Revista Matemática Iberoamericana


Full-Text PDF (244 KB) | Table of Contents | RMI summary
Volume 19, Issue 3, 2003, pp. 919–942
DOI: 10.4171/RMI/374

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

Sönke Blunck[1] and Peer Christian Kunstmann[2]

(1) Département de Mathématiques, Université de Cergy-Pontoise, 2, avenue A. Chauvin, 95302, CERGY-PONTOISE CEDEX, FRANCE
(2) Institut für Analysis, Karlsruher Institut für Technologie (KIT), Kaiserstr. 89-93 , 76133, KARLSRUHE, 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 S, Kunstmann P. 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