Differential and Integral Equations

Some improvements for a class of the Caffarelli-Kohn-Nirenberg inequalities

Megumi Sano and Futoshi Takahashi

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Abstract

In this paper, we concern a weighted version of the Hardy inequality, which is a special case of the more general Caffarelli-Kohn-Nirenberg inequalities. We improve the inequality on the whole space or on a bounded domain by adding various remainder terms. On the whole space, we show the existence of a remainder term which has the form of ratio of two weighted integrals. Also we give a simple derivation of the remainder term involving a distance from the manifold of the “virtual extremals”. Finally, on a bounded domain, we prove the existence of remainder terms involving the gradient of functions.

Article information

Source
Differential Integral Equations Volume 31, Number 1/2 (2018), 57-74.

Dates
First available in Project Euclid: 26 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.die/1509041401

Subjects
Primary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 26D10: Inequalities involving derivatives and differential and integral operators

Citation

Sano, Megumi; Takahashi, Futoshi. Some improvements for a class of the Caffarelli-Kohn-Nirenberg inequalities. Differential Integral Equations 31 (2018), no. 1/2, 57--74. https://projecteuclid.org/euclid.die/1509041401


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