Generalized Hardy operators

Abstract

Consider the operator on . Under the condition the operator is non negative and selfadjoint. We prove that fractional powers for s ∈ (0, 2] satisfy the estimates for suitable ranges of p. Our result fills the remaining gap in earlier results from Killip et al (2018 Math. Z. 288 1273–98); Merz (2021 Math. Z. 299 101–21); Frank et al (Int. Math. Res. Not. 2021 2284–303). The method of proof is based on square function estimates for operators whose heat kernel has a weak decay.