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Calderón-Zygmund theory for non-integral operators and the \(H^\infty\) functional calculus. (English) Zbl 1057.42010
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.

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47A60 Functional calculus for linear operators
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
42B25 Maximal functions, Littlewood-Paley theory
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