Let $L=-Δ+V$, where $V=V(x_1,\dots, x_d)$ be a nonnegative polynomial in $\mathbb{R}^d$ and $Δ$ denote the usual Laplacian. {\it J. Dziubański} and {\it J. Zienkiewicz} [Colloq. Math. 98, No. 1, 5‒38 (2003; Zbl 1083.42015)] showed that the Riesz transform $VL^{-1}$ is bounded on $L^{\infty}$. The main theme of this paper is to give a $L^{\infty}$-norm estimate independent of the dimension $d$, provided that $V$ satisfies the {\it C. Fefferman} condition [Bull. Am. Math. Soc., New Ser. 9, 129‒206 (1983; Zbl 0526.35080)]: For a fixed positive integer $D$, $|\{j \ne i:$ there exists a monomial of $V$ containing $x_i $ and $x_j$$\}| \le D$ for all $i$. As a corollary, it also presents a $L^p$-norm estimate free of the dimension for the operator $|\nabla L^{-1/2}|$, where $\nabla$ is the usual gradient and $1 Reviewer: Miyeon Kwon (Platteville, WI)