ISRAEL JOURNAL OF MATHEMATICS 173 (2009), 157–176 DOI: 10.1007/s11856-009-0086-x DIMENSION FREE ESTIMATES FOR RIESZ TRANSFORMS OF SOME SCHR ¨ ODINGER OPERATORS * BY Roman Urban ∗∗ and Jacek Zienkiewicz Institute of Mathematics, Wroclaw University Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland e-mail: urban@math.uni.wroc.pl, zenek@math.uni.wroc.pl In memory of Andrzey Hulanicki ABSTRACT We prove dimension free L ∞ → L ∞ -estimates for the Riesz transform T = VL -1 ,L = -Δ+ V, where Δ is the Laplacian in R d , and the polynomial V ≥ 0 satisfies C. L. Fefferman conditions; see [7]. As a corollary we get dimension free L p → L p (ℓ 2 )-estimates, 1 <p< ∞, for the vector of Riesz transforms. 1. Introduction Let L = −Δ+ V where V is nonnegative polynomial in R d , and Δ is the Laplacian ∑ d i=1 ∂ 2 xi . It is well-known that L generates a Markovian semigroup e −tL of operators with nonnegative kernels K t (see, e.g., [3, Theorem 1.8.1]). By the results of [5] the operator VL −1 f = V  ∞ 0 K t fdt is bounded on L ∞ . The main aim of this paper is to prove that if V satisfies the C. Fefferman conditions [7] then we get L ∞ → L ∞ bound independent on the dimension. Since trivially for any V ≥ 0, we have ‖VL −1 ‖ L 1 →L 1 ≤ 1, the result implies the dimension * Research supported in part by the European Commission Marie Curie Host Fel- lowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389. ** The first author was also supported by the MNiSW research grant N201 012 31/1020. Received June 6, 2007 and in revised form December 11, 2007 157