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Calderón-Zygmund theory for non-integral operators and the H 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 calculus, we show as an application the L r -boundedness of the H calculus for all r(p,q), provided the semigroup (e -tA ) satisfies suitable weighted L p L q -norm estimates with 2(p,q).

This generalizes results due to Duong, McIntosh and Robinson for the special case (p,q)=(1,) 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:
42B20Singular and oscillatory integrals, several variables
47A60Functional calculus of operators
47F05Partial differential operators
42B25Maximal functions, Littlewood-Paley theory