The Kato-Ponce Inequality

• Published in: Communications in partial differential equations Vol. 39; no. 6; pp. 1128 - 1157
• Main Authors: Grafakos, Loukas, Oh, Seungly
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis Group 03.06.2014; Taylor & Francis Ltd
• Subjects: Estimates; Fractional Leibniz rule; Inequalities; Inequality; Kato-Ponce; Norms; Partial differential equations; Primary 42B20; Secondary 46E35
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2013.822885

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.