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

Sönke Blunck

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

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

Abstract

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Hörmander’s famous Fourier multiplier theorem ensures the -boundedness of whenever for some , where we denote by the set of functions satisfying the Hörmander condition for derivatives. Spectral multiplier theorems are extensions of this result to more general operators and yield the -boundedness of provided for some sufficiently large. The harmonic oscillator shows that in general is not sufficient even if has a heat kernel satisfying gaussian estimates. In this paper, we prove the -boundedness of whenever for some , provided satisfies generalized gaussian estimates. This assumption allows to treat even operators 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.

How to cite

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Blunck, 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 $s&gt;\frac\{D\}\{2\}$, 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 $s&gt; \frac\{D\}\{2\}$ 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 $s&gt;\frac\{D+1\}\{2\}$, 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 $s&gt;\frac{D}{2}$, 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 $s&gt; \frac{D}{2}$ 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 $s&gt;\frac{D+1}{2}$, 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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