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 L p -boundedness of F ( - Δ ℝ D ) whenever F ∈ ℋ ( s ) for some s > 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 ≥ 0 and yield the L p -boundedness of F ( A ) provided F ∈ ℋ ( s ) for some s sufficiently large. The harmonic oscillator A = - Δ ℝ + x 2 shows that in general s > 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 ∈ ℋ ( s ) for some s > 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.

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 -

References

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