Given $p>1$, $\epsilon>0,c_{0}>0$ and $||f||_{p}<c_{0},||g||_{p}<c_{0}$ satisfying the normalizing condition in 0.1 there exists $c(\epsilon,c_{0})$ with

$$||\varphi-\psi||_{\infty}\leq c(\epsilon,c_{0})||f-g||_{1}^{1/(n+3+\epsilon)}$$

for suitably normalized $\varphi$ and $\psi$.