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 condition for arbitrary operators. Given an operator on with a bounded calculus, we show as an application the -boundedness of the calculus for all , provided the semigroup satisfies suitable weighted -norm estimates with . This generalizes results due to Duong, McIntosh and Robinson for the special case where these weighted norm estimates are equivalent to Poisson-type heat kernel bounds for the semigroup . Their results fail to apply in many situations where our improvement is still applicable, e.g. if 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 functional calculus. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 919–942DOI 10.4171/RMI/374