroups, C-semigroups and the abstract Cauchy problem’, Semigroup Forum 41 (1990), 83–95.CrossRefGoogle Scholar
[14]Duong, X. T. and McIntosh, A., ‘Singular integral operators with non-smooth kernels on irregular domains’, Rev. Mat. Iberoamericana 15 (1999), 233–265.CrossRefGoogle Scholar
[15]Duong, X. T., Sikora, A. and Ouhabaz, E. M., ‘Plancherel type estimates and sharp spectral multipliers’, J. Funct. Anal. 196 (2002), 443–485.CrossRefGoogle Scholar
[16]Grigor'yan, A., ‘Gaussian upper bounds for the heat kernel on an arbitrary manifolds’, J. Differential Geom. 45 (1997), 33–52.Google Scholar
[17]Hofmann, S. and Martell, J. M., ‘Lp bounds for Riesz transforms and square roots associated to second order elliptic operators’, Publ. Mat. 47 (2003), 497–515.CrossRefGoogle Scholar
[18]Hörmander, L., ‘Estimates for translation invariant operators in Lp spaces’, Acta Math. 104 (1960), 93–140.CrossRefGoogle Scholar
[19]Kunstmann, P. C., Lp-spectral properties of elliptic differential operators Habilitationsschrift, (Karlsruhe, 2002).Google Scholar
[20]Lanconelli, E., ‘Valutazioni in Lp (Rp) della soluzione del problema di Cauchy per l'equazione di Schrödinger’, Boll. Un. Mat. Ital. (4) 1 (1968), 591–607.Google Scholar
[21]Liskevich, V., Sobol, Z. and Vogt, H., ‘On Lp-theory of C 0-semigroups associated with second order elliptic operators II’, J. Funct. Anal. 193 (2002), 55–76.CrossRefGoogle Scholar
[22]McIntosh, A., ‘Operators which have an H ∞ functional calculus’, in: Miniconference on operator theory and partial differential equations (North Ryde), Proc. Center Math. Appl. Austral. Nat. Univ. 14 (Australian National Univ., Canberra, 1986) pp. 210–231.Google Scholar
[23]El Mennaoui, O., Trace des semi-groupes holomorphes singuliers à l'origine et comportement asymptotique (Ph.D. Thesis, University of Franche-Comté, France, 1992).Google Scholar
[24]Ouhabaz, E. M., ‘Gaussian estimates and holomorphy of semigroups’, Proc. Amer. Math. Soc. 123 (1995), 1465–1474.CrossRefGoogle Scholar
[25]Schreieck, G. and Voigt, J., ‘Stability of the Lp-spectrum of generalized Schrödinger operators with form small negative part of the potential’, in: Functional Analysis (eds. Pietsch, ?., Ruess, ?. and Vogt, H.), Proc. Essen 1991, Bierstedt (Marcel-Dekker, New York, 1994).Google Scholar
[26]Sjöstrand, S., ‘On the Riesz means of the solutions of the Schrödinger equation’, Ann. Scuola Norm. Sup. Pisa (3) 24 (1970), 331–348.Google Scholar