Calderón-Zygmund theory for non-integral operators and the H ∞ functional calculus

Abstract

We modify Hörmander's well-known weak typo (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 L2 with a bounded H∞ calculus, we show as an application the Lr-boundedness of the H∞ calculus for all r ∈ (p,q), provided the semigroup (e-tA) satisfies suitable weighted Lp → Lq-norm estimates with 2 ∈ (p, q). This generalizes results due to Duong, McIntosh and Robinson for the special case (p, q) = (1, ∞) where those 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.