The Kato-Ponce Inequality

By Loukas Grafakos, Seungly Oh. 2013

323 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d44f82fd2e18bd28fe3d7f112b36574a60d94c5e

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In Applied Mathematics

1. Wenhui Chen
2. Ryo Ikehata
3. Jesse Railo
4. Keijo Monkkonen
5. Giovanni Covi
6. Jarod Hart
7. Xinfeng Wu
8. Rodolfo H. Torres

In Mathematical Physics

1. Wenhui Chen
2. Jesse Railo
3. Keijo Monkkonen
4. Giovanni Covi
5. Gunther Uhlmann
6. Ryo Ikehata
7. Rodolfo H. Torres
8. Jarod Hart

Citations in the corpus

• The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case (2020, background): “While in the estimate of the second norm, the main tools are the fractional Leibniz rule and the fractional chain rule (see [20] and [44]) carrying a stronger condition p > 1 + ⌈s − 2⌉ > 2.”
• The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, background): “3 can be found in [17] (see also [16, 18]).”
• On Kato–Ponce and fractional Leibniz (2016, methodology): “To deal with this situation, we shall use the approach in Grafakos-Oh [11] and write ( Ds( f jg̃ j) ) (x) = 1 (2π)2d ∫ 2 js · |2− j(ξ + η)|sχ(2− jξ)χ(2− jη) f̂ j(ξ)̂̃g j(η)ei(ξ+η)·xdξdη, where χ is a smooth cut-off function with support in {ξ : 2−m0 |ξ| 2m0 } for some integer m0 ≥ 1.”
• Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces (2017, background): “ule for the optimal range of exponents 1/2 &lt;p&lt;∞, 1 &lt;p1,p2 ≤ ∞, 1 p = 1 p1 + 1 p2 and s&gt;max(0,n p −n) or sa positive even integer, was finally settled by Muscalu-Schlag [51] and Grafakos-Oh [37]. We refer to [37] for previous works, additional weaktype estimates, and counterexamples for th”