Calderón-Zygmund theory for non-integral operators and the \(H^\infty\) functional calculus
type: Item
instance of: scholarly article
title: Calderón-Zygmund theory for non-integral operators and the \(H^\infty\) functional calculus
zbMATH Open document ID: 1057.42010
DOI: 10.4171/RMI/374
author: Peer Christian Kunstmann
author: Sönke Blunck
published in: Revista Matemática Iberoamericana
publication date: 2004-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/39597
review text: Summary: 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.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 2109871
zbMATH Keywords: Calderón-Zygmund theory
zbMATH Keywords: functional calculus
zbMATH Keywords: elliptic operators
MaRDI profile type: Publication
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OpenAlex ID: W1970592390
description: scientific article; zbMATH DE number 2109871