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

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 | b u | p |bu|^{p} through the integral of | D u | p |Du|^{p} for p > 1 p>1 . Formally their proof is valid only for d ≥ 3 d\geq 3 . We present here further generalization with a different proof in which D D is replaced with the fractional power of the Laplacian for any dimension d ≥ 2 d\geq 2 .