# Revista Matemática Iberoamericana

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**Volume 19, Issue 3, 2003, pp. 919–942**

**DOI: 10.4171/RMI/374**

Published online: 2003-12-31

Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus

Sönke Blunck^{[1]}and Peer Christian Kunstmann

^{[2]}(1) Université de Cergy-Pontoise, Cergy-Pontoise, France

(2) Karlsruher Institut für Technologie (KIT), Germany

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.

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

Blunck Sönke, Kunstmann Peer Christian: Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus. *Rev. Mat. Iberoamericana* 19 (2003), 919-942. doi: 10.4171/RMI/374