Generalized Hardy operators

Consider the operator on L² (R^d) La = (−Δ)^α/2 + a|x|^−α with 0 < α < min {2,d} . Under the condition a ⩾ − 2αΓ((d+α)/4)² Γ(( d−α)/4)² the operator is non negative and selfadjoint. We prove that fractional powers L_a^s/² for s ∈ (0, 2] satisfy the estimates L_a^s/² f||_Lp ≲ (−Δ)^αs/4f||_Lp, ||(−Δ)^s/2f||_Lp ≲ L_a^αs/4f||_Lp 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.