The Kato-Ponce Inequality

In this article we revisit the inequalities of Kato and Ponce concerning the L-r norm of the Bessel potential J(s)=(1-)(s/2) (or Riesz potential D-s=(- )(s/2)) of the product of two functions in terms of the product of the L-p norm of one function and the L-q norm of the Bessel potential J(s) (resp. Riesz potential D-s) of the other function. Here the indices p, q, and r are related as in Holder's inequality 1/p+1/q=1/r and they satisfy 1p, q and 1/2r < and <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="lpde_a_822885_o_ilm0001.gif"></inline-graphic> . Also the estimate is of weak-type when either p or q is equal to 1. In the case r<1 we indicate via an example that when <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="lpde_a_822885_o_ilm0002.gif"></inline-graphic> the inequality fails. Furthermore, we extend these results to the multi-parameter case.