Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

Abstract

Let = −Δ + be a Schrödinger operator, where Δ is the laplacian on R and the nonnegative potential belongs to the reverse Hölder class 1 for some 1 ≥ ( /2). Assume that ∈ 1 (R ). Denote by 1 ( ) the weighted Hardy space related to the Schrödinger operator = −Δ + . Let R = [ , R] be the commutator generated by a function ∈ BMO (R ) and the Riesz transform R = ∇(−Δ + ) −(1/2) . Firstly, we show that the operator R is bounded from 1 ( ) into 1 weak ( ). Secondly, we obtain the endpoint estimates for the commutator [ , R]. Namely, it is bounded from the weighted Hardy space 1 ( ) into 1 weak ( ).

Estimates for the commutators of operator $V^{\alpha }\nabla (-\Delta +V)^{-\beta }$

“…Harboure and Salinas [10] obtained the L p -boundedness of [b, T 1/2 ] and Liu, Sheng and Wang [13] proved that [b, T 1/2 ] is bounded from H 1 L (R n ) to weak L 1 (R n ). More boundedness of commutator [b, T 1/2 ] can be found in [14] and [15].…”