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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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