Calderón-Zygmund theory for non-integral operators and the $H^{\infty }$ functional calculus

Abstract

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.