We study the L p , 1⩽p⩽∞, boundedness for Riesz transforms of the form V a (-1 2Δ+V) -a , where a>0 and V is a non-negative potential. We prove that V a (-1 2Δ+V) -a is bounded on L p (ℝ d ) with 1p⩽2 whenever a⩽1/p. We demonstrate that the L ∞ (ℝ d ) boundedness holds if V satisfies an a-dependent integral condition that is resistant to small perturbations. Similar results with stronger assumptions on V are also obtained on L 1 (ℝ d ). In particular our L ∞ and L 1 results apply to non-negative locally bounded potentials V which globally have a power growth or an exponential growth. We also discuss a counterexample showing that the L ∞ (ℝ d ) boundedness may fail.
Kucharski et al. (Mon,) studied this question.