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December, 2003 Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus
Sönke Blunck, Peer Christian Kunstmann
Rev. Mat. Iberoamericana 19(3): 919-942 (December, 2003).


We modify Hörmander's well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIntosh) and present a weak type $(p,p)$ condition for arbitrary operators. Given an operator $A$ on $L_2$ with a bounded $H^\infty$ calculus, we show as an application the $L_r$-boundedness of the $H^\infty$ calculus for all $r\in(p,q)$, provided the semigroup $(e^{-tA})$ satisfies suitable weighted $L_p\to L_q$-norm estimates with $2\in(p,q)$. This generalizes results due to Duong, McIntosh and Robinson for the special case $(p,q)=(1,\infty)$ where these weighted norm estimates are equivalent to Poisson-type heat kernel bounds for the semigroup $(e^{-tA})$. Their results fail to apply in many situations where our improvement is still applicable, e.g. if $A$ is a Schrödinger operator with a singular potential, an elliptic higher order operator with bounded measurable coefficients or an elliptic second order operator with singular lower order terms.


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Sönke Blunck. Peer Christian Kunstmann. "Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus." Rev. Mat. Iberoamericana 19 (3) 919 - 942, December, 2003.


Published: December, 2003
First available in Project Euclid: 20 February 2004

zbMATH: 1057.42010
MathSciNet: MR2053568

Primary: 42B20 , 47A60
Secondary: 42B25 , 47F05

Keywords: Calderón-Zygmund theory , elliptic operators , functional calculus

Rights: Copyright © 2003 Departamento de Matemáticas, Universidad Autónoma de Madrid


Vol.19 • No. 3 • December, 2003
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