A Hörmander-type spectral multiplier theorem for operators without heat kernel
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 2 (2003) no. 3, p. 449-459
Hörmander’s famous Fourier multiplier theorem ensures the ${L}_{p}$-boundedness of $F\left(-{\Delta }_{ℝ}D\right)$ whenever $F\in ℋ\left(s\right)$ for some $s>\frac{D}{2}$, where we denote by $ℋ\left(s\right)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A\ge 0$ and yield the ${L}_{p}$-boundedness of $F\left(A\right)$ provided $F\in ℋ\left(s\right)$ for some $s$ sufficiently large. The harmonic oscillator $A=-{\Delta }_{ℝ}+{x}^{2}$ shows that in general $s>\frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In this paper, we prove the ${L}_{p}$-boundedness of $F\left(A\right)$ whenever $F\in ℋ\left(s\right)$ for some $s>\frac{D+1}{2}$, provided $A$ satisfies generalized gaussian estimates. This assumption allows to treat even operators $A$ without heat kernel (e.g. operators of higher order and operators with complex or unbounded coefficients) which was impossible for all known spectral multiplier results.
Classification:  42B15,  42B20,  35G99,  35P99
@article{ASNSP_2003_5_2_3_449_0,
author = {Blunck, S\"onke},
title = {A H\"ormander-type spectral multiplier theorem for operators without heat kernel},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {3},
year = {2003},
pages = {449-459},
zbl = {1170.42301},
mrnumber = {2020856},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2003_5_2_3_449_0}
}

Blunck, Sönke. A Hörmander-type spectral multiplier theorem for operators without heat kernel. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 2 (2003) no. 3, pp. 449-459. http://www.numdam.org/item/ASNSP_2003_5_2_3_449_0/

[A] D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. AMS 73 (1967), 890-896. | MR 217444 | Zbl 0153.42002

[ACT] P. Auscher - T. Coulhon - P. Tchamitchian, Absence de principe du maximum pour certaines équations paraboliques complexes, Colloq. Math. 71 (1996), 87-95. | MR 1397370 | Zbl 0960.35011

[AT] P. Auscher - P. Tchamitchian, Square root problem for divergence operators and related topics, Soc. Math. de France, Astérisque 249 (1998). | MR 1651262 | Zbl 0909.35001

[B] S. Blunck, Generalized Gaussian estimates and Riesz means of Schrödinger groups, submitted. | Zbl 1116.43002

[BK1] S. Blunck - P. C. Kunstmann, Weighted norm estimates and maximal regularity, Adv. in Differential Eq. 7 (2002), 1513-1532. | MR 1920543 | Zbl 1045.34031

[BK2] S. Blunck - P. C. Kunstmann, Calderon-Zygmund theory for non-integral operators and the ${H}^{\infty }$ functional calculus, to appear in Rev. Mat. Iberoam. | MR 2053568 | Zbl 1057.42010

[BK3] S. Blunck - P. C. Kunstmann, Weak type $\left(p,p\right)$ estimates for Riesz transforms, to appear in Math. Z. | MR 2054523 | Zbl 1138.35315

[BK4] S. Blunck - P. C. Kunstmann, Generalized Gaussian estimates and the Legendre transform, submitted. | Zbl 1117.47020

[BC] S. Blunck - T. Coulhon, A note on local higher order Sobolev inequalities on Riemannian manifolds, in preparation.

[C] M. Christ, ${L}^{p}$ bounds for spectral multipliers on nilpotent groups, Trans. AMS 328 (1991), 73-81. | MR 1104196 | Zbl 0739.42010

[D1] E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), 141-169. | MR 1346221 | Zbl 0839.35034

[D3] E. B. Davies, Limits on ${L}^{p}$-regularity of self-adjoint elliptic operators, J. Differential Eq. 135 (1997), 83-102. | MR 1434916 | Zbl 0871.35020

[DM] X. T. Duong - A. Mcintosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoam (2) 15 (1999), 233-265. | MR 1715407 | Zbl 0980.42007

[DOS] X. T. Duong - A. Sikora - E. M. Ouhabaz, Plancherel type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), 443-485. | MR 1943098 | Zbl 1029.43006

[G] A. Grigor'Yan, Gaussian upper bounds for the heat kernel on an arbitrary manifolds, J. Differential Geom. 45 (1997), 33-52. | MR 1443330 | Zbl 0865.58042

[HM] S. Hofmann - J. M. Martell, ${L}^{p}$ bounds for Riesz transforms and square roots associated to second order elliptic operators, preprint, 2002. | MR 2006497 | Zbl 1074.35031

[LSV] V. Liskevich - Z. Sobol - H. Vogt, On ${L}_{p}$-theory of ${C}_{0}$-semigroups associated with second order elliptic operators II, J. Funct. Anal. 193 (2002), 55-76. | MR 1923628 | Zbl 1020.47029

[MM] G. Mauceri - S. Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoam 6 (1990), 141-164. | MR 1125759 | Zbl 0763.43005

[ScV] G. Schreieck - J. Voigt, Stability of the ${L}_{p}$-spectrum of generalized Schrödinger operators with form small negative part of the potential, In: “Functional Analysis", Biersted - Pietsch - Ruess - Vogt (eds.), Proc. Essen 1991, Marcel-Dekker, New York, 1994. | MR 1241673 | Zbl 0817.35065

[T] S. Thangavelu, Summability of Hermite expansions I, II, Trans. AMS (1989), 119-170. | MR 958904 | Zbl 0685.42016