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Spectral multiplier theorem for $H^1$ spaces associated with some Schrödinger operators

Author(s): Jacek Dziubanski
Journal: Proc. Amer. Math. Soc. 127 (1999), 3605-3613.
MSC (1991): Primary 42B30, 35J10; Secondary 42B15, 42B25, 43A80
Posted: July 23, 1999
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Abstract: Let $T_{t}$ be the semigroup of linear operators generated by a Schrödinger operator $-A=\Delta -V$, where $V$ is a nonnegative polynomial. We say that $f$ is an element of $H_{A}^{1}$ if the maximal function $\mathcal{M}f(x)=\sup _{t>0} |T_{t}f(x)|$ belongs to $L^{1}$. A criterion on functions $F$ which implies boundedness of the operators $F(A)$ on $H_{A}^{1}$ is given.

References:

[1]
M. Christ, $L^{p}$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991), 73-81. MR 92k:42017

[2]
M. Cowling, Harmonic analysis on semigroups, Ann. of Math. 117 (1983), 267-283. MR 84h:43004

[3]
L. De Michele and G. Mauceri, $H^{p}$ multipliers on stratified groups, Ann. Math. Pura Appl. 148 (1987), 353-366. MR 89e:43009

[4]
J. Dziuba\'{n}ski, A. Hulanicki and J.W. Jenkins, A nilpotent Lie algebra and eigenvalue estimates, Colloq. Math. 68 (1995), 7-16. MR 96c:22014

[5]
J. Dziuba\'{n}ski, A note on Schrödinger operators with polynomial potentials, Colloq. Math. 78 (1998), 149-161. CMP 99:04

[6]
J. Dziuba\'{n}ski and J. Zienkiewicz, Hardy spaces associated with some Schrödinger operators, Studia Math. 126 (1997), 149-160. MR 98k:42029

[7]
W. Hebisch, A multiplier theorem for Schrödinger operators, Coll. Math. 60/61 (1990), 659-664. MR 93f:47055

[8]
W. Hebisch, Functional calculus for slowly decaying kernels, preprint University of Wroc{\l}aw.

[9]
W. Hebisch and J. Zienkiewicz, Multiplier theorems on generalized Heisenberg groups II, Colloq. Math. 69 (1995), 29-36. MR 96h:43007

[10]
G. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982. MR 84h:43027

[11]
D. Müller and E. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. 73 (1994), 413-440. MR 96m:43004

[12]
E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton University Press, Princeton 1970. MR 40:6176

[13]
J. Zhong, Harmonic analysis for Schrödinger type operators, Ph. D. thesis, Princeton Univ. 1993.

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Additional Information:

Jacek Dziubanski
Affiliation: Institute of Mathematics, University of Wroclaw, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Email: jdziuban@math.uni.wroc.pl

DOI: 10.1090/S0002-9939-99-05413-1
PII: S 0002-9939(99)05413-1
Received by editor(s): February 17, 1998
Posted: July 23, 1999
Additional Notes: This research was partially supported by the European Commission via TMR network ``Harmonic Analysis", and by grant 2 P03A 058 14 from KBN, Poland.
Communicated by: Christopher D. Sogge
Copyright of article: Copyright 1999, American Mathematical Society

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