Generalized Hardy operators

Abstract

Abstract Consider the operator on L 2 ( R d ) L a = ( − Δ ) α / 2 + a | x | − α with 0 < α < m i n { 2 , d } . Under the condition a ⩾ − 2 α Γ ( ( 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 L p ≲ ( − Δ ) α s / 4 f L p , ( − Δ ) s / 2 f L p ≲ L a α s / 4 f L p 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.