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

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

- Volume: 2, Issue: 3, page 449-459
- ISSN: 0391-173X

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topBlunck, Sönke. "A Hörmander-type spectral multiplier theorem for operators without heat kernel." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.3 (2003): 449-459. <http://eudml.org/doc/84508>.

@article{Blunck2003,

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

author = {Blunck, Sönke},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

language = {eng},

number = {3},

pages = {449-459},

publisher = {Scuola normale superiore},

title = {A Hörmander-type spectral multiplier theorem for operators without heat kernel},

url = {http://eudml.org/doc/84508},

volume = {2},

year = {2003},

}

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

PB - Scuola normale superiore

VL - 2

IS - 3

SP - 449

EP - 459

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

LA - eng

UR - http://eudml.org/doc/84508

ER -

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