Sharp Estimates of the Sobolev Norm of u Times the Gradient of ¨

We give a new proof of the following inequality. In any dimension nG2 and for 1-p-nlet ss nqp.r2 p. Then 5u=¨5Lp Rn.FC5u5Lp,s Rn.5¨5Lp,s Rn., where Lp, s Rn. denotes the usual Sobolev space and =¨ denotes the gradient of ¨. The choice of s is optimal, as is the requirement that n)p. In addition, some Sobolev norms of u =¨ can be estimated.