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 − Δ) s/2 (or Riesz potential D s = (− Δ) 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 Hölder's inequality 1/p + 1/q = 1/r and they satisfy 1 ≤ p, q ≤ ∞ and 1/2 ≤ r < ∞ and . 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 the inequality fails. Furthermore, we extend these results to the multi-parameter case.