Spectral multiplier theorem for 𝐻¹ spaces associated with some Schrödinger operators

Dziubański, Jacek

doi:10.1090/S0002-9939-99-05413-1

Spectral multiplier theorem for $H^1$ spaces associated with some Schrödinger operators
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Proc. Amer. Math. Soc. 127 (1999), 3605-3613 Request permission

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

Michael Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991), no. 1, 73–81. MR 1104196, DOI 10.1090/S0002-9947-1991-1104196-7
Michael G. Cowling, Harmonic analysis on semigroups, Ann. of Math. (2) 117 (1983), no. 2, 267–283. MR 690846, DOI 10.2307/2007077
Leonede De Michele and Giancarlo Mauceri, $H^p$ multipliers on stratified groups, Ann. Mat. Pura Appl. (4) 148 (1987), 353–366. MR 932770, DOI 10.1007/BF01774295
Jacek Dziubański, Andrzej Hulanicki, and Joe Jenkins, A nilpotent Lie algebra and eigenvalue estimates, Colloq. Math. 68 (1995), no. 1, 7–16. MR 1311757, DOI 10.4064/cm-68-1-7-16
J. Dziubański, A note on Schrödinger operators with polynomial potentials, Colloq. Math. 78 (1998), 149–161.
Jacek Dziubański and Jacek Zienkiewicz, Hardy spaces associated with some Schrödinger operators, Studia Math. 126 (1997), no. 2, 149–160. MR 1472695, DOI 10.4064/sm-126-2-149-160
Waldemar Hebisch, A multiplier theorem for Schrödinger operators, Colloq. Math. 60/61 (1990), no. 2, 659–664. MR 1096404, DOI 10.4064/cm-60-61-2-659-664
W. Hebisch, Functional calculus for slowly decaying kernels, preprint University of Wrocław.
Waldemar Hebisch and Jacek Zienkiewicz, Multiplier theorem on generalized Heisenberg groups. II, Colloq. Math. 69 (1995), no. 1, 29–36. MR 1341678, DOI 10.4064/cm-69-1-29-36
G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
D. Müller and E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. (9) 73 (1994), no. 4, 413–440. MR 1290494
Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
J. Zhong, Harmonic analysis for Schrödinger type operators, Ph. D. thesis, Princeton Univ. 1993.