A remark on a paper of F. Chiarenza and M. Frasca

Abstract

In 1990 F. Chiarenza and M. Frasca published a paper in which they generalized a result of C. Fefferman on estimates of the integral of $|bu|^{p}$ through the integral of $|Du|^{p}$ for $p>1$. Formally their proof is valid only for $d\geq 3$. We present here further generalization with a different proof in which $D $ is replaced with the fractional power of the Laplacian for any dimension $d\geq 1$.