JournalsrmiVol. 19, No. 3pp. 919–942

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

  • Sönke Blunck

    Université de Cergy-Pontoise, Cergy-Pontoise, France
  • Peer Christian Kunstmann

    Karlsruher Institut für Technologie (KIT), Germany
Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus cover
Download PDF

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)(p,p) condition for arbitrary operators. Given an operator AA on L2L_2 with a bounded HH^\infty calculus, we show as an application the LrL_r-boundedness of the HH^\infty calculus for all r(p,q)r\in(p,q), provided the semigroup (etA)(e^{-tA}) satisfies suitable weighted LpLqL_p\to L_q-norm estimates with 2(p,q)2\in(p,q). This generalizes results due to Duong, McIntosh and Robinson for the special case (p,q)=(1,)(p,q)=(1,\infty) where these weighted norm estimates are equivalent to Poisson-type heat kernel bounds for the semigroup (etA)(e^{-tA}). Their results fail to apply in many situations where our improvement is still applicable, e.g. if AA 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.

Cite this article

Sönke Blunck, Peer Christian Kunstmann, Calderón-Zygmund theory for non-integral operators and the HH^{\infty} functional calculus. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 919–942

DOI 10.4171/RMI/374