*(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 $\left({e}^{-tA}\right)$ 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 $\left({e}^{-tA}\right)$. 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.