Calderón-Zygmund theory for non-integral operators and the \(H^\infty\) functional calculus
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.
scientific article; zbMATH DE number 2109871