Communications in Partial Differential Equations· 2013Q1
The Kato-Ponce Inequality
- 274citations
- Q1SCImago
- 2013year
Short summary
A new Kato-Ponce inequality is established for the Bessel potential of a product of two functions, proving it holds under specific Hölder-type conditions (1/p + 1/q = 1/r) for 1 ≤ p, q ≤ ∞ and 1/2 ≤ r < ∞.
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Key points
- Establishes a new Kato-Ponce inequality for the Bessel potential (J^s) of a product of two functions.
- The inequality holds for exponents p, q, r satisfying 1/p + 1/q = 1/r, with 1 ≤ p, q ≤ ∞ and 1/2 ≤ r < ∞.
- The estimate is of weak-type when p or q equals 1.
- An example shows the inequality fails when r < 1.
- Results are extended to the multi-parameter case.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Communications in Partial Differential Equations, 2013 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics