A Hörmander-type spectral multiplier theorem for operators without heat kernel

Blunck, Sönke

Blunck, Sönke

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 449-459.

Résumé

Hörmander’s famous Fourier multiplier theorem ensures the $L_{p}$ -boundedness of $F (- Δ_{ℝ} D)$ whenever $F \in ℋ (s)$ for some $s > \frac{D}{2}$ , where we denote by $ℋ (s)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A \geq 0$ and yield the $L_{p}$ -boundedness of $F (A)$ provided $F \in ℋ (s)$ for some $s$ sufficiently large. The harmonic oscillator $A = - Δ_{ℝ} + x^{2}$ 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 $F (A)$ whenever $F \in ℋ (s)$ 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.

MR Zbl

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},
     pages = {449--459},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {3},
     year = {2003},
     mrnumber = {2020856},
     zbl = {1170.42301},
     language = {en},
     url = {https://numdam.org/item/ASNSP_2003_5_2_3_449_0/}
}

TY  - JOUR
AU  - Blunck, Sönke
TI  - A Hörmander-type spectral multiplier theorem for operators without heat kernel
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2003
SP  - 449
EP  - 459
VL  - 2
IS  - 3
PB  - Scuola normale superiore
UR  - https://numdam.org/item/ASNSP_2003_5_2_3_449_0/
LA  - en
ID  - ASNSP_2003_5_2_3_449_0
ER  -

%0 Journal Article
%A Blunck, Sönke
%T A Hörmander-type spectral multiplier theorem for operators without heat kernel
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2003
%P 449-459
%V 2
%N 3
%I Scuola normale superiore
%U https://numdam.org/item/ASNSP_2003_5_2_3_449_0/
%G en
%F 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, Tome 2 (2003) no. 3, pp. 449-459. https://numdam.org/item/ASNSP_2003_5_2_3_449_0/

Bibliographie
Cité par

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

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

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